Optimal. Leaf size=88 \[ \frac {a}{(a+b x) (b c-a d)^2}+\frac {c}{(c+d x) (b c-a d)^2}+\frac {(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac {(a d+b c) \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} \frac {a}{(a+b x) (b c-a d)^2}+\frac {c}{(c+d x) (b c-a d)^2}+\frac {(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac {(a d+b c) \log (c+d x)}{(b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx &=\int \left (-\frac {a b}{(b c-a d)^2 (a+b x)^2}+\frac {b (b c+a d)}{(b c-a d)^3 (a+b x)}-\frac {c d}{(b c-a d)^2 (c+d x)^2}-\frac {d (b c+a d)}{(b c-a d)^3 (c+d x)}\right ) \, dx\\ &=\frac {a}{(b c-a d)^2 (a+b x)}+\frac {c}{(b c-a d)^2 (c+d x)}+\frac {(b c+a d) \log (a+b x)}{(b c-a d)^3}-\frac {(b c+a d) \log (c+d x)}{(b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 75, normalized size = 0.85 \begin {gather*} \frac {\frac {a (b c-a d)}{a+b x}+\frac {c (b c-a d)}{c+d x}+(a d+b c) \log (a+b x)-(a d+b c) \log (c+d x)}{(b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.01, size = 283, normalized size = 3.22 \begin {gather*} \frac {2 \, a b c^{2} - 2 \, a^{2} c d + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x + {\left (a b c^{2} + a^{2} c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2} + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) - {\left (a b c^{2} + a^{2} c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2} + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 167, normalized size = 1.90 \begin {gather*} \frac {\frac {a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (b^{3} c + a b^{2} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{2} c d}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 1.34 \begin {gather*} -\frac {a d \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}+\frac {a d \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {b c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}+\frac {b c \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {a}{\left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {c}{\left (a d -b c \right )^{2} \left (d x +c \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.16, size = 218, normalized size = 2.48 \begin {gather*} \frac {{\left (b c + a d\right )} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {{\left (b c + a d\right )} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, a c + {\left (b c + a d\right )} x}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 147, normalized size = 1.67 \begin {gather*} \frac {\frac {2\,a\,c}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {x\,\left (a\,d+b\,c\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a\,d+b\,c+2\,b\,d\,x\right )}{{\left (a\,d-b\,c\right )}^3}\right )\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.53, size = 483, normalized size = 5.49 \begin {gather*} \frac {2 a c + x \left (a d + b c\right )}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} + \frac {\left (a d + b c\right ) \log {\left (x + \frac {- \frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} - \frac {\left (a d + b c\right ) \log {\left (x + \frac {\frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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