3.4.81 \(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^3} \, dx\) [381]

Optimal. Leaf size=246 \[ -\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}} \]

[Out]

-3/4*(b^2*e^2-12*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(1/2)+3/4*(5*b^2*e^2-20*b*c*d*e+16*c
^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))*c^(1/2)/b^5/(-b*e+c*d)^(1/2)-1/2*(b*d+(-b*e+2*c*d)*x)*
(e*x+d)^(1/2)/b^2/(c*x^2+b*x)^2+1/4*(b*(-7*b*e+12*c*d)*(-b*e+c*d)+12*c*(-b*e+c*d)*(-b*e+2*c*d)*x)*(e*x+d)^(1/2
)/b^4/(-b*e+c*d)/(c*x^2+b*x)

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 836, 840, 1180, 214} \begin {gather*} \frac {\sqrt {d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left (b x+c x^2\right ) (c d-b e)}-\frac {\sqrt {d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(b*x + c*x^2)^3,x]

[Out]

-1/2*(Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b*(12*c*d - 7*b*e)*(c*d -
b*e) + 12*c*(c*d - b*e)*(2*c*d - b*e)*x))/(4*b^4*(c*d - b*e)*(b*x + c*x^2)) - (3*(16*c^2*d^2 - 12*b*c*d*e + b
^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*Sqrt[d]) + (3*Sqrt[c]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*Arc
Tanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c*d - b*e])

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 836

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

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)^{3/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} d (12 c d-7 b e)+\frac {5}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} d (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b e) (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-3 c d^2 e (c d-b e) (2 c d-b e)+\frac {3}{4} d e (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b e) (2 c d-b e) 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^4 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2-12 b c d e+b^2 e^2\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 )}{4 b^5}-\frac {\left (3 c \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\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 )}{4 b^5}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.98, size = 197, normalized size = 0.80 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (24 c^3 d x^3+b^2 c x (8 d-19 e x)-12 b c^2 x^2 (-3 d+e x)-b^3 (2 d+5 e x)\right )}{x^2 (b+c x)^2}-\frac {3 \sqrt {c} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {-c d+b e}}-\frac {3 \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^3,x]

[Out]

((b*Sqrt[d + e*x]*(24*c^3*d*x^3 + b^2*c*x*(8*d - 19*e*x) - 12*b*c^2*x^2*(-3*d + e*x) - b^3*(2*d + 5*e*x)))/(x^
2*(b + c*x)^2) - (3*Sqrt[c]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) +
b*e]])/Sqrt[-(c*d) + b*e] - (3*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d])/(
4*b^5)

________________________________________________________________________________________

Maple [A]
time = 0.50, size = 257, normalized size = 1.04

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {c \left (\frac {\left (\frac {7}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (3 b^{2} e^{2}-7 b c d e +4 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (5 b^{2} e^{2}-20 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {\frac {\frac {b e \left (5 b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {3}{2} b c \,d^{2} e -\frac {3}{8} b^{2} d \,e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}-12 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} e^{5}}\right )\)	\(257\)
default	\(2 e^{5} \left (-\frac {c \left (\frac {\left (\frac {7}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (3 b^{2} e^{2}-7 b c d e +4 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (5 b^{2} e^{2}-20 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {\frac {\frac {b e \left (5 b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {3}{2} b c \,d^{2} e -\frac {3}{8} b^{2} d \,e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}-12 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} e^{5}}\right )\)	\(257\)
risch	\(-\frac {\sqrt {e x +d}\, \left (5 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}-\frac {7 e^{2} c^{2} \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} \left (c e x +b e \right )^{2}}+\frac {3 e \,c^{3} \left (e x +d \right )^{\frac {3}{2}} d}{b^{4} \left (c e x +b e \right )^{2}}-\frac {9 e^{3} c \sqrt {e x +d}}{4 b^{2} \left (c e x +b e \right )^{2}}+\frac {21 e^{2} c^{2} \sqrt {e x +d}\, d}{4 b^{3} \left (c e x +b e \right )^{2}}-\frac {3 e \,c^{3} \sqrt {e x +d}\, d^{2}}{b^{4} \left (c e x +b e \right )^{2}}-\frac {15 e^{2} c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {15 e \,c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d}{b^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {12 c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2}}{b^{5} \sqrt {\left (b e -c d \right ) c}}-\frac {3 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} \sqrt {d}}+\frac {9 e \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4}}-\frac {12 d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5}}\)	\(371\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(-c/b^5/e^5*(((7/8*b^2*e^2*c-3/2*d*b*e*c^2)*(e*x+d)^(3/2)+3/8*b*e*(3*b^2*e^2-7*b*c*d*e+4*c^2*d^2)*(e*x+d
)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+3/8*(5*b^2*e^2-20*b*c*d*e+16*c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2
)/((b*e-c*d)*c)^(1/2)))-1/b^5/e^5*((1/8*b*e*(5*b*e-12*c*d)*(e*x+d)^(3/2)+(3/2*b*c*d^2*e-3/8*b^2*d*e^2)*(e*x+d)
^(1/2))/e^2/x^2+3/8*(b^2*e^2-12*b*c*d*e+16*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))))

