Optimal. Leaf size=123 \[ -\frac{b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{a (a+b x) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}+\frac{d^2}{c (c+d x) (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.248275, antiderivative size = 123, 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 \[ -\frac{b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{a (a+b x) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}+\frac{d^2}{c (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x)^2*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 43.1903, size = 107, normalized size = 0.87 \[ \frac{d^{2}}{c \left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{d^{2} \left (a d - 3 b c\right ) \log{\left (c + d x \right )}}{c^{2} \left (a d - b c\right )^{3}} + \frac{b^{2}}{a \left (a + b x\right ) \left (a d - b c\right )^{2}} - \frac{b^{2} \left (3 a d - b c\right ) \log{\left (a + b x \right )}}{a^{2} \left (a d - b c\right )^{3}} + \frac{\log{\left (x \right )}}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.33869, size = 120, normalized size = 0.98 \[ \frac{b^2 (b c-3 a d) \log (a+b x)}{a^2 (a d-b c)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{a (a+b x) (b c-a d)^2}+\frac{d^2 (a d-3 b c) \log (c+d x)}{c^2 (b c-a d)^3}+\frac{d^2}{c (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x)^2*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.022, size = 158, normalized size = 1.3 \[{\frac{{d}^{2}}{c \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{d}^{3}\ln \left ( dx+c \right ) a}{{c}^{2} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{d}^{2}\ln \left ( dx+c \right ) b}{c \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}{c}^{2}}}+{\frac{{b}^{2}}{ \left ( ad-bc \right ) ^{2}a \left ( bx+a \right ) }}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{3}a}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{3}{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x+a)^2/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.35011, size = 382, normalized size = 3.11 \[ -\frac{{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac{{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}} + \frac{b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x}{a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x} + \frac{\log \left (x\right )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 5.00338, size = 707, normalized size = 5.75 \[ \frac{a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x -{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) -{\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (d x + c\right ) +{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (x\right )}{a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} +{\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{2} +{\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2*x),x, algorithm="fricas")
[Out]
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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/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.29681, size = 269, normalized size = 2.19 \[{\left (\frac{b^{4}}{{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )}{\left (b x + a\right )}} - \frac{{\left (3 \, b c d^{2} - a d^{3}\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3}} - \frac{d^{3}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} c} + \frac{{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b c^{2}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2*x),x, algorithm="giac")
[Out]