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

Optimal. Leaf size=144 \[ -\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{2 a d+b c}{a^2 c^3 x}+\frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4}+\frac{d^3 (4 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^2}-\frac{d^3}{c^3 (c+d x) (b c-a d)}-\frac{1}{2 a c^2 x^2} \]

[Out]

-1/(2*a*c^2*x^2) + (b*c + 2*a*d)/(a^2*c^3*x) - d^3/(c^3*(b*c - a*d)*(c + d*x)) +
((b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^3*c^4) - (b^4*Log[a + b*x])/(a^3*
(b*c - a*d)^2) + (d^3*(4*b*c - 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^2)

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Rubi [A]  time = 0.303904, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{2 a d+b c}{a^2 c^3 x}+\frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4}+\frac{d^3 (4 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^2}-\frac{d^3}{c^3 (c+d x) (b c-a d)}-\frac{1}{2 a c^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)*(c + d*x)^2),x]

[Out]

-1/(2*a*c^2*x^2) + (b*c + 2*a*d)/(a^2*c^3*x) - d^3/(c^3*(b*c - a*d)*(c + d*x)) +
((b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^3*c^4) - (b^4*Log[a + b*x])/(a^3*
(b*c - a*d)^2) + (d^3*(4*b*c - 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^2)

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Rubi in Sympy [A]  time = 69.598, size = 134, normalized size = 0.93 \[ \frac{d^{3}}{c^{3} \left (c + d x\right ) \left (a d - b c\right )} - \frac{d^{3} \left (3 a d - 4 b c\right ) \log{\left (c + d x \right )}}{c^{4} \left (a d - b c\right )^{2}} - \frac{1}{2 a c^{2} x^{2}} + \frac{2 a d + b c}{a^{2} c^{3} x} - \frac{b^{4} \log{\left (a + b x \right )}}{a^{3} \left (a d - b c\right )^{2}} + \frac{\left (3 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log{\left (x \right )}}{a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3/(c**3*(c + d*x)*(a*d - b*c)) - d**3*(3*a*d - 4*b*c)*log(c + d*x)/(c**4*(a*d
- b*c)**2) - 1/(2*a*c**2*x**2) + (2*a*d + b*c)/(a**2*c**3*x) - b**4*log(a + b*x
)/(a**3*(a*d - b*c)**2) + (3*a**2*d**2 + 2*a*b*c*d + b**2*c**2)*log(x)/(a**3*c**
4)

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Mathematica [A]  time = 1.10869, size = 143, normalized size = 0.99 \[ -\frac{b^4 \log (a+b x)}{a^3 (b c-a d)^2}+\frac{c \left (\frac{2 b c}{a^2 x}+\frac{2 d^3}{(c+d x) (a d-b c)}-\frac{c-4 d x}{a x^2}\right )+\frac{2 d^3 (4 b c-3 a d) \log (c+d x)}{(b c-a d)^2}}{2 c^4}+\frac{\log (x) \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^3 c^4} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^3*c^4) - (b^4*Log[a + b*x])/(a^3*(
b*c - a*d)^2) + (c*((2*b*c)/(a^2*x) - (c - 4*d*x)/(a*x^2) + (2*d^3)/((-(b*c) + a
*d)*(c + d*x))) + (2*d^3*(4*b*c - 3*a*d)*Log[c + d*x])/(b*c - a*d)^2)/(2*c^4)

