Optimal. Leaf size=160 \[ \frac{b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac{d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}-\frac{\log (x) (3 a d+b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.353016, antiderivative size = 160, 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 \[ \frac{b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac{d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}-\frac{\log (x) (3 a d+b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x)*(c + d*x)^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 55.8341, size = 148, normalized size = 0.92 \[ - \frac{d^{2}}{2 c^{2} \left (c + d x\right )^{2} \left (a d - b c\right )} - \frac{d^{2} \left (2 a d - 3 b c\right )}{c^{3} \left (c + d x\right ) \left (a d - b c\right )^{2}} + \frac{d^{2} \left (3 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{c^{4} \left (a d - b c\right )^{3}} - \frac{1}{a c^{3} x} - \frac{b^{4} \log{\left (a + b x \right )}}{a^{2} \left (a d - b c\right )^{3}} - \frac{\left (3 a d + b c\right ) \log{\left (x \right )}}{a^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x+a)/(d*x+c)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.281628, size = 163, normalized size = 1.02 \[ -\frac{b^4 \log (a+b x)}{a^2 (a d-b c)^3}-\frac{\left (3 a^2 d^4-8 a b c d^3+6 b^2 c^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}+\frac{\log (x) (-3 a d-b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x)*(c + d*x)^3),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.021, size = 216, normalized size = 1.4 \[ -{\frac{{d}^{2}}{2\,{c}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}-2\,{\frac{{d}^{3}a}{{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+3\,{\frac{{d}^{2}b}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+3\,{\frac{{d}^{4}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{3}}}-8\,{\frac{{d}^{3}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{3}}}+6\,{\frac{{d}^{2}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{1}{a{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{a{c}^{4}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}{c}^{3}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x+a)/(d*x+c)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.37048, size = 477, normalized size = 2.98 \[ \frac{b^{4} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac{{\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac{2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{2} d^{2} - 5 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} +{\left (4 \, b^{2} c^{3} d - 15 \, a b c^{2} d^{2} + 9 \, a^{2} c d^{3}\right )} x}{2 \,{\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{3} + 2 \,{\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2} +{\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )}} - \frac{{\left (b c + 3 \, a d\right )} \log \left (x\right )}{a^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)^3*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 10.2643, size = 845, normalized size = 5.28 \[ -\frac{2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \,{\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x - 2 \,{\left (b^{4} c^{4} d^{2} x^{3} + 2 \, b^{4} c^{5} d x^{2} + b^{4} c^{6} x\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (6 \, a^{2} b^{2} c^{2} d^{4} - 8 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{3} + 2 \,{\left (6 \, a^{2} b^{2} c^{3} d^{3} - 8 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (6 \, a^{2} b^{2} c^{4} d^{2} - 8 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \left (d x + c\right ) + 2 \,{\left ({\left (b^{4} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{4} + 8 \, a^{3} b c d^{5} - 3 \, a^{4} d^{6}\right )} x^{3} + 2 \,{\left (b^{4} c^{5} d - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (b^{4} c^{6} - 6 \, a^{2} b^{2} c^{4} d^{2} + 8 \, a^{3} b c^{3} d^{3} - 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{2} b^{3} c^{7} d^{2} - 3 \, a^{3} b^{2} c^{6} d^{3} + 3 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}\right )} x^{3} + 2 \,{\left (a^{2} b^{3} c^{8} d - 3 \, a^{3} b^{2} c^{7} d^{2} + 3 \, a^{4} b c^{6} d^{3} - a^{5} c^{5} d^{4}\right )} x^{2} +{\left (a^{2} b^{3} c^{9} - 3 \, a^{3} b^{2} c^{8} d + 3 \, a^{4} b c^{7} d^{2} - a^{5} c^{6} d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)^3*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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**2/(b*x+a)/(d*x+c)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.330652, size = 452, normalized size = 2.82 \[ \frac{b^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}} - \frac{{\left (6 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}} - \frac{{\left (b c + 3 \, a d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2} c^{4}} - \frac{2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \,{\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} a^{2} c^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)^3*x^2),x, algorithm="giac")
[Out]