\(\int \frac {A+B x}{(d+e x)^{9/2} (b x+c x^2)} \, dx\) [1237]

Optimal result

Integrand size = 26, antiderivative size = 301 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}}-\frac {2 c^{7/2} (b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{9/2}} \]

[Out]

2/7*(-A*e+B*d)/d/(-b*e+c*d)/(e*x+d)^(7/2)+2/5*(B*c*d^2-A*e*(-b*e+2*c*d))/d^2/(-b*e+c*d)^2/(e*x+d)^(5/2)+2/3*(B
*c^2*d^3-A*e*(b^2*e^2-3*b*c*d*e+3*c^2*d^2))/d^3/(-b*e+c*d)^3/(e*x+d)^(3/2)-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/
b/d^(9/2)-2*c^(7/2)*(-A*c+B*b)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b/(-b*e+c*d)^(9/2)+2*(B*c^3*d^4
-A*e*(-b^3*e^3+4*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*d^3))/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {842, 840, 1180, 214} \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=-\frac {2 c^{7/2} (b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{9/2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 \sqrt {d+e x} (c d-b e)^4}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)} \]

[In]

Int[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]

[Out]

(2*(B*d - A*e))/(7*d*(c*d - b*e)*(d + e*x)^(7/2)) + (2*(B*c*d^2 - A*e*(2*c*d - b*e)))/(5*d^2*(c*d - b*e)^2*(d
+ e*x)^(5/2)) + (2*(B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2)))/(3*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))
+ (2*(B*c^3*d^4 - A*e*(4*c^3*d^3 - 6*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)))/(d^4*(c*d - b*e)^4*Sqrt[d + e*x]
) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(9/2)) - (2*c^(7/2)*(b*B - A*c)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/
Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(9/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {\int \frac {A (c d-b e)+c (B d-A e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{d (c d-b e)} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {\int \frac {A (c d-b e)^2+c \left (B c d^2-A e (2 c d-b e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{d^2 (c d-b e)^2} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {\int \frac {A (c d-b e)^3+c \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{d^3 (c d-b e)^3} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}+\frac {\int \frac {A (c d-b e)^4+c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{d^4 (c d-b e)^4} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}+\frac {2 \text {Subst}\left (\int \frac {A e (c d-b e)^4-c d \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )+c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{d^4 (c d-b e)^4} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}+\frac {(2 A c) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b d^4}+\frac {\left (2 c^4 (b B-A c)\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b (c d-b e)^4} \\ & = \frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}}-\frac {2 c^{7/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 \left (B d^4 \left (-15 b^3 e^3+3 b^2 c e^2 (22 d+7 e x)-b c^2 e \left (122 d^2+112 d e x+35 e^2 x^2\right )+c^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )\right )+A e \left (15 b c^2 d^2 e \left (66 d^3+161 d^2 e x+140 d e^2 x^2+42 e^3 x^3\right )+b^3 e^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )-3 c^3 d^3 \left (194 d^3+504 d^2 e x+455 d e^2 x^2+140 e^3 x^3\right )-b^2 c d e^2 \left (689 d^3+1624 d^2 e x+1400 d e^2 x^2+420 e^3 x^3\right )\right )\right )}{105 d^4 (c d-b e)^4 (d+e x)^{7/2}}-\frac {2 c^{7/2} (-b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b (-c d+b e)^{9/2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}} \]

[In]

Integrate[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]

[Out]

(2*(B*d^4*(-15*b^3*e^3 + 3*b^2*c*e^2*(22*d + 7*e*x) - b*c^2*e*(122*d^2 + 112*d*e*x + 35*e^2*x^2) + c^3*(176*d^
3 + 406*d^2*e*x + 350*d*e^2*x^2 + 105*e^3*x^3)) + A*e*(15*b*c^2*d^2*e*(66*d^3 + 161*d^2*e*x + 140*d*e^2*x^2 +
42*e^3*x^3) + b^3*e^3*(176*d^3 + 406*d^2*e*x + 350*d*e^2*x^2 + 105*e^3*x^3) - 3*c^3*d^3*(194*d^3 + 504*d^2*e*x
+ 455*d*e^2*x^2 + 140*e^3*x^3) - b^2*c*d*e^2*(689*d^3 + 1624*d^2*e*x + 1400*d*e^2*x^2 + 420*e^3*x^3))))/(105*
d^4*(c*d - b*e)^4*(d + e*x)^(7/2)) - (2*c^(7/2)*(-(b*B) + A*c)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*
e]])/(b*(-(c*d) + b*e)^(9/2)) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(9/2))

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {\frac {2 A e}{7}-\frac {2 B d}{7}}{d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {\frac {2}{5} A b \,e^{2}-\frac {4}{5} A c d e +\frac {2}{5} B c \,d^{2}}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {\frac {2}{3} A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e -\frac {2}{3} B \,c^{2} d^{3}}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {\left (2 b^{3} e^{4}-8 b^{2} c d \,e^{3}+12 b \,c^{2} d^{2} e^{2}-8 c^{3} d^{3} e \right ) A +2 B \,c^{3} d^{4}}{\sqrt {e x +d}\, d^{4} \left (b e -c d \right )^{4}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}\)	\(286\)
derivativedivides	\(-\frac {2 \left (-A e +B d \right )}{7 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (-A b \,e^{2}+2 A c d e -B c \,d^{2}\right )}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-A \,b^{2} e^{3}+3 A b c d \,e^{2}-3 A \,c^{2} d^{2} e +B \,c^{2} d^{3}\right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-A \,b^{3} e^{4}+4 A \,b^{2} c d \,e^{3}-6 A b \,c^{2} d^{2} e^{2}+4 A \,c^{3} d^{3} e -B \,c^{3} d^{4}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}\)	\(290\)
default	\(-\frac {2 \left (-A e +B d \right )}{7 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (-A b \,e^{2}+2 A c d e -B c \,d^{2}\right )}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-A \,b^{2} e^{3}+3 A b c d \,e^{2}-3 A \,c^{2} d^{2} e +B \,c^{2} d^{3}\right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-A \,b^{3} e^{4}+4 A \,b^{2} c d \,e^{3}-6 A b \,c^{2} d^{2} e^{2}+4 A \,c^{3} d^{3} e -B \,c^{3} d^{4}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}\)	\(290\)

[In]

int((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

2/7*(A*e-B*d)/d/(b*e-c*d)/(e*x+d)^(7/2)+2/5*(A*b*e^2-2*A*c*d*e+B*c*d^2)/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)+2/3*(A*b
^2*e^3-3*A*b*c*d*e^2+3*A*c^2*d^2*e-B*c^2*d^3)/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)+((2*b^3*e^4-8*b^2*c*d*e^3+12*b*c^2
*d^2*e^2-8*c^3*d^3*e)*A+2*B*c^3*d^4)/(e*x+d)^(1/2)/d^4/(b*e-c*d)^4-2/(b*e-c*d)^4*c^4*(A*c-B*b)/b/((b*e-c*d)*c)
^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(9/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (271) = 542\).

