3.4.69 \(\int \frac {(d+e x)^{9/2}}{(b x+c x^2)^2} \, dx\) [369]

Optimal. Leaf size=251 \[ \frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} (4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 838, 840, 1180, 214} \begin {gather*} -\frac {(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac {e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{5/2} (2 c d-b e)}{b^2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^3) + (e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*
e^2)*(d + e*x)^(3/2))/(3*b^2*c^2) + (e*(2*c*d - b*e)*(d + e*x)^(5/2))/(b^2*c) - ((d + e*x)^(7/2)*(b*d + (2*c*d
- b*e)*x))/(b^2*(b*x + c*x^2)) + (d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^
(7/2)*(4*c*d + 5*b*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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

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 {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (4 c d-9 b e)-\frac {5}{2} e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{b^2}\\ &=\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} c d^2 (4 c d-9 b e)-\frac {1}{2} e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c^2 d^3 (4 c d-9 b e)-\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} c^3 d^4 (4 c d-9 b e)+\frac {1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} c^3 d^4 e (4 c d-9 b e)-\frac {1}{2} d e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right )+\frac {1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\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 (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\left (c d^4 (4 c d-9 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)^4 (4 c d+5 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 c^3}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} (4 c d+5 b e) \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 = 0.56, size = 203, normalized size = 0.81 \begin {gather*} \frac {-\frac {b \sqrt {d+e x} \left (6 c^4 d^4 x+15 b^4 e^4 x+3 b c^3 d^3 (d-4 e x)+2 b^3 c e^3 x (-19 d+5 e x)-2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )\right )}{c^3 x (b+c x)}+\frac {3 (-c d+b e)^{7/2} (4 c d+5 b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{7/2}}+3 d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*Sqrt[d + e*x]*(6*c^4*d^4*x + 15*b^4*e^4*x + 3*b*c^3*d^3*(d - 4*e*x) + 2*b^3*c*e^3*x*(-19*d + 5*e*x) - 2*
b^2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2*x^2)))/(c^3*x*(b + c*x))) + (3*(-(c*d) + b*e)^(7/2)*(4*c*d + 5*b*e)*Arc
Tan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(7/2) + 3*d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqr
t[d]])/(3*b^3)

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Maple [A]
time = 0.52, size = 290, normalized size = 1.16

method	result	size
derivativedivides	\(2 e^{3} \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}}{c^{3}}-\frac {d^{4} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (9 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\frac {\left (-\frac {1}{2} b^{5} e^{5}+2 b^{4} c d \,e^{4}-3 b^{3} c^{2} d^{2} e^{3}+2 b^{2} c^{3} d^{3} e^{2}-\frac {1}{2} b \,c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 b^{5} e^{5}-16 b^{4} c d \,e^{4}+14 b^{3} c^{2} d^{2} e^{3}+4 b^{2} c^{3} d^{3} e^{2}-11 b \,c^{4} d^{4} e +4 c^{5} d^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} b^{3} e^{3}}\right )\)	\(290\)
default	\(2 e^{3} \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}}{c^{3}}-\frac {d^{4} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (9 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\frac {\left (-\frac {1}{2} b^{5} e^{5}+2 b^{4} c d \,e^{4}-3 b^{3} c^{2} d^{2} e^{3}+2 b^{2} c^{3} d^{3} e^{2}-\frac {1}{2} b \,c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 b^{5} e^{5}-16 b^{4} c d \,e^{4}+14 b^{3} c^{2} d^{2} e^{3}+4 b^{2} c^{3} d^{3} e^{2}-11 b \,c^{4} d^{4} e +4 c^{5} d^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} b^{3} e^{3}}\right )\)	\(290\)
