3.13.39 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^2} \, dx\) [1239]

Optimal. Leaf size=292 \[ \frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{5/2} (2 b B d-4 A c d+7 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{5/2} \left (2 b B c d-4 A c^2 d+5 b^2 B e-3 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}} \]

[Out]

1/3*e*(6*A*c^2*d+5*b^2*B*e-3*b*c*(A*e+B*d))*(e*x+d)^(3/2)/b^2/c^2-(e*x+d)^(5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*
c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)-d^(5/2)*(7*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3+(-b*e+c
*d)^(5/2)*(-3*A*b*c*e-4*A*c^2*d+5*B*b^2*e+2*B*b*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(7/
2)+e*(2*A*c^3*d^2-5*b^3*B*e^2-b*c^2*d*(2*A*e+B*d)+b^2*c*e*(3*A*e+8*B*d))*(e*x+d)^(1/2)/b^2/c^3

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Rubi [A]
time = 0.59, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 838, 840, 1180, 214} \begin {gather*} -\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac {(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}+\frac {e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]

[Out]

(e*(2*A*c^3*d^2 - 5*b^3*B*e^2 - b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(8*B*d + 3*A*e))*Sqrt[d + e*x])/(b^2*c^3) + (e
*(6*A*c^2*d + 5*b^2*B*e - 3*b*c*(B*d + A*e))*(d + e*x)^(3/2))/(3*b^2*c^2) - ((d + e*x)^(5/2)*(A*b*c*d + (2*A*c
^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*(b*x + c*x^2)) - (d^(5/2)*(2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[S
qrt[d + e*x]/Sqrt[d]])/b^3 + ((c*d - b*e)^(5/2)*(2*b*B*c*d - 4*A*c^2*d + 5*b^2*B*e - 3*A*b*c*e)*ArcTanh[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(7/2))

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], 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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 838

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

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 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*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} c d (2 b B d-4 A c d+7 A b e)+\frac {1}{2} e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c^2 d^2 (2 b B d-4 A c d+7 A b e)+\frac {1}{2} e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{2} c^3 d^3 (2 b B d-4 A c d+7 A b e)-\frac {1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} c^3 d^3 e (2 b B d-4 A c d+7 A b e)+\frac {1}{2} d e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right )-\frac {1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\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 )}{b^2 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\left (c d^3 (2 b B d-4 A c d+7 A b e)\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^3}+\frac {\left ((c d-b e)^3 \left (4 A c^2 d-5 b^2 B e-b c (2 B d-3 A e)\right )\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^3 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{5/2} (2 b B d-4 A c d+7 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{5/2} \left (2 b B c d-4 A c^2 d+5 b^2 B e-3 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 262, normalized size = 0.90 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (-3 A c \left (2 c^3 d^3 x-3 b^3 e^3 x+b c^2 d^2 (d-3 e x)+b^2 c e^2 x (3 d-2 e x)\right )+b B x \left (3 c^3 d^3-15 b^3 e^3+b^2 c e^2 (29 d-10 e x)+b c^2 e \left (-9 d^2+20 d e x+2 e^2 x^2\right )\right )\right )}{c^3 x (b+c x)}-\frac {3 (-c d+b e)^{5/2} \left (-2 b B c d+4 A c^2 d-5 b^2 B e+3 A b c e\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{7/2}}-3 d^{5/2} (2 b B d-4 A c d+7 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]

[Out]

((b*Sqrt[d + e*x]*(-3*A*c*(2*c^3*d^3*x - 3*b^3*e^3*x + b*c^2*d^2*(d - 3*e*x) + b^2*c*e^2*x*(3*d - 2*e*x)) + b*
B*x*(3*c^3*d^3 - 15*b^3*e^3 + b^2*c*e^2*(29*d - 10*e*x) + b*c^2*e*(-9*d^2 + 20*d*e*x + 2*e^2*x^2))))/(c^3*x*(b
+ c*x)) - (3*(-(c*d) + b*e)^(5/2)*(-2*b*B*c*d + 4*A*c^2*d - 5*b^2*B*e + 3*A*b*c*e)*ArcTan[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[-(c*d) + b*e]])/c^(7/2) - 3*d^(5/2)*(2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3
*b^3)