Time = 15.31 (sec) , antiderivative size = 4757, normalized size of antiderivative = 15.80 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-1/105*(105*((B*b*c^3 - A*c^4)*d^5*e^4*x^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^
2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^3 - A*c^4)*d^9)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d
- b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 105*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2
- 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*
e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*
d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x
^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(d)*
log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(176*B*b*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*
A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3
- 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 6
0*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*
b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115
*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3
+ b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 +
4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11
*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c
^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x), -1/105*(210*((B*b*c^3 - A*c^4)*d^5*e^4*x
^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^
3 - A*c^4)*d^9)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 1
05*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4
*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e
^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b
^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5
*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(176*B*b
*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (1
5*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^
2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 60*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e
^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*
(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^
2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3 + b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^
3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 + 4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5
- 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d
^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*
d^8*e^5)*x), 1/105*(210*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4
+ (A*c^4*d^4*e^4 - 4*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*
e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*
b*c^3*d^5*e^3 + 6*A*b^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*
e^2 + 6*A*b^2*c^2*d^5*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) -
105*((B*b*c^3 - A*c^4)*d^5*e^4*x^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*
b*c^3 - A*c^4)*d^8*e*x + (B*b*c^3 - A*c^4)*d^9)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*s
qrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(176*B*b*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*
A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3
- 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 6
0*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*
b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115
*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3
+ b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 +
4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11
*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c
^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x), -2/105*(105*((B*b*c^3 - A*c^4)*d^5*e^4*x
^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^
3 - A*c^4)*d^9)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 1
05*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4
*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e
^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b
^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5
*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (176*B*b*c^3*d^8 + 17
6*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689
*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4
*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 60*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c
^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2
+ 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e
+ 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3 + b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6
- 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 + 4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^
7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*
d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x)]

Sympy [A] (verification not implemented)

Time = 11.78 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {A e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{4} \sqrt {- d}} - \frac {e \left (- A e + B d\right )}{7 d \left (d + e x\right )^{\frac {7}{2}} \left (b e - c d\right )} + \frac {e \left (A b e^{2} - 2 A c d e + B c d^{2}\right )}{5 d^{2} \left (d + e x\right )^{\frac {5}{2}} \left (b e - c d\right )^{2}} + \frac {e \left (A b^{2} e^{3} - 3 A b c d e^{2} + 3 A c^{2} d^{2} e - B c^{2} d^{3}\right )}{3 d^{3} \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )^{3}} + \frac {e \left (A b^{3} e^{4} - 4 A b^{2} c d e^{3} + 6 A b c^{2} d^{2} e^{2} - 4 A c^{3} d^{3} e + B c^{3} d^{4}\right )}{d^{4} \sqrt {d + e x} \left (b e - c d\right )^{4}} + \frac {c^{3} e \left (- A c + B b\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {B \log {\left (b x + c x^{2} \right )}}{2 c} + \left (A - \frac {B b}{2 c}\right ) \left (- \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\- \frac {\log {\left (b - 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b}\right )}{d^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((B*x+A)/(e*x+d)**(9/2)/(c*x**2+b*x),x)

[Out]

Piecewise((2*(A*e*atan(sqrt(d + e*x)/sqrt(-d))/(b*d**4*sqrt(-d)) - e*(-A*e + B*d)/(7*d*(d + e*x)**(7/2)*(b*e -
c*d)) + e*(A*b*e**2 - 2*A*c*d*e + B*c*d**2)/(5*d**2*(d + e*x)**(5/2)*(b*e - c*d)**2) + e*(A*b**2*e**3 - 3*A*b
*c*d*e**2 + 3*A*c**2*d**2*e - B*c**2*d**3)/(3*d**3*(d + e*x)**(3/2)*(b*e - c*d)**3) + e*(A*b**3*e**4 - 4*A*b**
2*c*d*e**3 + 6*A*b*c**2*d**2*e**2 - 4*A*c**3*d**3*e + B*c**3*d**4)/(d**4*sqrt(d + e*x)*(b*e - c*d)**4) + c**3*
e*(-A*c + B*b)*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**4))/e, Ne(e, 0)), (
(B*log(b*x + c*x**2)/(2*c) + (A - B*b/(2*c))*(-2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), (-log(b - 2*c*(b/(2*
c) + x))/(2*c), True))/b - 2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), (log(b + 2*c*(b/(2*c) + x))/(2*c), True)
)/b))/d**(9/2), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (271) = 542\).

Time = 0.29 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 \, {\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} \sqrt {-c^{2} d + b c e}} + \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{3} B c^{3} d^{4} + 35 \, {\left (e x + d\right )}^{2} B c^{3} d^{5} + 21 \, {\left (e x + d\right )} B c^{3} d^{6} + 15 \, B c^{3} d^{7} - 420 \, {\left (e x + d\right )}^{3} A c^{3} d^{3} e - 35 \, {\left (e x + d\right )}^{2} B b c^{2} d^{4} e - 105 \, {\left (e x + d\right )}^{2} A c^{3} d^{4} e - 42 \, {\left (e x + d\right )} B b c^{2} d^{5} e - 42 \, {\left (e x + d\right )} A c^{3} d^{5} e - 45 \, B b c^{2} d^{6} e - 15 \, A c^{3} d^{6} e + 630 \, {\left (e x + d\right )}^{3} A b c^{2} d^{2} e^{2} + 210 \, {\left (e x + d\right )}^{2} A b c^{2} d^{3} e^{2} + 21 \, {\left (e x + d\right )} B b^{2} c d^{4} e^{2} + 105 \, {\left (e x + d\right )} A b c^{2} d^{4} e^{2} + 45 \, B b^{2} c d^{5} e^{2} + 45 \, A b c^{2} d^{5} e^{2} - 420 \, {\left (e x + d\right )}^{3} A b^{2} c d e^{3} - 140 \, {\left (e x + d\right )}^{2} A b^{2} c d^{2} e^{3} - 84 \, {\left (e x + d\right )} A b^{2} c d^{3} e^{3} - 15 \, B b^{3} d^{4} e^{3} - 45 \, A b^{2} c d^{4} e^{3} + 105 \, {\left (e x + d\right )}^{3} A b^{3} e^{4} + 35 \, {\left (e x + d\right )}^{2} A b^{3} d e^{4} + 21 \, {\left (e x + d\right )} A b^{3} d^{2} e^{4} + 15 \, A b^{3} d^{3} e^{4}\right )}}{105 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}} + \frac {2 \, A \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{4}} \]

