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

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) + (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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Rubi [A]  time = 0.538298, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{b e+3 c d}{b^4 d^2 x}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}-\frac{1}{2 b^3 d x^2}+\frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]

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)

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Rubi in Sympy [A]  time = 69.3058, size = 184, normalized size = 0.95 \[ - \frac{e^{5} \log{\left (d + e x \right )}}{d^{3} \left (b e - c d\right )^{3}} - \frac{c^{3}}{2 b^{3} \left (b + c x\right )^{2} \left (b e - c d\right )} - \frac{1}{2 b^{3} d x^{2}} - \frac{c^{3} \left (4 b e - 3 c d\right )}{b^{4} \left (b + c x\right ) \left (b e - c d\right )^{2}} + \frac{b e + 3 c d}{b^{4} d^{2} x} + \frac{c^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} \left (b e - c d\right )^{3}} + \frac{\left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x \right )}}{b^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.360456, 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*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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Maple [A]  time = 0.023, size = 254, normalized size = 1.3 \[ -{\frac{1}{2\,d{b}^{3}{x}^{2}}}+{\frac{e}{{d}^{2}{b}^{3}x}}+3\,{\frac{c}{d{b}^{4}x}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}{b}^{3}}}+3\,{\frac{\ln \left ( x \right ) ce}{{d}^{2}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{d{b}^{5}}}-{\frac{{c}^{3}}{ \left ( 2\,be-2\,cd \right ){b}^{3} \left ( cx+b \right ) ^{2}}}-4\,{\frac{{c}^{3}e}{ \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{4}d}{ \left ( be-cd \right ) ^{2}{b}^{4} \left ( cx+b \right ) }}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{3}{b}^{4}}}+6\,{\frac{{c}^{5}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{5}}}-{\frac{{e}^{5}\ln \left ( ex+d \right ) }{{d}^{3} \left ( be-cd \right ) ^{3}}} \]

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

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Maxima [A]  time = 0.717404, size = 593, normalized size = 3.07 \[ \frac{e^{5} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac{{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac{b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \,{\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} -{\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \,{\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} +{\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)),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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Fricas [A]  time = 38.7888, size = 967, normalized size = 5.01 \[ -\frac{b^{4} c^{3} d^{5} - 3 \, b^{5} c^{2} d^{4} e + 3 \, b^{6} c d^{3} e^{2} - b^{7} d^{2} e^{3} - 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{5} c^{2} d e^{4}\right )} x^{3} -{\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} + b^{5} c^{2} d^{2} e^{3} - 4 \, b^{6} c d e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e + 3 \, b^{5} c^{2} d^{3} e^{2} + b^{6} c d^{2} e^{3} - b^{7} d e^{4}\right )} x + 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left (b^{5} c^{2} e^{5} x^{4} + 2 \, b^{6} c e^{5} x^{3} + b^{7} e^{5} x^{2}\right )} \log \left (e x + d\right ) - 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (b^{5} c^{5} d^{6} - 3 \, b^{6} c^{4} d^{5} e + 3 \, b^{7} c^{3} d^{4} e^{2} - b^{8} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \,{\left (b^{6} c^{4} d^{6} - 3 \, b^{7} c^{3} d^{5} e + 3 \, b^{8} c^{2} d^{4} e^{2} - b^{9} c d^{3} e^{3}\right )} x^{3} +{\left (b^{7} c^{3} d^{6} - 3 \, b^{8} c^{2} d^{5} e + 3 \, b^{9} c d^{4} e^{2} - b^{10} d^{3} e^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)),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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.210496, size = 559, normalized size = 2.9 \[ -\frac{{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac{e^{6}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac{b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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