Optimal. Leaf size=121 \[ \frac {a b}{(a+b x) (b c-a d)^3}+\frac {a d+b c}{(c+d x) (b c-a d)^3}+\frac {c}{2 (c+d x)^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac {b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]
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Rubi [A] time = 0.10, antiderivative size = 121, 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} \[ \frac {a b}{(a+b x) (b c-a d)^3}+\frac {a d+b c}{(c+d x) (b c-a d)^3}+\frac {c}{2 (c+d x)^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac {b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (-\frac {a b^2}{(b c-a d)^3 (a+b x)^2}+\frac {b^2 (b c+2 a d)}{(b c-a d)^4 (a+b x)}-\frac {c d}{(b c-a d)^2 (c+d x)^3}-\frac {d (b c+a d)}{(b c-a d)^3 (c+d x)^2}-\frac {b d (b c+2 a d)}{(b c-a d)^4 (c+d x)}\right ) \, dx\\ &=\frac {a b}{(b c-a d)^3 (a+b x)}+\frac {c}{2 (b c-a d)^2 (c+d x)^2}+\frac {b c+a d}{(b c-a d)^3 (c+d x)}+\frac {b (b c+2 a d) \log (a+b x)}{(b c-a d)^4}-\frac {b (b c+2 a d) \log (c+d x)}{(b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 111, normalized size = 0.92 \[ \frac {\frac {c (b c-a d)^2}{(c+d x)^2}+\frac {2 a b (b c-a d)}{a+b x}+\frac {2 (a d+b c) (b c-a d)}{c+d x}+2 b (2 a d+b c) \log (a+b x)-2 b (2 a d+b c) \log (c+d x)}{2 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 586, normalized size = 4.84 \[ \frac {5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x + 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 255, normalized size = 2.11 \[ \frac {\frac {2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac {2 \, {\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 170, normalized size = 1.40 \[ \frac {2 a b d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {2 a b d \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {b^{2} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {b^{2} c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a b}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {a d}{\left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {b c}{\left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 406, normalized size = 3.36 \[ \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {5 \, a b c^{2} + a^{2} c d + 2 \, {\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{2} + {\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 377, normalized size = 3.12 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (2\,a\,d+b\,c\right )\,\left (a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b^2\,d\,x\,\left (2\,a\,d+b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {d\,a^2\,c+5\,b\,a\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (2\,a\,d+b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d\,x^2\,\left (2\,a\,d+b\,c\right )}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.35, size = 774, normalized size = 6.40 \[ - \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {- \frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {\frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- a^{2} c d - 5 a b c^{2} + x^{2} \left (- 4 a b d^{2} - 2 b^{2} c d\right ) + x \left (- 2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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