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

2*(B*b*c^4 - A*c^5)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^4*d^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^2
*e^2 - 4*b^4*c*d*e^3 + b^5*e^4)*sqrt(-c^2*d + b*c*e)) + 2/105*(105*(e*x + d)^3*B*c^3*d^4 + 35*(e*x + d)^2*B*c^
3*d^5 + 21*(e*x + d)*B*c^3*d^6 + 15*B*c^3*d^7 - 420*(e*x + d)^3*A*c^3*d^3*e - 35*(e*x + d)^2*B*b*c^2*d^4*e - 1
05*(e*x + d)^2*A*c^3*d^4*e - 42*(e*x + d)*B*b*c^2*d^5*e - 42*(e*x + d)*A*c^3*d^5*e - 45*B*b*c^2*d^6*e - 15*A*c
^3*d^6*e + 630*(e*x + d)^3*A*b*c^2*d^2*e^2 + 210*(e*x + d)^2*A*b*c^2*d^3*e^2 + 21*(e*x + d)*B*b^2*c*d^4*e^2 +
105*(e*x + d)*A*b*c^2*d^4*e^2 + 45*B*b^2*c*d^5*e^2 + 45*A*b*c^2*d^5*e^2 - 420*(e*x + d)^3*A*b^2*c*d*e^3 - 140*
(e*x + d)^2*A*b^2*c*d^2*e^3 - 84*(e*x + d)*A*b^2*c*d^3*e^3 - 15*B*b^3*d^4*e^3 - 45*A*b^2*c*d^4*e^3 + 105*(e*x
+ d)^3*A*b^3*e^4 + 35*(e*x + d)^2*A*b^3*d*e^4 + 21*(e*x + d)*A*b^3*d^2*e^4 + 15*A*b^3*d^3*e^4)/((c^4*d^8 - 4*b
*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(e*x + d)^(7/2)) + 2*A*arctan(sqrt(e*x + d)/sq
rt(-d))/(b*sqrt(-d)*d^4)

Mupad [B] (verification not implemented)

Time = 15.19 (sec) , antiderivative size = 11601, normalized size of antiderivative = 38.54 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

int((A + B*x)/((b*x + c*x^2)*(d + e*x)^(9/2)),x)

[Out]

