Optimal. Leaf size=82 \[ \frac {b^2 \log (a+b x)}{(b c-a d)^3}-\frac {b^2 \log (c+d x)}{(b c-a d)^3}+\frac {b}{(c+d x) (b c-a d)^2}+\frac {1}{2 (c+d x)^2 (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 82, 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} \begin {gather*} \frac {b^2 \log (a+b x)}{(b c-a d)^3}-\frac {b^2 \log (c+d x)}{(b c-a d)^3}+\frac {b}{(c+d x) (b c-a d)^2}+\frac {1}{2 (c+d x)^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d x)^3} \, dx &=\int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx\\ &=\frac {1}{2 (b c-a d) (c+d x)^2}+\frac {b}{(b c-a d)^2 (c+d x)}+\frac {b^2 \log (a+b x)}{(b c-a d)^3}-\frac {b^2 \log (c+d x)}{(b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.82 \begin {gather*} \frac {2 b^2 \log (a+b x)+\frac {(b c-a d) (-a d+3 b c+2 b d x)}{(c+d x)^2}-2 b^2 \log (c+d x)}{2 (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 {1}{(a+b x) (c+d x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.26, size = 242, normalized size = 2.95 \begin {gather*} \frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (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} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.90, size = 165, normalized size = 2.01 \begin {gather*} \frac {b^{3} \log \left ({\left | b x + a \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} d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} + \frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.99 \begin {gather*} -\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}+\frac {b^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {b}{\left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {1}{2 \left (a d -b c \right ) \left (d x +c \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.11, size = 202, normalized size = 2.46 \begin {gather*} \frac {b^{2} \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 {b^{2} \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 \, b d x + 3 \, b c - a d}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 183, normalized size = 2.23 \begin {gather*} -\frac {\frac {a\,d-3\,b\,c}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,d\,x}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{c^2+2\,c\,d\,x+d^2\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\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 = 1.57, size = 381, normalized size = 4.65 \begin {gather*} \frac {b^{2} \log {\left (x + \frac {- \frac {a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d - \frac {b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{\left (a d - b c\right )^{3}} - \frac {b^{2} \log {\left (x + \frac {\frac {a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d + \frac {b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a d + 3 b c + 2 b d x}{2 a^{2} c^{2} d^{2} - 4 a b c^{3} d + 2 b^{2} c^{4} + x^{2} \left (2 a^{2} d^{4} - 4 a b c d^{3} + 2 b^{2} c^{2} d^{2}\right ) + x \left (4 a^{2} c d^{3} - 8 a b c^{2} d^{2} + 4 b^{2} c^{3} d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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