________________________________________________________________________________________

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)^(3/2)/(c*x^2+b*x)^3,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

________________________________________________________________________________________

Fricas [A]
time = 2.65, size = 1752, normalized size = 7.12 \begin {gather*} \left [\frac {3 \, {\left (16 \, c^{4} d^{3} x^{4} + 32 \, b c^{3} d^{3} x^{3} + 16 \, b^{2} c^{2} d^{3} x^{2} + 5 \, {\left (b^{2} c^{2} d x^{4} + 2 \, b^{3} c d x^{3} + b^{4} d x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{3} + b^{3} c d^{2} x^{2}\right )} e\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d + 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} - {\left (12 \, b^{2} c^{2} d x^{3} + 19 \, b^{3} c d x^{2} + 5 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )}}, \frac {6 \, {\left (16 \, c^{4} d^{3} x^{4} + 32 \, b c^{3} d^{3} x^{3} + 16 \, b^{2} c^{2} d^{3} x^{2} + 5 \, {\left (b^{2} c^{2} d x^{4} + 2 \, b^{3} c d x^{3} + b^{4} d x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{3} + b^{3} c d^{2} x^{2}\right )} e\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} - {\left (12 \, b^{2} c^{2} d x^{3} + 19 \, b^{3} c d x^{2} + 5 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )}}, \frac {6 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + 3 \, {\left (16 \, c^{4} d^{3} x^{4} + 32 \, b c^{3} d^{3} x^{3} + 16 \, b^{2} c^{2} d^{3} x^{2} + 5 \, {\left (b^{2} c^{2} d x^{4} + 2 \, b^{3} c d x^{3} + b^{4} d x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{3} + b^{3} c d^{2} x^{2}\right )} e\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d + 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} - {\left (12 \, b^{2} c^{2} d x^{3} + 19 \, b^{3} c d x^{2} + 5 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )}}, \frac {3 \, {\left (16 \, c^{4} d^{3} x^{4} + 32 \, b c^{3} d^{3} x^{3} + 16 \, b^{2} c^{2} d^{3} x^{2} + 5 \, {\left (b^{2} c^{2} d x^{4} + 2 \, b^{3} c d x^{3} + b^{4} d x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{3} + b^{3} c d^{2} x^{2}\right )} e\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} - {\left (12 \, b^{2} c^{2} d x^{3} + 19 \, b^{3} c d x^{2} + 5 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*(16*c^4*d^3*x^4 + 32*b*c^3*d^3*x^3 + 16*b^2*c^2*d^3*x^2 + 5*(b^2*c^2*d*x^4 + 2*b^3*c*d*x^3 + b^4*d*x^2
)*e^2 - 20*(b*c^3*d^2*x^4 + 2*b^2*c^2*d^2*x^3 + b^3*c*d^2*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d + 2*(c*d - b*
e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2
*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)
*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2
*x - 2*b^4*d^2 - (12*b^2*c^2*d*x^3 + 19*b^3*c*d*x^2 + 5*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*
x^3 + b^7*d*x^2), 1/8*(6*(16*c^4*d^3*x^4 + 32*b*c^3*d^3*x^3 + 16*b^2*c^2*d^3*x^2 + 5*(b^2*c^2*d*x^4 + 2*b^3*c*
d*x^3 + b^4*d*x^2)*e^2 - 20*(b*c^3*d^2*x^4 + 2*b^2*c^2*d^2*x^3 + b^3*c*d^2*x^2)*e)*sqrt(-c/(c*d - b*e))*arctan
(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^
2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e
)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^
2*x - 2*b^4*d^2 - (12*b^2*c^2*d*x^3 + 19*b^3*c*d*x^2 + 5*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d
*x^3 + b^7*d*x^2), 1/8*(6*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3
+ b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d
) + 3*(16*c^4*d^3*x^4 + 32*b*c^3*d^3*x^3 + 16*b^2*c^2*d^3*x^2 + 5*(b^2*c^2*d*x^4 + 2*b^3*c*d*x^3 + b^4*d*x^2)*
e^2 - 20*(b*c^3*d^2*x^4 + 2*b^2*c^2*d^2*x^3 + b^3*c*d^2*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d + 2*(c*d - b*e)
*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^
3*c*d^2*x - 2*b^4*d^2 - (12*b^2*c^2*d*x^3 + 19*b^3*c*d*x^2 + 5*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*d*x^4 + 2*b
^6*c*d*x^3 + b^7*d*x^2), 1/4*(3*(16*c^4*d^3*x^4 + 32*b*c^3*d^3*x^3 + 16*b^2*c^2*d^3*x^2 + 5*(b^2*c^2*d*x^4 + 2
*b^3*c*d*x^3 + b^4*d*x^2)*e^2 - 20*(b*c^3*d^2*x^4 + 2*b^2*c^2*d^2*x^3 + b^3*c*d^2*x^2)*e)*sqrt(-c/(c*d - b*e))
*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3
+ 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d
*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2*x -
2*b^4*d^2 - (12*b^2*c^2*d*x^3 + 19*b^3*c*d*x^2 + 5*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 +
b^7*d*x^2)]