(A*atan((B^2*b^2*c^19*d^41*(d + e*x)^(1/2)*1i + A^2*b^21*d^20*e^21*(d + e*x)^(1/2)*1i - A^2*b^20*c*d^21*e^20*(
d + e*x)^(1/2)*21i - B^2*b^3*c^18*d^40*e*(d + e*x)^(1/2)*12i - A*B*b*c^20*d^41*(d + e*x)^(1/2)*2i - A^2*b^2*c^
19*d^39*e^2*(d + e*x)^(1/2)*144i + A^2*b^3*c^18*d^38*e^3*(d + e*x)^(1/2)*1110i - A^2*b^4*c^17*d^37*e^4*(d + e*
x)^(1/2)*5490i + A^2*b^5*c^16*d^36*e^5*(d + e*x)^(1/2)*19557i - A^2*b^6*c^15*d^35*e^6*(d + e*x)^(1/2)*53340i +
A^2*b^7*c^14*d^34*e^7*(d + e*x)^(1/2)*115488i - A^2*b^8*c^13*d^33*e^8*(d + e*x)^(1/2)*202995i + A^2*b^9*c^12*
d^32*e^9*(d + e*x)^(1/2)*293710i - A^2*b^10*c^11*d^31*e^10*(d + e*x)^(1/2)*352650i + A^2*b^11*c^10*d^30*e^11*(
d + e*x)^(1/2)*352704i - A^2*b^12*c^9*d^29*e^12*(d + e*x)^(1/2)*293929i + A^2*b^13*c^8*d^28*e^13*(d + e*x)^(1/
2)*203490i - A^2*b^14*c^7*d^27*e^14*(d + e*x)^(1/2)*116280i + A^2*b^15*c^6*d^26*e^15*(d + e*x)^(1/2)*54264i -
A^2*b^16*c^5*d^25*e^16*(d + e*x)^(1/2)*20349i + A^2*b^17*c^4*d^24*e^17*(d + e*x)^(1/2)*5985i - A^2*b^18*c^3*d^
23*e^18*(d + e*x)^(1/2)*1330i + A^2*b^19*c^2*d^22*e^19*(d + e*x)^(1/2)*210i + B^2*b^4*c^17*d^39*e^2*(d + e*x)^
(1/2)*66i - B^2*b^5*c^16*d^38*e^3*(d + e*x)^(1/2)*220i + B^2*b^6*c^15*d^37*e^4*(d + e*x)^(1/2)*495i - B^2*b^7*
c^14*d^36*e^5*(d + e*x)^(1/2)*792i + B^2*b^8*c^13*d^35*e^6*(d + e*x)^(1/2)*924i - B^2*b^9*c^12*d^34*e^7*(d + e
*x)^(1/2)*792i + B^2*b^10*c^11*d^33*e^8*(d + e*x)^(1/2)*495i - B^2*b^11*c^10*d^32*e^9*(d + e*x)^(1/2)*220i + B
^2*b^12*c^9*d^31*e^10*(d + e*x)^(1/2)*66i - B^2*b^13*c^8*d^30*e^11*(d + e*x)^(1/2)*12i + B^2*b^14*c^7*d^29*e^1
2*(d + e*x)^(1/2)*1i + A^2*b*c^20*d^40*e*(d + e*x)^(1/2)*9i - A*B*b^3*c^18*d^39*e^2*(d + e*x)^(1/2)*132i + A*B
*b^4*c^17*d^38*e^3*(d + e*x)^(1/2)*440i - A*B*b^5*c^16*d^37*e^4*(d + e*x)^(1/2)*990i + A*B*b^6*c^15*d^36*e^5*(
d + e*x)^(1/2)*1584i - A*B*b^7*c^14*d^35*e^6*(d + e*x)^(1/2)*1848i + A*B*b^8*c^13*d^34*e^7*(d + e*x)^(1/2)*158
4i - A*B*b^9*c^12*d^33*e^8*(d + e*x)^(1/2)*990i + A*B*b^10*c^11*d^32*e^9*(d + e*x)^(1/2)*440i - A*B*b^11*c^10*
d^31*e^10*(d + e*x)^(1/2)*132i + A*B*b^12*c^9*d^30*e^11*(d + e*x)^(1/2)*24i - A*B*b^13*c^8*d^29*e^12*(d + e*x)
^(1/2)*2i + A*B*b^2*c^19*d^40*e*(d + e*x)^(1/2)*24i)/(d^9*(d^9)^(1/2)*(d^9*(d^9*(d^9*(9*A^2*b*c^20*e + B^2*b^2
*c^19*d - 12*B^2*b^3*c^18*e - 2*A*B*b*c^20*d + 24*A*B*b^2*c^19*e) - 352650*A^2*b^10*c^11*e^10 + 66*B^2*b^12*c^
9*e^10 - 144*A^2*b^2*c^19*d^8*e^2 + 1110*A^2*b^3*c^18*d^7*e^3 - 5490*A^2*b^4*c^17*d^6*e^4 + 19557*A^2*b^5*c^16
*d^5*e^5 - 53340*A^2*b^6*c^15*d^4*e^6 + 115488*A^2*b^7*c^14*d^3*e^7 - 202995*A^2*b^8*c^13*d^2*e^8 + 66*B^2*b^4
*c^17*d^8*e^2 - 220*B^2*b^5*c^16*d^7*e^3 + 495*B^2*b^6*c^15*d^6*e^4 - 792*B^2*b^7*c^14*d^5*e^5 + 924*B^2*b^8*c
^13*d^4*e^6 - 792*B^2*b^9*c^12*d^3*e^7 + 495*B^2*b^10*c^11*d^2*e^8 - 132*A*B*b^11*c^10*e^10 + 293710*A^2*b^9*c
^12*d*e^9 - 220*B^2*b^11*c^10*d*e^9 + 440*A*B*b^10*c^11*d*e^9 - 132*A*B*b^3*c^18*d^8*e^2 + 440*A*B*b^4*c^17*d^
7*e^3 - 990*A*B*b^5*c^16*d^6*e^4 + 1584*A*B*b^6*c^15*d^5*e^5 - 1848*A*B*b^7*c^14*d^4*e^6 + 1584*A*B*b^8*c^13*d
^3*e^7 - 990*A*B*b^9*c^12*d^2*e^8) + 210*A^2*b^19*c^2*e^19 + 352704*A^2*b^11*c^10*d^8*e^11 - 293929*A^2*b^12*c
^9*d^7*e^12 + 203490*A^2*b^13*c^8*d^6*e^13 - 116280*A^2*b^14*c^7*d^5*e^14 + 54264*A^2*b^15*c^6*d^4*e^15 - 2034
9*A^2*b^16*c^5*d^3*e^16 + 5985*A^2*b^17*c^4*d^2*e^17 - 12*B^2*b^13*c^8*d^8*e^11 + B^2*b^14*c^7*d^7*e^12 - 1330
*A^2*b^18*c^3*d*e^18 + 24*A*B*b^12*c^9*d^8*e^11 - 2*A*B*b^13*c^8*d^7*e^12) + A^2*b^21*d^7*e^21 - 21*A^2*b^20*c
*d^8*e^20)))*2i)/(b*(d^9)^(1/2)) - ((2*(A*e - B*d))/(7*(c*d^2 - b*d*e)) - (2*(d + e*x)^3*(A*b^3*e^4 + B*c^3*d^
4 - 4*A*c^3*d^3*e + 6*A*b*c^2*d^2*e^2 - 4*A*b^2*c*d*e^3))/(c*d^2 - b*d*e)^4 + (2*(d + e*x)^2*(A*b^2*e^3 - B*c^
2*d^3 + 3*A*c^2*d^2*e - 3*A*b*c*d*e^2))/(3*(c*d^2 - b*d*e)^3) - (2*(d + e*x)*(A*b*e^2 + B*c*d^2 - 2*A*c*d*e))/
(5*(c*d^2 - b*d*e)^2))/(d + e*x)^(7/2) + (atan((((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A
^2*c^23*d^32*e^2 + 2048*A^2*b^2*c^21*d^30*e^4 - 10880*A^2*b^3*c^20*d^29*e^5 + 42720*A^2*b^4*c^19*d^28*e^6 - 13