risch	\(-\frac {d^{4} \sqrt {e x +d}}{b^{2} x}-\frac {e^{5} b^{2} \sqrt {e x +d}}{c^{3} \left (c e x +b e \right )}+\frac {4 e^{4} b \sqrt {e x +d}\, d}{c^{2} \left (c e x +b e \right )}-\frac {6 e^{3} \sqrt {e x +d}\, d^{2}}{c \left (c e x +b e \right )}+\frac {4 e^{2} \sqrt {e x +d}\, d^{3}}{b \left (c e x +b e \right )}-\frac {e c \sqrt {e x +d}\, d^{4}}{b^{2} \left (c e x +b e \right )}+\frac {5 e^{5} b^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {16 e^{4} b \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d}{c^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {14 e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2}}{c \sqrt {\left (b e -c d \right ) c}}+\frac {4 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{3}}{b \sqrt {\left (b e -c d \right ) c}}-\frac {11 e c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{4}}{b^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {4 c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{5}}{b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {2 e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2}}-\frac {4 e^{4} b \sqrt {e x +d}}{c^{3}}+\frac {8 e^{3} d \sqrt {e x +d}}{c^{2}}-\frac {9 e \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 d^{\frac {9}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{3}}\)	\(515\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*(-1/c^3*(-1/3*c*(e*x+d)^(3/2)+2*b*e*(e*x+d)^(1/2)-4*c*d*(e*x+d)^(1/2))-d^4/b^3/e^3*(1/2*b*(e*x+d)^(1/2)/
x+1/2*(9*b*e-4*c*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+1/c^3/b^3/e^3*((-1/2*b^5*e^5+2*b^4*c*d*e^4-3*b^3*c
^2*d^2*e^3+2*b^2*c^3*d^3*e^2-1/2*b*c^4*d^4*e)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(5*b^5*e^5-16*b^4*c*d*e^4+
14*b^3*c^2*d^2*e^3+4*b^2*c^3*d^3*e^2-11*b*c^4*d^4*e+4*c^5*d^5)/((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((e*x+d)^(9/2)/(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 = 3.68, size = 1592, normalized size = 6.34 \begin {gather*} \left [-\frac {3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 5 \, {\left (b^{4} c x^{2} + b^{5} x\right )} e^{4} + 11 \, {\left (b^{3} c^{2} d x^{2} + b^{4} c d x\right )} e^{3} - 3 \, {\left (b^{2} c^{3} d^{2} x^{2} + b^{3} c^{2} d^{2} x\right )} e^{2} - 7 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + 3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 9 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (6 \, b c^{4} d^{4} x - 12 \, b^{2} c^{3} d^{3} x e + 3 \, b^{2} c^{3} d^{4} + 18 \, b^{3} c^{2} d^{2} x e^{2} - {\left (2 \, b^{3} c^{2} x^{3} - 10 \, b^{4} c x^{2} - 15 \, b^{5} x\right )} e^{4} - 2 \, {\left (13 \, b^{3} c^{2} d x^{2} + 19 \, b^{4} c d x\right )} e^{3}\right )} \sqrt {x e + d}}{6 \, {\left (b^{3} c^{4} x^{2} + b^{4} c^{3} x\right )}}, -\frac {6 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 5 \, {\left (b^{4} c x^{2} + b^{5} x\right )} e^{4} + 11 \, {\left (b^{3} c^{2} d x^{2} + b^{4} c d x\right )} e^{3} - 3 \, {\left (b^{2} c^{3} d^{2} x^{2} + b^{3} c^{2} d^{2} x\right )} e^{2} - 7 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + 3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 9 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (6 \, b c^{4} d^{4} x - 12 \, b^{2} c^{3} d^{3} x e + 3 \, b^{2} c^{3} d^{4} + 18 \, b^{3} c^{2} d^{2} x e^{2} - {\left (2 \, b^{3} c^{2} x^{3} - 10 \, b^{4} c x^{2} - 15 \, b^{5} x\right )} e^{4} - 2 \, {\left (13 \, b^{3} c^{2} d x^{2} + 19 \, b^{4} c d x\right )} e^{3}\right )} \sqrt {x e + d}}{6 \, {\left (b^{3} c^{4} x^{2} + b^{4} c^{3} x\right )}}, -\frac {6 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 9 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + 3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 5 \, {\left (b^{4} c x^{2} + b^{5} x\right )} e^{4} + 11 \, {\left (b^{3} c^{2} d x^{2} + b^{4} c d x\right )} e^{3} - 3 \, {\left (b^{2} c^{3} d^{2} x^{2} + b^{3} c^{2} d^{2} x\right )} e^{2} - 7 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (6 \, b c^{4} d^{4} x - 12 \, b^{2} c^{3} d^{3} x e + 3 \, b^{2} c^{3} d^{4} + 18 \, b^{3} c^{2} d^{2} x e^{2} - {\left (2 \, b^{3} c^{2} x^{3} - 10 \, b^{4} c x^{2} - 15 \, b^{5} x\right )} e^{4} - 2 \, {\left (13 \, b^{3} c^{2} d x^{2} + 19 \, b^{4} c d x\right )} e^{3}\right )} \sqrt {x e + d}}{6 \, {\left (b^{3} c^{4} x^{2} + b^{4} c^{3} x\right )}}, -\frac {3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 5 \, {\left (b^{4} c x^{2} + b^{5} x\right )} e^{4} + 11 \, {\left (b^{3} c^{2} d x^{2} + b^{4} c d x\right )} e^{3} - 3 \, {\left (b^{2} c^{3} d^{2} x^{2} + b^{3} c^{2} d^{2} x\right )} e^{2} - 7 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + 3 \, {\left (4 \, c^{5} d^{4} x^{2} + 4 \, b c^{4} d^{4} x - 9 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (6 \, b c^{4} d^{4} x - 12 \, b^{2} c^{3} d^{3} x e + 3 \, b^{2} c^{3} d^{4} + 18 \, b^{3} c^{2} d^{2} x e^{2} - {\left (2 \, b^{3} c^{2} x^{3} - 10 \, b^{4} c x^{2} - 15 \, b^{5} x\right )} e^{4} - 2 \, {\left (13 \, b^{3} c^{2} d x^{2} + 19 \, b^{4} c d x\right )} e^{3}\right )} \sqrt {x e + d}}{3 \, {\left (b^{3} c^{4} x^{2} + b^{4} c^{3} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 5*(b^4*c*x^2 + b^5*x)*e^4 + 11*(b^3*c^2*d*x^2 + b^4*c*d*x)*e^3 - 3*(
b^2*c^3*d^2*x^2 + b^3*c^2*d^2*x)*e^2 - 7*(b*c^4*d^3*x^2 + b^2*c^3*d^3*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*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 9*(b*c^4*d
^3*x^2 + b^2*c^3*d^3*x)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(6*b*c^4*d^4*x - 12*b^2*c^
3*d^3*x*e + 3*b^2*c^3*d^4 + 18*b^3*c^2*d^2*x*e^2 - (2*b^3*c^2*x^3 - 10*b^4*c*x^2 - 15*b^5*x)*e^4 - 2*(13*b^3*c
^2*d*x^2 + 19*b^4*c*d*x)*e^3)*sqrt(x*e + d))/(b^3*c^4*x^2 + b^4*c^3*x), -1/6*(6*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x
- 5*(b^4*c*x^2 + b^5*x)*e^4 + 11*(b^3*c^2*d*x^2 + b^4*c*d*x)*e^3 - 3*(b^2*c^3*d^2*x^2 + b^3*c^2*d^2*x)*e^2 -
7*(b*c^4*d^3*x^2 + b^2*c^3*d^3*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*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 9*(b*c^4*d^3*x^2 + b^2*c^3*d^3*x)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e
+ d)*sqrt(d) + 2*d)/x) + 2*(6*b*c^4*d^4*x - 12*b^2*c^3*d^3*x*e + 3*b^2*c^3*d^4 + 18*b^3*c^2*d^2*x*e^2 - (2*b^3
*c^2*x^3 - 10*b^4*c*x^2 - 15*b^5*x)*e^4 - 2*(13*b^3*c^2*d*x^2 + 19*b^4*c*d*x)*e^3)*sqrt(x*e + d))/(b^3*c^4*x^2
+ b^4*c^3*x), -1/6*(6*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 9*(b*c^4*d^3*x^2 + b^2*c^3*d^3*x)*e)*sqrt(-d)*arctan(s
qrt(x*e + d)*sqrt(-d)/d) + 3*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 5*(b^4*c*x^2 + b^5*x)*e^4 + 11*(b^3*c^2*d*x^2 +
b^4*c*d*x)*e^3 - 3*(b^2*c^3*d^2*x^2 + b^3*c^2*d^2*x)*e^2 - 7*(b*c^4*d^3*x^2 + b^2*c^3*d^3*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*(6*b*c^4*d^4*x - 12*b^2
*c^3*d^3*x*e + 3*b^2*c^3*d^4 + 18*b^3*c^2*d^2*x*e^2 - (2*b^3*c^2*x^3 - 10*b^4*c*x^2 - 15*b^5*x)*e^4 - 2*(13*b^
3*c^2*d*x^2 + 19*b^4*c*d*x)*e^3)*sqrt(x*e + d))/(b^3*c^4*x^2 + b^4*c^3*x), -1/3*(3*(4*c^5*d^4*x^2 + 4*b*c^4*d^
4*x - 5*(b^4*c*x^2 + b^5*x)*e^4 + 11*(b^3*c^2*d*x^2 + b^4*c*d*x)*e^3 - 3*(b^2*c^3*d^2*x^2 + b^3*c^2*d^2*x)*e^2
- 7*(b*c^4*d^3*x^2 + b^2*c^3*d^3*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*(4*c^5*d^4*x^2 + 4*b*c^4*d^4*x - 9*(b*c^4*d^3*x^2 + b^2*c^3*d^3*x)*e)*sqrt(-d)*arctan(sqrt(x*e +
d)*sqrt(-d)/d) + (6*b*c^4*d^4*x - 12*b^2*c^3*d^3*x*e + 3*b^2*c^3*d^4 + 18*b^3*c^2*d^2*x*e^2 - (2*b^3*c^2*x^3 -
10*b^4*c*x^2 - 15*b^5*x)*e^4 - 2*(13*b^3*c^2*d*x^2 + 19*b^4*c*d*x)*e^3)*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((e*x+d)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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Giac [A]
time = 1.30, size = 436, normalized size = 1.74 \begin {gather*} -\frac {{\left (4 \, c d^{5} - 9 \, b d^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{5} d^{5} - 11 \, b c^{4} d^{4} e + 4 \, b^{2} c^{3} d^{3} e^{2} + 14 \, b^{3} c^{2} d^{2} e^{3} - 16 \, b^{4} c d e^{4} + 5 \, b^{5} e^{5}\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}} c^{4} e^{3} + 12 \, \sqrt {x e + d} c^{4} d e^{3} - 6 \, \sqrt {x e + d} b c^{3} e^{4}\right )}}{3 \, c^{6}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e - 2 \, \sqrt {x e + d} c^{4} d^{5} e - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{2} + 5 \, \sqrt {x e + d} b c^{3} d^{4} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{3} - 6 \, \sqrt {x e + d} b^{2} c^{2} d^{3} e^{3} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d e^{4} + 4 \, \sqrt {x e + d} b^{3} c d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{5} - \sqrt {x e + d} b^{4} d e^{5}}{{\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((e*x+d)^(9/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

-(4*c*d^5 - 9*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^5*d^5 - 11*b*c^4*d^4*e + 4*b^2*c^3
