3.283 \(\int \frac {1}{(d+e x) (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=193 \[ \frac {c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac {b e+3 c d}{b^4 d^2 x}+\frac {c^3}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {1}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac {c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]

[Out]

-1/2/b^3/d/x^2+(b*e+3*c*d)/b^4/d^2/x+1/2*c^3/b^3/(-b*e+c*d)/(c*x+b)^2+c^3*(-4*b*e+3*c*d)/b^4/(-b*e+c*d)^2/(c*x
+b)+(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(x)/b^5/d^3-c^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)*ln(c*x+b)/b^5/(-b*e+c*d)
^3+e^5*ln(e*x+d)/d^3/(-b*e+c*d)^3

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Rubi [A]  time = 0.23, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ \frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac {c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac {c^3}{2 b^3 (b+c x)^2 (c d-b e)}+\frac {b e+3 c d}{b^4 d^2 x}-\frac {1}{2 b^3 d x^2}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^3),x]

[Out]

-1/(2*b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) + c^3/(2*b^3*(c*d - b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(b^
4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) - (c^3*(6*c^2*d^2 - 15*b*c*d
*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {1}{b^3 d x^3}+\frac {-3 c d-b e}{b^4 d^2 x^2}+\frac {6 c^2 d^2+3 b c d e+b^2 e^2}{b^5 d^3 x}+\frac {c^4}{b^3 (-c d+b e) (b+c x)^3}+\frac {c^4 (-3 c d+4 b e)}{b^4 (-c d+b e)^2 (b+c x)^2}+\frac {c^4 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 (-c d+b e)^3 (b+c x)}+\frac {e^6}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 192, normalized size = 0.99 \[ \frac {c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac {b e+3 c d}{b^4 d^2 x}-\frac {c^3}{2 b^3 (b+c x)^2 (b e-c d)}-\frac {1}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}+\frac {c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (b e-c d)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)^3),x]

[Out]

-1/2*1/(b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) - c^3/(2*b^3*(-(c*d) + b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e)
)/(b^4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) + (c^3*(6*c^2*d^2 - 15*
b*c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

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fricas [B]  time = 68.86, size = 716, normalized size = 3.71 \[ -\frac {b^{4} c^{3} d^{5} - 3 \, b^{5} c^{2} d^{4} e + 3 \, b^{6} c d^{3} e^{2} - b^{7} d^{2} e^{3} - 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{5} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} + b^{5} c^{2} d^{2} e^{3} - 4 \, b^{6} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e + 3 \, b^{5} c^{2} d^{3} e^{2} + b^{6} c d^{2} e^{3} - b^{7} d e^{4}\right )} x + 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left (b^{5} c^{2} e^{5} x^{4} + 2 \, b^{6} c e^{5} x^{3} + b^{7} e^{5} x^{2}\right )} \log \left (e x + d\right ) - 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left ({\left (b^{5} c^{5} d^{6} - 3 \, b^{6} c^{4} d^{5} e + 3 \, b^{7} c^{3} d^{4} e^{2} - b^{8} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (b^{6} c^{4} d^{6} - 3 \, b^{7} c^{3} d^{5} e + 3 \, b^{8} c^{2} d^{4} e^{2} - b^{9} c d^{3} e^{3}\right )} x^{3} + {\left (b^{7} c^{3} d^{6} - 3 \, b^{8} c^{2} d^{5} e + 3 \, b^{9} c d^{4} e^{2} - b^{10} d^{3} e^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^3*d^5 - 3*b^5*c^2*d^4*e + 3*b^6*c*d^3*e^2 - b^7*d^2*e^3 - 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b
^3*c^4*d^3*e^2 - b^5*c^2*d*e^4)*x^3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^3*d^3*e^2 + b^5*c^2*d^2*e^
3 - 4*b^6*c*d*e^4)*x^2 - 2*(2*b^3*c^4*d^5 - 5*b^4*c^3*d^4*e + 3*b^5*c^2*d^3*e^2 + b^6*c*d^2*e^3 - b^7*d*e^4)*x
+ 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d
^3*e^2)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b^4*c^3*d^3*e^2)*x^2)*log(c*x + b) - 2*(b^5*c^2*e^5*x^4 +
2*b^6*c*e^5*x^3 + b^7*e^5*x^2)*log(e*x + d) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 - b^5*c^2*e
^5)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*c*e^5)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^
4*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x^2)*log(x))/((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b^8
*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 -
3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3*e^3)*x^2)

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giac [B]  time = 0.18, size = 414, normalized size = 2.15 \[ -\frac {{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac {e^{6} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-(6*c^6*d^2 - 15*b*c^5*d*e + 10*b^2*c^4*e^2)*log(abs(c*x + b))/(b^5*c^4*d^3 - 3*b^6*c^3*d^2*e + 3*b^7*c^2*d*e^
2 - b^8*c*e^3) + e^6*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4) + (6*c^2*
d^2 + 3*b*c*d*e + b^2*e^2)*log(abs(x))/(b^5*d^3) - 1/2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*
d^2*e^3 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - b^4*c^2*d*e^4)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4
*d^4*e + 30*b^3*c^3*d^3*e^2 + b^4*c^2*d^2*e^3 - 4*b^5*c*d*e^4)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e + 3*b^
4*c^2*d^3*e^2 + b^5*c*d^2*e^3 - b^6*d*e^4)*x)/((c*d - b*e)^3*(c*x + b)^2*b^4*d^3*x^2)