0368*A^2*b^5*c^18*d^27*e^7 + 317472*A^2*b^6*c^17*d^26*e^8 - 626496*A^2*b^7*c^16*d^25*e^9 + 1011720*A^2*b^8*c^1
5*d^24*e^10 - 1345440*A^2*b^9*c^14*d^23*e^11 + 1478576*A^2*b^10*c^13*d^22*e^12 - 1343776*A^2*b^11*c^12*d^21*e^
13 + 1007768*A^2*b^12*c^11*d^20*e^14 - 620160*A^2*b^13*c^10*d^19*e^15 + 310080*A^2*b^14*c^9*d^18*e^16 - 124032
*A^2*b^15*c^8*d^17*e^17 + 38760*A^2*b^16*c^7*d^16*e^18 - 9120*A^2*b^17*c^6*d^15*e^19 + 1520*A^2*b^18*c^5*d^14*
e^20 - 160*A^2*b^19*c^4*d^13*e^21 + 8*A^2*b^20*c^3*d^12*e^22 + 8*B^2*b^2*c^21*d^32*e^2 - 96*B^2*b^3*c^20*d^31*
e^3 + 528*B^2*b^4*c^19*d^30*e^4 - 1760*B^2*b^5*c^18*d^29*e^5 + 3960*B^2*b^6*c^17*d^28*e^6 - 6336*B^2*b^7*c^16*
d^27*e^7 + 7392*B^2*b^8*c^15*d^26*e^8 - 6336*B^2*b^9*c^14*d^25*e^9 + 3960*B^2*b^10*c^13*d^24*e^10 - 1760*B^2*b
^11*c^12*d^23*e^11 + 528*B^2*b^12*c^11*d^22*e^12 - 96*B^2*b^13*c^10*d^21*e^13 + 8*B^2*b^14*c^9*d^20*e^14 - 256
*A^2*b*c^22*d^31*e^3 - 16*A*B*b*c^22*d^32*e^2 + 192*A*B*b^2*c^21*d^31*e^3 - 1056*A*B*b^3*c^20*d^30*e^4 + 3520*
A*B*b^4*c^19*d^29*e^5 - 7920*A*B*b^5*c^18*d^28*e^6 + 12672*A*B*b^6*c^17*d^27*e^7 - 14784*A*B*b^7*c^16*d^26*e^8
+ 12672*A*B*b^8*c^15*d^25*e^9 - 7920*A*B*b^9*c^14*d^24*e^10 + 3520*A*B*b^10*c^13*d^23*e^11 - 1056*A*B*b^11*c^
12*d^22*e^12 + 192*A*B*b^12*c^11*d^21*e^13 - 16*A*B*b^13*c^10*d^20*e^14) - ((-c^7*(b*e - c*d)^9)^(1/2)*(A*c -
B*b)*(((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^23*d^41*e^2 - 328*b^3*c^22*d^40*e^3 +
3200*b^4*c^21*d^39*e^4 - 19760*b^5*c^20*d^38*e^5 + 86640*b^6*c^19*d^37*e^6 - 286824*b^7*c^18*d^36*e^7 + 744192
*b^8*c^17*d^35*e^8 - 1550400*b^9*c^16*d^34*e^9 + 2635680*b^10*c^15*d^33*e^10 - 3695120*b^11*c^14*d^32*e^11 + 4
299776*b^12*c^13*d^31*e^12 - 4165408*b^13*c^12*d^30*e^13 + 3359200*b^14*c^11*d^29*e^14 - 2248080*b^15*c^10*d^2
8*e^15 + 1240320*b^16*c^9*d^27*e^16 - 558144*b^17*c^8*d^26*e^17 + 201552*b^18*c^7*d^25*e^18 - 57000*b^19*c^6*d
^24*e^19 + 12160*b^20*c^5*d^23*e^20 - 1840*b^21*c^4*d^22*e^21 + 176*b^22*c^3*d^21*e^22 - 8*b^23*c^2*d^20*e^23)
)/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 12
6*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8) - 40*A*b^2*c^22*d^36*e^3 + 720*A*
b^3*c^21*d^35*e^4 - 6160*A*b^4*c^20*d^34*e^5 + 33320*A*b^5*c^19*d^33*e^6 - 127848*A*b^6*c^18*d^32*e^7 + 370048
*A*b^7*c^17*d^31*e^8 - 838720*A*b^8*c^16*d^30*e^9 + 1524960*A*b^9*c^15*d^29*e^10 - 2259920*A*b^10*c^14*d^28*e^
11 + 2757664*A*b^11*c^13*d^27*e^12 - 2786784*A*b^12*c^12*d^26*e^13 + 2336880*A*b^13*c^11*d^25*e^14 - 1623440*A
*b^14*c^10*d^24*e^15 + 929280*A*b^15*c^9*d^23*e^16 - 433984*A*b^16*c^8*d^22*e^17 + 162784*A*b^17*c^7*d^21*e^18
- 47880*A*b^18*c^6*d^20*e^19 + 10640*A*b^19*c^5*d^19*e^20 - 1680*A*b^20*c^4*d^18*e^21 + 168*A*b^21*c^3*d^17*e
^22 - 8*A*b^22*c^2*d^16*e^23 + 8*B*b^2*c^22*d^37*e^2 - 128*B*b^3*c^21*d^36*e^3 + 960*B*b^4*c^20*d^35*e^4 - 448
0*B*b^5*c^19*d^34*e^5 + 14560*B*b^6*c^18*d^33*e^6 - 34944*B*b^7*c^17*d^32*e^7 + 64064*B*b^8*c^16*d^31*e^8 - 91
520*B*b^9*c^15*d^30*e^9 + 102960*B*b^10*c^14*d^29*e^10 - 91520*B*b^11*c^13*d^28*e^11 + 64064*B*b^12*c^12*d^27*
e^12 - 34944*B*b^13*c^11*d^26*e^13 + 14560*B*b^14*c^10*d^25*e^14 - 4480*B*b^15*c^9*d^24*e^15 + 960*B*b^16*c^8*
d^23*e^16 - 128*B*b^17*c^7*d^22*e^17 + 8*B*b^18*c^6*d^21*e^18))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b
^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*
c^2*d^2*e^7 - 9*b^9*c*d*e^8))*1i)/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^
6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8) +
((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^23*d^32*e^2 + 2048*A^2*b^2*c^21*d^30*e^4 -
10880*A^2*b^3*c^20*d^29*e^5 + 42720*A^2*b^4*c^19*d^28*e^6 - 130368*A^2*b^5*c^18*d^27*e^7 + 317472*A^2*b^6*c^1
7*d^26*e^8 - 626496*A^2*b^7*c^16*d^25*e^9 + 1011720*A^2*b^8*c^15*d^24*e^10 - 1345440*A^2*b^9*c^14*d^23*e^11 +
1478576*A^2*b^10*c^13*d^22*e^12 - 1343776*A^2*b^11*c^12*d^21*e^13 + 1007768*A^2*b^12*c^11*d^20*e^14 - 620160*A
^2*b^13*c^10*d^19*e^15 + 310080*A^2*b^14*c^9*d^18*e^16 - 124032*A^2*b^15*c^8*d^17*e^17 + 38760*A^2*b^16*c^7*d^
16*e^18 - 9120*A^2*b^17*c^6*d^15*e^19 + 1520*A^2*b^18*c^5*d^14*e^20 - 160*A^2*b^19*c^4*d^13*e^21 + 8*A^2*b^20*