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Maple [A]
time = 0.76, size = 405, normalized size = 1.39

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {\frac {B c \left (e x +d \right )^{\frac {3}{2}}}{3}+A c e \sqrt {e x +d}-2 B b e \sqrt {e x +d}+3 B c d \sqrt {e x +d}}{c^{3}}-\frac {d^{3} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e -4 A c d +2 B b d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}-\frac {\frac {\left (-\frac {1}{2} A \,b^{4} c \,e^{4}+\frac {3}{2} A \,b^{3} c^{2} d \,e^{3}-\frac {3}{2} A \,b^{2} c^{3} d^{2} e^{2}+\frac {1}{2} A \,c^{4} d^{3} e b +\frac {1}{2} b^{5} B \,e^{4}-\frac {3}{2} B \,b^{4} c d \,e^{3}+\frac {3}{2} B \,b^{3} c^{2} d^{2} e^{2}-\frac {1}{2} B \,b^{2} c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A \,b^{4} c \,e^{4}-5 A \,b^{3} c^{2} d \,e^{3}-3 A \,b^{2} c^{3} d^{2} e^{2}+9 A \,c^{4} d^{3} e b -4 A \,d^{4} c^{5}-5 b^{5} B \,e^{4}+13 B \,b^{4} c d \,e^{3}-9 B \,b^{3} c^{2} d^{2} e^{2}-B \,b^{2} c^{3} d^{3} e +2 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} e^{2} b^{3}}\right )\)	\(405\)
default	\(2 e^{2} \left (\frac {\frac {B c \left (e x +d \right )^{\frac {3}{2}}}{3}+A c e \sqrt {e x +d}-2 B b e \sqrt {e x +d}+3 B c d \sqrt {e x +d}}{c^{3}}-\frac {d^{3} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e -4 A c d +2 B b d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}-\frac {\frac {\left (-\frac {1}{2} A \,b^{4} c \,e^{4}+\frac {3}{2} A \,b^{3} c^{2} d \,e^{3}-\frac {3}{2} A \,b^{2} c^{3} d^{2} e^{2}+\frac {1}{2} A \,c^{4} d^{3} e b +\frac {1}{2} b^{5} B \,e^{4}-\frac {3}{2} B \,b^{4} c d \,e^{3}+\frac {3}{2} B \,b^{3} c^{2} d^{2} e^{2}-\frac {1}{2} B \,b^{2} c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A \,b^{4} c \,e^{4}-5 A \,b^{3} c^{2} d \,e^{3}-3 A \,b^{2} c^{3} d^{2} e^{2}+9 A \,c^{4} d^{3} e b -4 A \,d^{4} c^{5}-5 b^{5} B \,e^{4}+13 B \,b^{4} c d \,e^{3}-9 B \,b^{3} c^{2} d^{2} e^{2}-B \,b^{2} c^{3} d^{3} e +2 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} e^{2} b^{3}}\right )\)	\(405\)
risch	\(-\frac {4 e^{3} b B \sqrt {e x +d}}{c^{3}}+\frac {2 e^{2} B \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2}}+\frac {2 e^{3} A \sqrt {e x +d}}{c^{2}}-\frac {9 e c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A \,d^{3}}{b^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {13 e^{3} b \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B d}{c^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {e c \sqrt {e x +d}\, A \,d^{3}}{b^{2} \left (c e x +b e \right )}+\frac {3 e^{3} b \sqrt {e x +d}\, B d}{c^{2} \left (c e x +b e \right )}+\frac {6 e^{2} B d \sqrt {e x +d}}{c^{2}}-\frac {d^{3} A \sqrt {e x +d}}{b^{2} x}-\frac {7 e \,d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) A}{b^{2}}+\frac {4 d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) A c}{b^{3}}+\frac {5 e^{4} b^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B}{c^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {2 d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) B}{b^{2}}+\frac {5 e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A d}{c \sqrt {\left (b e -c d \right ) c}}+\frac {4 c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A \,d^{4}}{b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {9 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B \,d^{2}}{c \sqrt {\left (b e -c d \right ) c}}-\frac {2 c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B \,d^{4}}{b^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {3 e^{3} \sqrt {e x +d}\, A d}{c \left (c e x +b e \right )}-\frac {3 e^{2} \sqrt {e x +d}\, B \,d^{2}}{c \left (c e x +b e \right )}+\frac {3 e^{2} \sqrt {e x +d}\, A \,d^{2}}{b \left (c e x +b e \right )}+\frac {e \sqrt {e x +d}\, B \,d^{3}}{b \left (c e x +b e \right )}+\frac {3 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A \,d^{2}}{b \sqrt {\left (b e -c d \right ) c}}+\frac {e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B \,d^{3}}{b \sqrt {\left (b e -c d \right ) c}}+\frac {e^{4} b \sqrt {e x +d}\, A}{c^{2} \left (c e x +b e \right )}-\frac {e^{4} b^{2} \sqrt {e x +d}\, B}{c^{3} \left (c e x +b e \right )}-\frac {3 e^{4} b \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A}{c^{2} \sqrt {\left (b e -c d \right ) c}}\)	\(823\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^2*(1/c^3*(1/3*B*c*(e*x+d)^(3/2)+A*c*e*(e*x+d)^(1/2)-2*B*b*e*(e*x+d)^(1/2)+3*B*c*d*(e*x+d)^(1/2))-d^3/e^2/b
^3*(1/2*A*b*(e*x+d)^(1/2)/x+1/2*(7*A*b*e-4*A*c*d+2*B*b*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))-1/c^3/e^2/b^
3*((-1/2*A*b^4*c*e^4+3/2*A*b^3*c^2*d*e^3-3/2*A*b^2*c^3*d^2*e^2+1/2*A*c^4*d^3*e*b+1/2*b^5*B*e^4-3/2*B*b^4*c*d*e
^3+3/2*B*b^3*c^2*d^2*e^2-1/2*B*b^2*c^3*d^3*e)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(3*A*b^4*c*e^4-5*A*b^3*c^2
*d*e^3-3*A*b^2*c^3*d^2*e^2+9*A*b*c^4*d^3*e-4*A*c^5*d^4-5*B*b^5*e^4+13*B*b^4*c*d*e^3-9*B*b^3*c^2*d^2*e^2-B*b^2*
c^3*d^3*e+2*B*b*c^4*d^4)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 149.64, size = 2145, normalized size = 7.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[-1/6*(3*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + ((5*B*b^4*c - 3*A*b^3*c^2)*x^2 + (
5*B*b^5 - 3*A*b^4*c)*x)*e^3 - 2*((4*B*b^3*c^2 - A*b^2*c^3)*d*x^2 + (4*B*b^4*c - A*b^3*c^2)*d*x)*e^2 + ((B*b^2*
c^3 + 5*A*b*c^4)*d^2*x^2 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*x)*e)*sqrt((c*d - b*e)/c)*log((2*c*d - 2*sqrt(x*e + d
)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) - 3*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^