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maple [A]  time = 0.06, size = 254, normalized size = 1.32 \[ \frac {10 c^{3} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {15 c^{4} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{4}}+\frac {6 c^{5} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}-\frac {4 c^{3} e}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{3}}+\frac {3 c^{4} d}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{4}}-\frac {c^{3}}{2 \left (b e -c d \right ) \left (c x +b \right )^{2} b^{3}}+\frac {e^{2} \ln \relax (x )}{b^{3} d^{3}}+\frac {3 c e \ln \relax (x )}{b^{4} d^{2}}+\frac {6 c^{2} \ln \relax (x )}{b^{5} d}+\frac {e}{b^{3} d^{2} x}+\frac {3 c}{b^{4} d x}-\frac {1}{2 b^{3} d \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x)^3,x)

[Out]

-1/2*c^3/(b*e-c*d)/b^3/(c*x+b)^2-4*c^3/(b*e-c*d)^2/b^3/(c*x+b)*e+3*c^4/(b*e-c*d)^2/b^4/(c*x+b)*d+10*c^3/(b*e-c
*d)^3/b^3*ln(c*x+b)*e^2-15*c^4/(b*e-c*d)^3/b^4*ln(c*x+b)*d*e+6*c^5/(b*e-c*d)^3/b^5*ln(c*x+b)*d^2-e^5/(b*e-c*d)
^3/d^3*ln(e*x+d)-1/2/b^3/d/x^2+1/b^3/d^2/x*e+3/b^4/d/x*c+1/b^3/d^3*ln(x)*e^2+3/b^4/d^2*ln(x)*c*e+6/b^5/d*ln(x)
*c^2

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maxima [B]  time = 1.64, size = 439, normalized size = 2.27 \[ \frac {e^{5} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \, {\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} - {\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \relax (x)}{b^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

e^5*log(e*x + d)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3) - (6*c^5*d^2 - 15*b*c^4*d*e + 10*b^
2*c^3*e^2)*log(c*x + b)/(b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3) - 1/2*(b^3*c^2*d^3 - 2*b^4*c
*d^2*e + b^5*d*e^2 - 2*(6*c^5*d^3 - 9*b*c^4*d^2*e + b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^3 - (18*b*c^4*d^3 - 27*b^2*
c^3*d^2*e + 3*b^3*c^2*d*e^2 + 4*b^4*c*e^3)*x^2 - 2*(2*b^2*c^3*d^3 - 3*b^3*c^2*d^2*e + b^5*e^3)*x)/((b^4*c^4*d^
4 - 2*b^5*c^3*d^3*e + b^6*c^2*d^2*e^2)*x^4 + 2*(b^5*c^3*d^4 - 2*b^6*c^2*d^3*e + b^7*c*d^2*e^2)*x^3 + (b^6*c^2*
d^4 - 2*b^7*c*d^3*e + b^8*d^2*e^2)*x^2) + (6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^3)

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mupad [B]  time = 0.83, size = 331, normalized size = 1.72 \[ \frac {\frac {x\,\left (b\,e+2\,c\,d\right )}{b^2\,d^2}-\frac {1}{2\,b\,d}+\frac {x^2\,\left (4\,b^3\,c\,e^3+3\,b^2\,c^2\,d\,e^2-27\,b\,c^3\,d^2\,e+18\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^3\,\left (b^3\,c^2\,e^3+b^2\,c^3\,d\,e^2-9\,b\,c^4\,d^2\,e+6\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (10\,b^2\,c^3\,e^2-15\,b\,c^4\,d\,e+6\,c^5\,d^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}-\frac {e^5\,\ln \left (d+e\,x\right )}{d^3\,{\left (b\,e-c\,d\right )}^3}+\frac {\ln \relax (x)\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)),x)

[Out]

((x*(b*e + 2*c*d))/(b^2*d^2) - 1/(2*b*d) + (x^2*(18*c^4*d^3 + 4*b^3*c*e^3 + 3*b^2*c^2*d*e^2 - 27*b*c^3*d^2*e))
/(2*b^3*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)) + (x^3*(6*c^5*d^3 + b^3*c^2*e^3 + b^2*c^3*d*e^2 - 9*b*c^4*d^2*e))
/(b^4*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(b + c*x)*(6*c^5*d^2 + 10*b^
2*c^3*e^2 - 15*b*c^4*d*e))/(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2) - (e^5*log(d + e*x))/(d^3
*(b*e - c*d)^3) + (log(x)*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(b^5*d^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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