c^3*d^12*e^22 + 8*B^2*b^2*c^21*d^32*e^2 - 96*B^2*b^3*c^20*d^31*e^3 + 528*B^2*b^4*c^19*d^30*e^4 - 1760*B^2*b^5*
c^18*d^29*e^5 + 3960*B^2*b^6*c^17*d^28*e^6 - 6336*B^2*b^7*c^16*d^27*e^7 + 7392*B^2*b^8*c^15*d^26*e^8 - 6336*B^
2*b^9*c^14*d^25*e^9 + 3960*B^2*b^10*c^13*d^24*e^10 - 1760*B^2*b^11*c^12*d^23*e^11 + 528*B^2*b^12*c^11*d^22*e^1
2 - 96*B^2*b^13*c^10*d^21*e^13 + 8*B^2*b^14*c^9*d^20*e^14 - 256*A^2*b*c^22*d^31*e^3 - 16*A*B*b*c^22*d^32*e^2 +
192*A*B*b^2*c^21*d^31*e^3 - 1056*A*B*b^3*c^20*d^30*e^4 + 3520*A*B*b^4*c^19*d^29*e^5 - 7920*A*B*b^5*c^18*d^28*
e^6 + 12672*A*B*b^6*c^17*d^27*e^7 - 14784*A*B*b^7*c^16*d^26*e^8 + 12672*A*B*b^8*c^15*d^25*e^9 - 7920*A*B*b^9*c
^14*d^24*e^10 + 3520*A*B*b^10*c^13*d^23*e^11 - 1056*A*B*b^11*c^12*d^22*e^12 + 192*A*B*b^12*c^11*d^21*e^13 - 16
*A*B*b^13*c^10*d^20*e^14) - ((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*(((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*(
d + e*x)^(1/2)*(16*b^2*c^23*d^41*e^2 - 328*b^3*c^22*d^40*e^3 + 3200*b^4*c^21*d^39*e^4 - 19760*b^5*c^20*d^38*e^
5 + 86640*b^6*c^19*d^37*e^6 - 286824*b^7*c^18*d^36*e^7 + 744192*b^8*c^17*d^35*e^8 - 1550400*b^9*c^16*d^34*e^9
+ 2635680*b^10*c^15*d^33*e^10 - 3695120*b^11*c^14*d^32*e^11 + 4299776*b^12*c^13*d^31*e^12 - 4165408*b^13*c^12*
d^30*e^13 + 3359200*b^14*c^11*d^29*e^14 - 2248080*b^15*c^10*d^28*e^15 + 1240320*b^16*c^9*d^27*e^16 - 558144*b^
17*c^8*d^26*e^17 + 201552*b^18*c^7*d^25*e^18 - 57000*b^19*c^6*d^24*e^19 + 12160*b^20*c^5*d^23*e^20 - 1840*b^21
*c^4*d^22*e^21 + 176*b^22*c^3*d^21*e^22 - 8*b^23*c^2*d^20*e^23))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*
b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8
*c^2*d^2*e^7 - 9*b^9*c*d*e^8) + 40*A*b^2*c^22*d^36*e^3 - 720*A*b^3*c^21*d^35*e^4 + 6160*A*b^4*c^20*d^34*e^5 -
33320*A*b^5*c^19*d^33*e^6 + 127848*A*b^6*c^18*d^32*e^7 - 370048*A*b^7*c^17*d^31*e^8 + 838720*A*b^8*c^16*d^30*e
^9 - 1524960*A*b^9*c^15*d^29*e^10 + 2259920*A*b^10*c^14*d^28*e^11 - 2757664*A*b^11*c^13*d^27*e^12 + 2786784*A*
b^12*c^12*d^26*e^13 - 2336880*A*b^13*c^11*d^25*e^14 + 1623440*A*b^14*c^10*d^24*e^15 - 929280*A*b^15*c^9*d^23*e
^16 + 433984*A*b^16*c^8*d^22*e^17 - 162784*A*b^17*c^7*d^21*e^18 + 47880*A*b^18*c^6*d^20*e^19 - 10640*A*b^19*c^
5*d^19*e^20 + 1680*A*b^20*c^4*d^18*e^21 - 168*A*b^21*c^3*d^17*e^22 + 8*A*b^22*c^2*d^16*e^23 - 8*B*b^2*c^22*d^3
7*e^2 + 128*B*b^3*c^21*d^36*e^3 - 960*B*b^4*c^20*d^35*e^4 + 4480*B*b^5*c^19*d^34*e^5 - 14560*B*b^6*c^18*d^33*e
^6 + 34944*B*b^7*c^17*d^32*e^7 - 64064*B*b^8*c^16*d^31*e^8 + 91520*B*b^9*c^15*d^30*e^9 - 102960*B*b^10*c^14*d^
29*e^10 + 91520*B*b^11*c^13*d^28*e^11 - 64064*B*b^12*c^12*d^27*e^12 + 34944*B*b^13*c^11*d^26*e^13 - 14560*B*b^
14*c^10*d^25*e^14 + 4480*B*b^15*c^9*d^24*e^15 - 960*B*b^16*c^8*d^23*e^16 + 128*B*b^17*c^7*d^22*e^17 - 8*B*b^18
*c^6*d^21*e^18))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c
^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8))*1i)/(b^10*e^9 - b
*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e
^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8))/(((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*((d +
e*x)^(1/2)*(16*A^2*c^23*d^32*e^2 + 2048*A^2*b^2*c^21*d^30*e^4 - 10880*A^2*b^3*c^20*d^29*e^5 + 42720*A^2*b^4*c^
19*d^28*e^6 - 130368*A^2*b^5*c^18*d^27*e^7 + 317472*A^2*b^6*c^17*d^26*e^8 - 626496*A^2*b^7*c^16*d^25*e^9 + 101
1720*A^2*b^8*c^15*d^24*e^10 - 1345440*A^2*b^9*c^14*d^23*e^11 + 1478576*A^2*b^10*c^13*d^22*e^12 - 1343776*A^2*b
^11*c^12*d^21*e^13 + 1007768*A^2*b^12*c^11*d^20*e^14 - 620160*A^2*b^13*c^10*d^19*e^15 + 310080*A^2*b^14*c^9*d^
18*e^16 - 124032*A^2*b^15*c^8*d^17*e^17 + 38760*A^2*b^16*c^7*d^16*e^18 - 9120*A^2*b^17*c^6*d^15*e^19 + 1520*A^
2*b^18*c^5*d^14*e^20 - 160*A^2*b^19*c^4*d^13*e^21 + 8*A^2*b^20*c^3*d^12*e^22 + 8*B^2*b^2*c^21*d^32*e^2 - 96*B^
2*b^3*c^20*d^31*e^3 + 528*B^2*b^4*c^19*d^30*e^4 - 1760*B^2*b^5*c^18*d^29*e^5 + 3960*B^2*b^6*c^17*d^28*e^6 - 63
36*B^2*b^7*c^16*d^27*e^7 + 7392*B^2*b^8*c^15*d^26*e^8 - 6336*B^2*b^9*c^14*d^25*e^9 + 3960*B^2*b^10*c^13*d^24*e