________________________________________________________________________________________

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)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.50, size = 392, normalized size = 1.59 \begin {gather*} -\frac {3 \, {\left (16 \, c^{3} d^{2} - 20 \, b c^{2} d e + 5 \, b^{2} c e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5}} + \frac {3 \, {\left (16 \, c^{2} d^{2} - 12 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 24 \, \sqrt {x e + d} c^{3} d^{4} e - 12 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{2} + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{2} - 108 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 48 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} - 19 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{3} + 46 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 27 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} + 3 \, \sqrt {x e + d} b^{3} d e^{4}}{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 )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^3*d^2 - 20*b*c^2*d*e + 5*b^2*c*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c
*e)*b^5) + 3/4*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*(x*
e + d)^(7/2)*c^3*d*e - 72*(x*e + d)^(5/2)*c^3*d^2*e + 72*(x*e + d)^(3/2)*c^3*d^3*e - 24*sqrt(x*e + d)*c^3*d^4*
e - 12*(x*e + d)^(7/2)*b*c^2*e^2 + 72*(x*e + d)^(5/2)*b*c^2*d*e^2 - 108*(x*e + d)^(3/2)*b*c^2*d^2*e^2 + 48*sqr
t(x*e + d)*b*c^2*d^3*e^2 - 19*(x*e + d)^(5/2)*b^2*c*e^3 + 46*(x*e + d)^(3/2)*b^2*c*d*e^3 - 27*sqrt(x*e + d)*b^
2*c*d^2*e^3 - 5*(x*e + d)^(3/2)*b^3*e^4 + 3*sqrt(x*e + d)*b^3*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)^2*b^4)