4)*d^3*x + 7*(A*b*c^4*d^2*x^2 + A*b^2*c^3*d^2*x)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(
3*A*b^2*c^3*d^3 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*x*e - (2*B*b^3*c^2*x^3 - 2*(
5*B*b^4*c - 3*A*b^3*c^2)*x^2 - 3*(5*B*b^5 - 3*A*b^4*c)*x)*e^3 - (20*B*b^3*c^2*d*x^2 + (29*B*b^4*c - 9*A*b^3*c^
2)*d*x)*e^2)*sqrt(x*e + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/6*(6*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 -
2*A*b*c^4)*d^3*x + ((5*B*b^4*c - 3*A*b^3*c^2)*x^2 + (5*B*b^5 - 3*A*b^4*c)*x)*e^3 - 2*((4*B*b^3*c^2 - A*b^2*c^3
)*d*x^2 + (4*B*b^4*c - A*b^3*c^2)*d*x)*e^2 + ((B*b^2*c^3 + 5*A*b*c^4)*d^2*x^2 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*
x)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(x*e + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*(2*(B*b*c^4 - 2*A*c^5
)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + 7*(A*b*c^4*d^2*x^2 + A*b^2*c^3*d^2*x)*e)*sqrt(d)*log((x*e - 2*sq
rt(x*e + d)*sqrt(d) + 2*d)/x) - 2*(3*A*b^2*c^3*d^3 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + 9*(B*b^3*c^2 - A*b^2*c^
3)*d^2*x*e - (2*B*b^3*c^2*x^3 - 2*(5*B*b^4*c - 3*A*b^3*c^2)*x^2 - 3*(5*B*b^5 - 3*A*b^4*c)*x)*e^3 - (20*B*b^3*c
^2*d*x^2 + (29*B*b^4*c - 9*A*b^3*c^2)*d*x)*e^2)*sqrt(x*e + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/6*(6*(2*(B*b*c^4 -
2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + 7*(A*b*c^4*d^2*x^2 + A*b^2*c^3*d^2*x)*e)*sqrt(-d)*arctan
(sqrt(x*e + d)*sqrt(-d)/d) - 3*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + ((5*B*b^4*c
- 3*A*b^3*c^2)*x^2 + (5*B*b^5 - 3*A*b^4*c)*x)*e^3 - 2*((4*B*b^3*c^2 - A*b^2*c^3)*d*x^2 + (4*B*b^4*c - A*b^3*c^
2)*d*x)*e^2 + ((B*b^2*c^3 + 5*A*b*c^4)*d^2*x^2 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*x)*e)*sqrt((c*d - b*e)/c)*log((
2*c*d - 2*sqrt(x*e + d)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) - 2*(3*A*b^2*c^3*d^3 - 3*(B*b^2*c^3 -
2*A*b*c^4)*d^3*x + 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*x*e - (2*B*b^3*c^2*x^3 - 2*(5*B*b^4*c - 3*A*b^3*c^2)*x^2 - 3*
(5*B*b^5 - 3*A*b^4*c)*x)*e^3 - (20*B*b^3*c^2*d*x^2 + (29*B*b^4*c - 9*A*b^3*c^2)*d*x)*e^2)*sqrt(x*e + d))/(b^3*
c^4*x^2 + b^4*c^3*x), 1/3*(3*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + ((5*B*b^4*c -
3*A*b^3*c^2)*x^2 + (5*B*b^5 - 3*A*b^4*c)*x)*e^3 - 2*((4*B*b^3*c^2 - A*b^2*c^3)*d*x^2 + (4*B*b^4*c - A*b^3*c^2)
*d*x)*e^2 + ((B*b^2*c^3 + 5*A*b*c^4)*d^2*x^2 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*x)*e)*sqrt(-(c*d - b*e)/c)*arctan
(-sqrt(x*e + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*(2*(B*b*c^4 - 2*A*c^5)*d^3*x^2 + 2*(B*b^2*c^3 - 2*A*b*
c^4)*d^3*x + 7*(A*b*c^4*d^2*x^2 + A*b^2*c^3*d^2*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) - (3*A*b^2*c^3