^10 - 1760*B^2*b^11*c^12*d^23*e^11 + 528*B^2*b^12*c^11*d^22*e^12 - 96*B^2*b^13*c^10*d^21*e^13 + 8*B^2*b^14*c^9
*d^20*e^14 - 256*A^2*b*c^22*d^31*e^3 - 16*A*B*b*c^22*d^32*e^2 + 192*A*B*b^2*c^21*d^31*e^3 - 1056*A*B*b^3*c^20*
d^30*e^4 + 3520*A*B*b^4*c^19*d^29*e^5 - 7920*A*B*b^5*c^18*d^28*e^6 + 12672*A*B*b^6*c^17*d^27*e^7 - 14784*A*B*b
^7*c^16*d^26*e^8 + 12672*A*B*b^8*c^15*d^25*e^9 - 7920*A*B*b^9*c^14*d^24*e^10 + 3520*A*B*b^10*c^13*d^23*e^11 -
1056*A*B*b^11*c^12*d^22*e^12 + 192*A*B*b^12*c^11*d^21*e^13 - 16*A*B*b^13*c^10*d^20*e^14) - ((-c^7*(b*e - c*d)^
9)^(1/2)*(A*c - B*b)*(((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^23*d^41*e^2 - 328*b^3*
c^22*d^40*e^3 + 3200*b^4*c^21*d^39*e^4 - 19760*b^5*c^20*d^38*e^5 + 86640*b^6*c^19*d^37*e^6 - 286824*b^7*c^18*d
^36*e^7 + 744192*b^8*c^17*d^35*e^8 - 1550400*b^9*c^16*d^34*e^9 + 2635680*b^10*c^15*d^33*e^10 - 3695120*b^11*c^
14*d^32*e^11 + 4299776*b^12*c^13*d^31*e^12 - 4165408*b^13*c^12*d^30*e^13 + 3359200*b^14*c^11*d^29*e^14 - 22480
80*b^15*c^10*d^28*e^15 + 1240320*b^16*c^9*d^27*e^16 - 558144*b^17*c^8*d^26*e^17 + 201552*b^18*c^7*d^25*e^18 -
57000*b^19*c^6*d^24*e^19 + 12160*b^20*c^5*d^23*e^20 - 1840*b^21*c^4*d^22*e^21 + 176*b^22*c^3*d^21*e^22 - 8*b^2
3*c^2*d^20*e^23))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*
c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8) - 40*A*b^2*c^22*d
^36*e^3 + 720*A*b^3*c^21*d^35*e^4 - 6160*A*b^4*c^20*d^34*e^5 + 33320*A*b^5*c^19*d^33*e^6 - 127848*A*b^6*c^18*d
^32*e^7 + 370048*A*b^7*c^17*d^31*e^8 - 838720*A*b^8*c^16*d^30*e^9 + 1524960*A*b^9*c^15*d^29*e^10 - 2259920*A*b
^10*c^14*d^28*e^11 + 2757664*A*b^11*c^13*d^27*e^12 - 2786784*A*b^12*c^12*d^26*e^13 + 2336880*A*b^13*c^11*d^25*
e^14 - 1623440*A*b^14*c^10*d^24*e^15 + 929280*A*b^15*c^9*d^23*e^16 - 433984*A*b^16*c^8*d^22*e^17 + 162784*A*b^
17*c^7*d^21*e^18 - 47880*A*b^18*c^6*d^20*e^19 + 10640*A*b^19*c^5*d^19*e^20 - 1680*A*b^20*c^4*d^18*e^21 + 168*A
*b^21*c^3*d^17*e^22 - 8*A*b^22*c^2*d^16*e^23 + 8*B*b^2*c^22*d^37*e^2 - 128*B*b^3*c^21*d^36*e^3 + 960*B*b^4*c^2
0*d^35*e^4 - 4480*B*b^5*c^19*d^34*e^5 + 14560*B*b^6*c^18*d^33*e^6 - 34944*B*b^7*c^17*d^32*e^7 + 64064*B*b^8*c^
16*d^31*e^8 - 91520*B*b^9*c^15*d^30*e^9 + 102960*B*b^10*c^14*d^29*e^10 - 91520*B*b^11*c^13*d^28*e^11 + 64064*B
*b^12*c^12*d^27*e^12 - 34944*B*b^13*c^11*d^26*e^13 + 14560*B*b^14*c^10*d^25*e^14 - 4480*B*b^15*c^9*d^24*e^15 +
960*B*b^16*c^8*d^23*e^16 - 128*B*b^17*c^7*d^22*e^17 + 8*B*b^18*c^6*d^21*e^18))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*
c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d
^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8)))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 +
84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b
^9*c*d*e^8) - ((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^23*d^32*e^2 + 2048*A^2*b^2*c^
21*d^30*e^4 - 10880*A^2*b^3*c^20*d^29*e^5 + 42720*A^2*b^4*c^19*d^28*e^6 - 130368*A^2*b^5*c^18*d^27*e^7 + 31747
2*A^2*b^6*c^17*d^26*e^8 - 626496*A^2*b^7*c^16*d^25*e^9 + 1011720*A^2*b^8*c^15*d^24*e^10 - 1345440*A^2*b^9*c^14
*d^23*e^11 + 1478576*A^2*b^10*c^13*d^22*e^12 - 1343776*A^2*b^11*c^12*d^21*e^13 + 1007768*A^2*b^12*c^11*d^20*e^
14 - 620160*A^2*b^13*c^10*d^19*e^15 + 310080*A^2*b^14*c^9*d^18*e^16 - 124032*A^2*b^15*c^8*d^17*e^17 + 38760*A^
2*b^16*c^7*d^16*e^18 - 9120*A^2*b^17*c^6*d^15*e^19 + 1520*A^2*b^18*c^5*d^14*e^20 - 160*A^2*b^19*c^4*d^13*e^21
+ 8*A^2*b^20*c^3*d^12*e^22 + 8*B^2*b^2*c^21*d^32*e^2 - 96*B^2*b^3*c^20*d^31*e^3 + 528*B^2*b^4*c^19*d^30*e^4 -
1760*B^2*b^5*c^18*d^29*e^5 + 3960*B^2*b^6*c^17*d^28*e^6 - 6336*B^2*b^7*c^16*d^27*e^7 + 7392*B^2*b^8*c^15*d^26*
e^8 - 6336*B^2*b^9*c^14*d^25*e^9 + 3960*B^2*b^10*c^13*d^24*e^10 - 1760*B^2*b^11*c^12*d^23*e^11 + 528*B^2*b^12*
c^11*d^22*e^12 - 96*B^2*b^13*c^10*d^21*e^13 + 8*B^2*b^14*c^9*d^20*e^14 - 256*A^2*b*c^22*d^31*e^3 - 16*A*B*b*c^
22*d^32*e^2 + 192*A*B*b^2*c^21*d^31*e^3 - 1056*A*B*b^3*c^20*d^30*e^4 + 3520*A*B*b^4*c^19*d^29*e^5 - 7920*A*B*b
^5*c^18*d^28*e^6 + 12672*A*B*b^6*c^17*d^27*e^7 - 14784*A*B*b^7*c^16*d^26*e^8 + 12672*A*B*b^8*c^15*d^25*e^9 - 7