________________________________________________________________________________________

Mupad [B]
time = 0.79, size = 1880, normalized size = 7.64 \begin {gather*} \frac {\frac {3\,\sqrt {d+e\,x}\,\left (b^3\,d\,e^4-9\,b^2\,c\,d^2\,e^3+16\,b\,c^2\,d^3\,e^2-8\,c^3\,d^4\,e\right )}{4\,b^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (5\,b^3\,e^4-46\,b^2\,c\,d\,e^3+108\,b\,c^2\,d^2\,e^2-72\,c^3\,d^3\,e\right )}{4\,b^4}-\frac {e\,{\left (d+e\,x\right )}^{5/2}\,\left (19\,b^2\,c\,e^2-72\,b\,c^2\,d\,e+72\,c^3\,d^2\right )}{4\,b^4}+\frac {3\,c\,e\,\left (2\,c^2\,d-b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{b^4}}{c^2\,{\left (d+e\,x\right )}^4-\left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )-\left (4\,c^2\,d-2\,b\,c\,e\right )\,{\left (d+e\,x\right )}^3+{\left (d+e\,x\right )}^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )+c^2\,d^4+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e}-\frac {3\,\mathrm {atanh}\left (\frac {27\,c^2\,e^9\,\sqrt {d+e\,x}}{32\,d^{3/2}\,\left (\frac {27\,c^2\,e^9}{32\,d}-\frac {81\,c^3\,e^8}{8\,b}+\frac {27\,c^4\,d\,e^7}{2\,b^2}\right )}-\frac {81\,c^3\,e^8\,\sqrt {d+e\,x}}{8\,\sqrt {d}\,\left (\frac {27\,b\,c^2\,e^9}{32\,d}-\frac {81\,c^3\,e^8}{8}+\frac {27\,c^4\,d\,e^7}{2\,b}\right )}+\frac {27\,c^4\,\sqrt {d}\,e^7\,\sqrt {d+e\,x}}{2\,\left (\frac {27\,c^4\,d\,e^7}{2}-\frac {81\,b\,c^3\,e^8}{8}+\frac {27\,b^2\,c^2\,e^9}{32\,d}\right )}\right )\,\left (b^2\,e^2-12\,b\,c\,d\,e+16\,c^2\,d^2\right )}{4\,b^5\,\sqrt {d}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {d+e\,x}\,\left (117\,b^4\,c^3\,e^6-1008\,b^3\,c^4\,d\,e^5+3312\,b^2\,c^5\,d^2\,e^4-4608\,b\,c^6\,d^3\,e^3+2304\,c^7\,d^4\,e^2\right )}{4\,b^8}-\frac {3\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (\frac {3\,b^{12}\,c^2\,e^5-24\,b^{11}\,c^3\,d\,e^4+24\,b^{10}\,c^4\,d^2\,e^3}{b^{12}}-\frac {3\,\left (32\,b^{11}\,c^2\,e^3-64\,b^{10}\,c^3\,d\,e^2\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{32\,b^8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )\,3{}\mathrm {i}}{8\,\left (b^6\,e-b^5\,c\,d\right )}+\frac {\left (\frac {\sqrt {d+e\,x}\,\left (117\,b^4\,c^3\,e^6-1008\,b^3\,c^4\,d\,e^5+3312\,b^2\,c^5\,d^2\,e^4-4608\,b\,c^6\,d^3\,e^3+2304\,c^7\,d^4\,e^2\right )}{4\,b^8}+\frac {3\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (\frac {3\,b^{12}\,c^2\,e^5-24\,b^{11}\,c^3\,d\,e^4+24\,b^{10}\,c^4\,d^2\,e^3}{b^{12}}+\frac {3\,\left (32\,b^{11}\,c^2\,e^3-64\,b^{10}\,c^3\,d\,e^2\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{32\,b^8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )\,3{}\mathrm {i}}{8\,\left (b^6\,e-b^5\,c\,d\right )}}{\frac {\frac {135\,b^5\,c^3\,e^8}{8}-\frac {1215\,b^4\,c^4\,d\,e^7}{4}+1674\,b^3\,c^5\,d^2\,e^6-3996\,b^2\,c^6\,d^3\,e^5+4320\,b\,c^7\,d^4\,e^4-1728\,c^8\,d^5\,e^3}{b^{12}}-\frac {3\,\left (\frac {\sqrt {d+e\,x}\,\left (117\,b^4\,c^3\,e^6-1008\,b^3\,c^4\,d\,e^5+3312\,b^2\,c^5\,d^2\,e^4-4608\,b\,c^6\,d^3\,e^3+2304\,c^7\,d^4\,e^2\right )}{4\,b^8}-\frac {3\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (\frac {3\,b^{12}\,c^2\,e^5-24\,b^{11}\,c^3\,d\,e^4+24\,b^{10}\,c^4\,d^2\,e^3}{b^{12}}-\frac {3\,\left (32\,b^{11}\,c^2\,e^3-64\,b^{10}\,c^3\,d\,e^2\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{32\,b^8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}+\frac {3\,\left (\frac {\sqrt {d+e\,x}\,\left (117\,b^4\,c^3\,e^6-1008\,b^3\,c^4\,d\,e^5+3312\,b^2\,c^5\,d^2\,e^4-4608\,b\,c^6\,d^3\,e^3+2304\,c^7\,d^4\,e^2\right )}{4\,b^8}+\frac {3\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (\frac {3\,b^{12}\,c^2\,e^5-24\,b^{11}\,c^3\,d\,e^4+24\,b^{10}\,c^4\,d^2\,e^3}{b^{12}}+\frac {3\,\left (32\,b^{11}\,c^2\,e^3-64\,b^{10}\,c^3\,d\,e^2\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{32\,b^8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{8\,\left (b^6\,e-b^5\,c\,d\right )}}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )\,3{}\mathrm {i}}{4\,\left (b^6\,e-b^5\,c\,d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(b*x + c*x^2)^3,x)

