Optimal. Leaf size=123 \[ -\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}+\frac {\log (x)}{a^2 c^2}+\frac {b^2}{a (a+b x) (b c-a d)^2}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}+\frac {d^2}{c (c+d x) (b c-a d)^2} \]
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Rubi [A] time = 0.12, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ -\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}+\frac {\log (x)}{a^2 c^2}+\frac {b^2}{a (a+b x) (b c-a d)^2}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}+\frac {d^2}{c (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac {1}{a^2 c^2 x}-\frac {b^3}{a (-b c+a d)^2 (a+b x)^2}-\frac {b^3 (-b c+3 a d)}{a^2 (-b c+a d)^3 (a+b x)}-\frac {d^3}{c (b c-a d)^2 (c+d x)^2}-\frac {d^3 (3 b c-a d)}{c^2 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=\frac {b^2}{a (b c-a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a^2 c^2}-\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.23, 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.
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fricas [B] time = 9.97, size = 524, normalized size = 4.26 \[ \frac {a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x - {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) - {\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (d x + c\right ) + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \relax (x)}{a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} + {\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{2} + {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 199, normalized size = 1.62 \[ {\left (\frac {b^{4}}{{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3}} - \frac {d^{3}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} c} + \frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b c^{2}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 158, normalized size = 1.28 \[ -\frac {a \,d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{2}}-\frac {3 b^{2} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a}+\frac {b^{3} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{2}}+\frac {3 b \,d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c}+\frac {b^{2}}{\left (a d -b c \right )^{2} \left (b x +a \right ) a}+\frac {d^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c}+\frac {\ln \relax (x )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 283, normalized size = 2.30 \[ -\frac {{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac {{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}} + \frac {b^{2} c^{2} + a^{2} d^{2} + {\left (b^{2} c d + a b d^{2}\right )} x}{a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x} + \frac {\log \relax (x)}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 183, normalized size = 1.49 \[ \frac {\frac {a^2\,d^2+b^2\,c^2}{a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b\,d\,x\,\left (a\,d+b\,c\right )}{a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}+\frac {\ln \relax (x)}{a^2\,c^2}-\frac {b^2\,\ln \left (a+b\,x\right )\,\left (3\,a\,d-b\,c\right )}{a^2\,{\left (a\,d-b\,c\right )}^3}-\frac {d^2\,\ln \left (c+d\,x\right )\,\left (a\,d-3\,b\,c\right )}{c^2\,{\left (a\,d-b\,c\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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