920*A*B*b^9*c^14*d^24*e^10 + 3520*A*B*b^10*c^13*d^23*e^11 - 1056*A*B*b^11*c^12*d^22*e^12 + 192*A*B*b^12*c^11*d
^21*e^13 - 16*A*B*b^13*c^10*d^20*e^14) - ((-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*(((-c^7*(b*e - c*d)^9)^(1/2)*
(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^23*d^41*e^2 - 328*b^3*c^22*d^40*e^3 + 3200*b^4*c^21*d^39*e^4 - 19760*b^5
*c^20*d^38*e^5 + 86640*b^6*c^19*d^37*e^6 - 286824*b^7*c^18*d^36*e^7 + 744192*b^8*c^17*d^35*e^8 - 1550400*b^9*c
^16*d^34*e^9 + 2635680*b^10*c^15*d^33*e^10 - 3695120*b^11*c^14*d^32*e^11 + 4299776*b^12*c^13*d^31*e^12 - 41654
08*b^13*c^12*d^30*e^13 + 3359200*b^14*c^11*d^29*e^14 - 2248080*b^15*c^10*d^28*e^15 + 1240320*b^16*c^9*d^27*e^1
6 - 558144*b^17*c^8*d^26*e^17 + 201552*b^18*c^7*d^25*e^18 - 57000*b^19*c^6*d^24*e^19 + 12160*b^20*c^5*d^23*e^2
0 - 1840*b^21*c^4*d^22*e^21 + 176*b^22*c^3*d^21*e^22 - 8*b^23*c^2*d^20*e^23))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^
8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3
*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8) + 40*A*b^2*c^22*d^36*e^3 - 720*A*b^3*c^21*d^35*e^4 + 6160*A*b^4*c^2
0*d^34*e^5 - 33320*A*b^5*c^19*d^33*e^6 + 127848*A*b^6*c^18*d^32*e^7 - 370048*A*b^7*c^17*d^31*e^8 + 838720*A*b^
8*c^16*d^30*e^9 - 1524960*A*b^9*c^15*d^29*e^10 + 2259920*A*b^10*c^14*d^28*e^11 - 2757664*A*b^11*c^13*d^27*e^12
+ 2786784*A*b^12*c^12*d^26*e^13 - 2336880*A*b^13*c^11*d^25*e^14 + 1623440*A*b^14*c^10*d^24*e^15 - 929280*A*b^
15*c^9*d^23*e^16 + 433984*A*b^16*c^8*d^22*e^17 - 162784*A*b^17*c^7*d^21*e^18 + 47880*A*b^18*c^6*d^20*e^19 - 10
640*A*b^19*c^5*d^19*e^20 + 1680*A*b^20*c^4*d^18*e^21 - 168*A*b^21*c^3*d^17*e^22 + 8*A*b^22*c^2*d^16*e^23 - 8*B
*b^2*c^22*d^37*e^2 + 128*B*b^3*c^21*d^36*e^3 - 960*B*b^4*c^20*d^35*e^4 + 4480*B*b^5*c^19*d^34*e^5 - 14560*B*b^
6*c^18*d^33*e^6 + 34944*B*b^7*c^17*d^32*e^7 - 64064*B*b^8*c^16*d^31*e^8 + 91520*B*b^9*c^15*d^30*e^9 - 102960*B
*b^10*c^14*d^29*e^10 + 91520*B*b^11*c^13*d^28*e^11 - 64064*B*b^12*c^12*d^27*e^12 + 34944*B*b^13*c^11*d^26*e^13
- 14560*B*b^14*c^10*d^25*e^14 + 4480*B*b^15*c^9*d^24*e^15 - 960*B*b^16*c^8*d^23*e^16 + 128*B*b^17*c^7*d^22*e^
17 - 8*B*b^18*c^6*d^21*e^18))/(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^
3 - 126*b^5*c^5*d^5*e^4 + 126*b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8)))/(b^
10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*b^6
*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8) + 64*A^3*c^22*d^27*e^3 + 5440*A^3*b^2*
c^20*d^25*e^5 - 21200*A^3*b^3*c^19*d^24*e^6 + 57216*A^3*b^4*c^18*d^23*e^7 - 113344*A^3*b^5*c^17*d^22*e^8 + 170
368*A^3*b^6*c^16*d^21*e^9 - 198000*A^3*b^7*c^15*d^20*e^10 + 179520*A^3*b^8*c^14*d^19*e^11 - 127072*A^3*b^9*c^1
3*d^18*e^12 + 69696*A^3*b^10*c^12*d^17*e^13 - 29104*A^3*b^11*c^11*d^16*e^14 + 8960*A^3*b^12*c^10*d^15*e^15 - 1
920*A^3*b^13*c^9*d^14*e^16 + 256*A^3*b^14*c^8*d^13*e^17 - 16*A^3*b^15*c^7*d^12*e^18 - 16*A^2*B*c^22*d^28*e^2 -
864*A^3*b*c^21*d^26*e^4 - 192*A*B^2*b^2*c^20*d^27*e^3 + 1056*A*B^2*b^3*c^19*d^26*e^4 - 3520*A*B^2*b^4*c^18*d^
25*e^5 + 7920*A*B^2*b^5*c^17*d^24*e^6 - 12672*A*B^2*b^6*c^16*d^23*e^7 + 14784*A*B^2*b^7*c^15*d^22*e^8 - 12672*
A*B^2*b^8*c^14*d^21*e^9 + 7920*A*B^2*b^9*c^13*d^20*e^10 - 3520*A*B^2*b^10*c^12*d^19*e^11 + 1056*A*B^2*b^11*c^1
1*d^18*e^12 - 192*A*B^2*b^12*c^10*d^17*e^13 + 16*A*B^2*b^13*c^9*d^16*e^14 - 192*A^2*B*b^2*c^20*d^26*e^4 - 1920
*A^2*B*b^3*c^19*d^25*e^5 + 13280*A^2*B*b^4*c^18*d^24*e^6 - 44544*A^2*B*b^5*c^17*d^23*e^7 + 98560*A^2*B*b^6*c^1
6*d^22*e^8 - 157696*A^2*B*b^7*c^15*d^21*e^9 + 190080*A^2*B*b^8*c^14*d^20*e^10 - 176000*A^2*B*b^9*c^13*d^19*e^1
1 + 126016*A^2*B*b^10*c^12*d^18*e^12 - 69504*A^2*B*b^11*c^11*d^17*e^13 + 29088*A^2*B*b^12*c^10*d^16*e^14 - 896
0*A^2*B*b^13*c^9*d^15*e^15 + 1920*A^2*B*b^14*c^8*d^14*e^16 - 256*A^2*B*b^15*c^7*d^13*e^17 + 16*A^2*B*b^16*c^6*
d^12*e^18 + 16*A*B^2*b*c^21*d^28*e^2 + 128*A^2*B*b*c^21*d^27*e^3))*(-c^7*(b*e - c*d)^9)^(1/2)*(A*c - B*b)*2i)/
(b^10*e^9 - b*c^9*d^9 + 9*b^2*c^8*d^8*e - 36*b^3*c^7*d^7*e^2 + 84*b^4*c^6*d^6*e^3 - 126*b^5*c^5*d^5*e^4 + 126*
b^6*c^4*d^4*e^5 - 84*b^7*c^3*d^3*e^6 + 36*b^8*c^2*d^2*e^7 - 9*b^9*c*d*e^8)