*d^3 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^3*x + 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*x*e - (2*B*b^3*c^2*x^3 - 2*(5*B*b^4*c -
3*A*b^3*c^2)*x^2 - 3*(5*B*b^5 - 3*A*b^4*c)*x)*e^3 - (20*B*b^3*c^2*d*x^2 + (29*B*b^4*c - 9*A*b^3*c^2)*d*x)*e^2
)*sqrt(x*e + d))/(b^3*c^4*x^2 + b^4*c^3*x)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (286) = 572\).
time = 0.94, size = 639, normalized size = 2.19 \begin {gather*} \frac {{\left (2 \, B b d^{4} - 4 \, A c d^{4} + 7 \, A b d^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{4} d^{4} - 4 \, A c^{5} d^{4} - B b^{2} c^{3} d^{3} e + 9 \, A b c^{4} d^{3} e - 9 \, B b^{3} c^{2} d^{2} e^{2} - 3 \, A b^{2} c^{3} d^{2} e^{2} + 13 \, B b^{4} c d e^{3} - 5 \, A b^{3} c^{2} d e^{3} - 5 \, B b^{5} e^{4} + 3 \, A b^{4} c e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c^{4} e^{2} + 9 \, \sqrt {x e + d} B c^{4} d e^{2} - 6 \, \sqrt {x e + d} B b c^{3} e^{3} + 3 \, \sqrt {x e + d} A c^{4} e^{3}\right )}}{3 \, c^{6}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{3} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{4} d^{3} e - \sqrt {x e + d} B b c^{3} d^{4} e + 2 \, \sqrt {x e + d} A c^{4} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{2} d^{2} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{3} d^{2} e^{2} + 3 \, \sqrt {x e + d} B b^{2} c^{2} d^{3} e^{2} - 4 \, \sqrt {x e + d} A b c^{3} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c d e^{3} - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d e^{3} - 3 \, \sqrt {x e + d} B b^{3} c d^{2} e^{3} + 3 \, \sqrt {x e + d} A b^{2} c^{2} d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c e^{4} + \sqrt {x e + d} B b^{4} d e^{4} - \sqrt {x e + d} A b^{3} c d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(2*B*b*d^4 - 4*A*c*d^4 + 7*A*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^4*d^4 - 4*A*c^5
*d^4 - B*b^2*c^3*d^3*e + 9*A*b*c^4*d^3*e - 9*B*b^3*c^2*d^2*e^2 - 3*A*b^2*c^3*d^2*e^2 + 13*B*b^4*c*d*e^3 - 5*A*
b^3*c^2*d*e^3 - 5*B*b^5*e^4 + 3*A*b^4*c*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e
)*b^3*c^3) + 2/3*((x*e + d)^(3/2)*B*c^4*e^2 + 9*sqrt(x*e + d)*B*c^4*d*e^2 - 6*sqrt(x*e + d)*B*b*c^3*e^3 + 3*sq
rt(x*e + d)*A*c^4*e^3)/c^6 + ((x*e + d)^(3/2)*B*b*c^3*d^3*e - 2*(x*e + d)^(3/2)*A*c^4*d^3*e - sqrt(x*e + d)*B*
b*c^3*d^4*e + 2*sqrt(x*e + d)*A*c^4*d^4*e - 3*(x*e + d)^(3/2)*B*b^2*c^2*d^2*e^2 + 3*(x*e + d)^(3/2)*A*b*c^3*d^
2*e^2 + 3*sqrt(x*e + d)*B*b^2*c^2*d^3*e^2 - 4*sqrt(x*e + d)*A*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*B*b^3*c*d*e^3
- 3*(x*e + d)^(3/2)*A*b^2*c^2*d*e^3 - 3*sqrt(x*e + d)*B*b^3*c*d^2*e^3 + 3*sqrt(x*e + d)*A*b^2*c^2*d^2*e^3 - (x
*e + d)^(3/2)*B*b^4*e^4 + (x*e + d)^(3/2)*A*b^3*c*e^4 + sqrt(x*e + d)*B*b^4*d*e^4 - sqrt(x*e + d)*A*b^3*c*d*e^
4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^3)