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Maple [A]  time = 0.02, size = 171, normalized size = 1.2 \[{\frac{{d}^{3}}{{c}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) }}-3\,{\frac{{d}^{4}\ln \left ( dx+c \right ) a}{{c}^{4} \left ( ad-bc \right ) ^{2}}}+4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) b}{{c}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{1}{2\,a{c}^{2}{x}^{2}}}+2\,{\frac{d}{ax{c}^{3}}}+{\frac{b}{x{a}^{2}{c}^{2}}}+3\,{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{4}}}+2\,{\frac{b\ln \left ( x \right ) d}{{a}^{2}{c}^{3}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}{c}^{2}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.36997, size = 331, normalized size = 2.3 \[ -\frac{b^{4} \log \left (b x + a\right )}{a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}} + \frac{{\left (4 \, b c d^{3} - 3 \, a d^{4}\right )} \log \left (d x + c\right )}{b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}} - \frac{a b c^{3} - a^{2} c^{2} d - 2 \,{\left (b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} -{\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x}{2 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{3} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 6.41106, size = 473, normalized size = 3.28 \[ -\frac{a^{2} b^{2} c^{5} - 2 \, a^{3} b c^{4} d + a^{4} c^{3} d^{2} - 2 \,{\left (a b^{3} c^{4} d - 4 \, a^{3} b c^{2} d^{3} + 3 \, a^{4} c d^{4}\right )} x^{2} -{\left (2 \, a b^{3} c^{5} - a^{2} b^{2} c^{4} d - 4 \, a^{3} b c^{3} d^{2} + 3 \, a^{4} c^{2} d^{3}\right )} x + 2 \,{\left (b^{4} c^{4} d x^{3} + b^{4} c^{5} x^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (4 \, a^{3} b c d^{4} - 3 \, a^{4} d^{5}\right )} x^{3} +{\left (4 \, a^{3} b c^{2} d^{3} - 3 \, a^{4} c d^{4}\right )} x^{2}\right )} \log \left (d x + c\right ) - 2 \,{\left ({\left (b^{4} c^{4} d - 4 \, a^{3} b c d^{4} + 3 \, a^{4} d^{5}\right )} x^{3} +{\left (b^{4} c^{5} - 4 \, a^{3} b c^{2} d^{3} + 3 \, a^{4} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{3} b^{2} c^{6} d - 2 \, a^{4} b c^{5} d^{2} + a^{5} c^{4} d^{3}\right )} x^{3} +{\left (a^{3} b^{2} c^{7} - 2 \, a^{4} b c^{6} d + a^{5} c^{5} d^{2}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(a^2*b^2*c^5 - 2*a^3*b*c^4*d + a^4*c^3*d^2 - 2*(a*b^3*c^4*d - 4*a^3*b*c^2*d
^3 + 3*a^4*c*d^4)*x^2 - (2*a*b^3*c^5 - a^2*b^2*c^4*d - 4*a^3*b*c^3*d^2 + 3*a^4*c
^2*d^3)*x + 2*(b^4*c^4*d*x^3 + b^4*c^5*x^2)*log(b*x + a) - 2*((4*a^3*b*c*d^4 - 3
*a^4*d^5)*x^3 + (4*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*log(d*x + c) - 2*((b^4*c^4*
d - 4*a^3*b*c*d^4 + 3*a^4*d^5)*x^3 + (b^4*c^5 - 4*a^3*b*c^2*d^3 + 3*a^4*c*d^4)*x
^2)*log(x))/((a^3*b^2*c^6*d - 2*a^4*b*c^5*d^2 + a^5*c^4*d^3)*x^3 + (a^3*b^2*c^7
- 2*a^4*b*c^6*d + a^5*c^5*d^2)*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/x**3/(b*x+a)/(d*x+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275187, size = 285, normalized size = 1.98 \[ -\frac{d^{7}}{{\left (b c^{4} d^{4} - a c^{3} d^{5}\right )}{\left (d x + c\right )}} - \frac{b^{4} d{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{a^{3} b^{2} c^{2} d - 2 \, a^{4} b c d^{2} + a^{5} d^{3}} + \frac{{\left (b^{2} c^{2} d + 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )}{\rm ln}\left ({\left | -\frac{c}{d x + c} + 1 \right |}\right )}{a^{3} c^{4} d} + \frac{2 \, a b c d + 5 \, a^{2} d^{2} - \frac{2 \,{\left (a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )}}{{\left (d x + c\right )} d}}{2 \, a^{3} c^{4}{\left (\frac{c}{d x + c} - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d^7/((b*c^4*d^4 - a*c^3*d^5)*(d*x + c)) - b^4*d*ln(abs(b - b*c/(d*x + c) + a*d/
(d*x + c)))/(a^3*b^2*c^2*d - 2*a^4*b*c*d^2 + a^5*d^3) + (b^2*c^2*d + 2*a*b*c*d^2
+ 3*a^2*d^3)*ln(abs(-c/(d*x + c) + 1))/(a^3*c^4*d) + 1/2*(2*a*b*c*d + 5*a^2*d^2
- 2*(a*b*c^2*d^2 + 3*a^2*c*d^3)/((d*x + c)*d))/(a^3*c^4*(c/(d*x + c) - 1)^2)