Optimal. Leaf size=109 \[ \frac {d^2}{(c+d x) (b c-a d)^3}+\frac {3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac {3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac {2 b d}{(a+b x) (b c-a d)^3}-\frac {b}{2 (a+b x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {44} \[ \frac {d^2}{(c+d x) (b c-a d)^3}+\frac {3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac {3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac {2 b d}{(a+b x) (b c-a d)^3}-\frac {b}{2 (a+b x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^3 (c+d x)^2} \, dx &=\int \left (\frac {b^2}{(b c-a d)^2 (a+b x)^3}-\frac {2 b^2 d}{(b c-a d)^3 (a+b x)^2}+\frac {3 b^2 d^2}{(b c-a d)^4 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)^2}-\frac {3 b d^3}{(b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac {b}{2 (b c-a d)^2 (a+b x)^2}+\frac {2 b d}{(b c-a d)^3 (a+b x)}+\frac {d^2}{(b c-a d)^3 (c+d x)}+\frac {3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac {3 b d^2 \log (c+d x)}{(b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 0.90 \[ \frac {\frac {2 d^2 (b c-a d)}{c+d x}+\frac {4 b d (b c-a d)}{a+b x}-\frac {b (b c-a d)^2}{(a+b x)^2}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 494, normalized size = 4.53 \[ -\frac {b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4} + {\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 216, normalized size = 1.98 \[ \frac {3 \, b d^{3} \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} + \frac {d^{5}}{{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} {\left (d x + c\right )}} + \frac {5 \, b^{3} d^{2} - \frac {6 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )}}{{\left (d x + c\right )} d}}{2 \, {\left (b c - a d\right )}^{4} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 109, normalized size = 1.00 \[ \frac {3 b \,d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {3 b \,d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {2 b d}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {b}{2 \left (a d -b c \right )^{2} \left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.55, size = 386, normalized size = 3.54 \[ \frac {3 \, b d^{2} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {3 \, b d^{2} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{2 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 330, normalized size = 3.03 \[ \frac {6\,b\,d^2\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {2\,a^2\,d^2+5\,a\,b\,c\,d-b^2\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,d\,x\,\left (c\,b^2+3\,a\,d\,b\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (d\,a^2+2\,b\,c\,a\right )+a^2\,c+x^2\,\left (c\,b^2+2\,a\,d\,b\right )+b^2\,d\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.72, size = 634, normalized size = 5.82 \[ - \frac {3 b d^{2} \log {\left (x + \frac {- \frac {3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} + \frac {3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac {3 b d^{2} \log {\left (x + \frac {\frac {3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} - \frac {3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac {- 2 a^{2} d^{2} - 5 a b c d + b^{2} c^{2} - 6 b^{2} d^{2} x^{2} + x \left (- 9 a b d^{2} - 3 b^{2} c d\right )}{2 a^{5} c d^{3} - 6 a^{4} b c^{2} d^{2} + 6 a^{3} b^{2} c^{3} d - 2 a^{2} b^{3} c^{4} + x^{3} \left (2 a^{3} b^{2} d^{4} - 6 a^{2} b^{3} c d^{3} + 6 a b^{4} c^{2} d^{2} - 2 b^{5} c^{3} d\right ) + x^{2} \left (4 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 6 a^{2} b^{3} c^{2} d^{2} + 2 a b^{4} c^{3} d - 2 b^{5} c^{4}\right ) + x \left (2 a^{5} d^{4} - 2 a^{4} b c d^{3} - 6 a^{3} b^{2} c^{2} d^{2} + 10 a^{2} b^{3} c^{3} d - 4 a b^{4} c^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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