*d^3*e^2 + 14*b^3*c^2*d^2*e^3 - 16*b^4*c*d*e^4 + 5*b^5*e^5)*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)*c^4*e^3 + 12*sqrt(x*e + d)*c^4*d*e^3 - 6*sqrt(x*e + d)*b*c^3*
e^4)/c^6 - (2*(x*e + d)^(3/2)*c^4*d^4*e - 2*sqrt(x*e + d)*c^4*d^5*e - 4*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 5*sqrt
(x*e + d)*b*c^3*d^4*e^2 + 6*(x*e + d)^(3/2)*b^2*c^2*d^2*e^3 - 6*sqrt(x*e + d)*b^2*c^2*d^3*e^3 - 4*(x*e + d)^(3
/2)*b^3*c*d*e^4 + 4*sqrt(x*e + d)*b^3*c*d^2*e^4 + (x*e + d)^(3/2)*b^4*e^5 - sqrt(x*e + d)*b^4*d*e^5)/(((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 = 1.02, size = 2500, normalized size = 9.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(3/2)*(b^4*e^5 + 2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 6*b^2*c^2*d^2*e^3 - 4*b^3*c*d*e^4))/b^2 - ((d + e
*x)^(1/2)*(b^4*d*e^5 + 2*c^4*d^5*e - 5*b*c^3*d^4*e^2 - 4*b^3*c*d^2*e^4 + 6*b^2*c^2*d^3*e^3))/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*e^3*(d + e*x)^(3/2))/(3*c^2) + (2*e^3*(4*c^2*
d - 2*b*c*e)*(d + e*x)^(1/2))/c^4 - (atan(((((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 5
6*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(
1/2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*
c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c
^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(
9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3) - (((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*
b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/
2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^
10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5
*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(9*
b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3))/((((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^
8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)
*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10
*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d
^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(9*b*
e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(225*b^10*d^4*e^13 + 32*c^10*d^14*e^3 - 224*b*c^9*d^13*e^4 - 1540*b^9*c*d
^5*e^12 + 326*b^2*c^8*d^12*e^5 + 956*b^3*c^7*d^11*e^6 - 3430*b^4*c^6*d^10*e^7 + 3048*b^5*c^5*d^9*e^8 + 1659*b^
6*c^4*d^8*e^9 - 5256*b^7*c^3*d^7*e^10 + 4204*b^8*c^2*d^6*e^11))/(b^6*c^5) + (((((20*b^10*c^4*d*e^7 + 8*b^6*c^8
*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c
^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(d
+ e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5
- 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10
- 160*b^9*c*d*e^11))/(b^4*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3)))*(9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/b^3 + (
atan((((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 2
34*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408
*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5) + ((-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e +
4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)
/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b
^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d)*1i)/(2*b^3*c^7) + ((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*
b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*
e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11
))/(b^4*c^5) - ((-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^
7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e
- c*d)^7)^(1/2)*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d)*1i)/(2*b^3*c^7))/(
(2*(225*b^10*d^4*e^13 + 32*c^10*d^14*e^3 - 224*b*c^9*d^13*e^4 - 1540*b^9*c*d^5*e^12 + 326*b^2*c^8*d^12*e^5 + 9
56*b^3*c^7*d^11*e^6 - 3430*b^4*c^6*d^10*e^7 + 3048*b^5*c^5*d^9*e^8 + 1659*b^6*c^4*d^8*e^9 - 5256*b^7*c^3*d^7*e
^10 + 4204*b^8*c^2*d^6*e^11))/(b^6*c^5) - ((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c
^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^
5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5) + (
(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b
^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^7)^(1/2)
*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d))/(2*b^3*c^7) + ((-c^7*(b*e - c*d)^
7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32...

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