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Mupad [B]
time = 3.29, size = 2500, normalized size = 8.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x)

[Out]

(((d + e*x)^(3/2)*(B*b^4*e^4 - A*b^3*c*e^4 + 2*A*c^4*d^3*e - 3*A*b*c^3*d^2*e^2 + 3*A*b^2*c^2*d*e^3 + 3*B*b^2*c
^2*d^2*e^2 - B*b*c^3*d^3*e - 3*B*b^3*c*d*e^3))/b^2 - ((d + e*x)^(1/2)*(2*A*c^4*d^4*e + B*b^4*d*e^4 - 4*A*b*c^3
*d^3*e^2 - 3*B*b^3*c*d^2*e^3 + 3*A*b^2*c^2*d^2*e^3 + 3*B*b^2*c^2*d^3*e^2 - A*b^3*c*d*e^4 - B*b*c^3*d^4*e))/b^2
)/((2*c^4*d - b*c^3*e)*(d + e*x) - c^4*(d + e*x)^2 - c^4*d^2 + b*c^3*d*e) + ((2*A*e^3 - 2*B*d*e^2)/c^2 + (2*B*
e^2*(4*c^2*d - 2*b*c*e))/c^4)*(d + e*x)^(1/2) + (atan(((((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6
*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B
*b^9*c^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*
d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*B^2*b^10*
e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b
^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^
3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B
^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 -
32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c
^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/
2)*(7*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3) - (((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4
*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5
*d^2*e^5)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*
b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) - (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9
*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d
^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B
^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c
^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b
*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e
^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/2)*(7*A*
b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3))/((2*(32*A^3*c^10*d^11*e^3 + 50*B^3*b^10*d^4*e^10 + 262*A^3*b^2*c^8*d^9*e
^5 + 141*A^3*b^3*c^7*d^8*e^6 - 658*A^3*b^4*c^6*d^7*e^7 + 413*A^3*b^5*c^5*d^6*e^8 + 169*A^3*b^6*c^4*d^5*e^9 - 2
46*A^3*b^7*c^3*d^4*e^10 + 63*A^3*b^8*c^2*d^3*e^11 - 4*B^3*b^3*c^7*d^11*e^3 - 34*B^3*b^4*c^6*d^10*e^4 + 88*B^3*
b^5*c^5*d^9*e^5 + 90*B^3*b^6*c^4*d^8*e^6 - 448*B^3*b^7*c^3*d^7*e^7 + 518*B^3*b^8*c^2*d^6*e^8 + 175*A*B^2*b^10*
d^3*e^11 - 176*A^3*b*c^9*d^10*e^4 - 260*B^3*b^9*c*d^5*e^9 + 24*A*B^2*b^2*c^8*d^11*e^3 + 92*A*B^2*b^3*c^7*d^10*
e^4 - 605*A*B^2*b^4*c^6*d^9*e^5 + 594*A*B^2*b^5*c^5*d^8*e^6 + 1113*A*B^2*b^6*c^4*d^7*e^7 - 2912*A*B^2*b^7*c^3*
d^6*e^8 + 2589*A*B^2*b^8*c^2*d^5*e^9 + 40*A^2*B*b^2*c^8*d^10*e^4 + 727*A^2*B*b^3*c^7*d^9*e^5 - 2133*A^2*B*b^4*
c^6*d^8*e^6 + 1953*A^2*B*b^5*c^5*d^7*e^7 + 287*A^2*B*b^6*c^4*d^6*e^8 - 1650*A^2*B*b^7*c^3*d^5*e^9 + 1034*A^2*B
*b^8*c^2*d^4*e^10 - 1070*A*B^2*b^9*c*d^4*e^10 - 48*A^2*B*b*c^9*d^11*e^3 - 210*A^2*B*b^9*c*d^3*e^11))/(b^6*c^5)
+ (((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d
^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^
6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*
c*d + 2*B*b*d))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 15
4*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*
c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5
+ 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b
*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*
B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^
4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) + (((((12*
A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*
B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2
)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*...

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