[Out]

((3*(d + e*x)^(1/2)*(b^3*d*e^4 - 8*c^3*d^4*e + 16*b*c^2*d^3*e^2 - 9*b^2*c*d^2*e^3))/(4*b^4) - ((d + e*x)^(3/2)
*(5*b^3*e^4 - 72*c^3*d^3*e + 108*b*c^2*d^2*e^2 - 46*b^2*c*d*e^3))/(4*b^4) - (e*(d + e*x)^(5/2)*(72*c^3*d^2 + 1
9*b^2*c*e^2 - 72*b*c^2*d*e))/(4*b^4) + (3*c*e*(2*c^2*d - b*c*e)*(d + e*x)^(7/2))/b^4)/(c^2*(d + e*x)^4 - (d +
e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*
d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (3*atanh((27*c^2*e^9*(d + e*x)^(1/2))/(32*d^(3/2)*((
27*c^2*e^9)/(32*d) - (81*c^3*e^8)/(8*b) + (27*c^4*d*e^7)/(2*b^2))) - (81*c^3*e^8*(d + e*x)^(1/2))/(8*d^(1/2)*(
(27*b*c^2*e^9)/(32*d) - (81*c^3*e^8)/8 + (27*c^4*d*e^7)/(2*b))) + (27*c^4*d^(1/2)*e^7*(d + e*x)^(1/2))/(2*((27
*c^4*d*e^7)/2 - (81*b*c^3*e^8)/8 + (27*b^2*c^2*e^9)/(32*d))))*(b^2*e^2 + 16*c^2*d^2 - 12*b*c*d*e))/(4*b^5*d^(1
/2)) + (atan((((((d + e*x)^(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5
+ 3312*b^2*c^5*d^2*e^4))/(4*b^8) - (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c
^4*d^2*e^3)/b^12 - (3*(32*b^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2
+ 16*c^2*d^2 - 20*b*c*d*e))/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5
*c*d)))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(8*(b^6*e - b^5*c*d)) + ((((d + e*x)^
(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^4))/(
4*b^8) + (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 + (3*(32*b
^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))
/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d))^(1
/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(8*(b^6*e - b^5*c*d)))/(((135*b^5*c^3*e^8)/8 - 1728*c^8*d^5*e^3
+ 4320*b*c^7*d^4*e^4 - (1215*b^4*c^4*d*e^7)/4 - 3996*b^2*c^6*d^3*e^5 + 1674*b^3*c^5*d^2*e^6)/b^12 - (3*(((d +
e*x)^(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^
4))/(4*b^8) - (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 - (3*
(32*b^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*
d*e))/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d
))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)) + (3*(((d + e*x)^(1/2)*(117*b^4*c^3*e^6
+ 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^4))/(4*b^8) + (3*(-c*(b*e -
c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 + (3*(32*b^11*c^2*e^3 - 64*b^10*c
^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(32*b^8*(b^6*e - b^5*c
*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2
*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d))))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(
4*(b^6*e - b^5*c*d))

________________________________________________________________________________________