Optimal. Leaf size=85 \[ -\frac {a}{(c+d x) (b c-a d)^2}-\frac {c}{2 d (c+d x)^2 (b c-a d)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 85, 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} \[ -\frac {a}{(c+d x) (b c-a d)^2}-\frac {c}{2 d (c+d x)^2 (b c-a d)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x}{(a+b x) (c+d x)^3} \, dx &=\int \left (-\frac {a b^2}{(b c-a d)^3 (a+b x)}+\frac {c}{(b c-a d) (c+d x)^3}+\frac {a d}{(-b c+a d)^2 (c+d x)^2}-\frac {a b d}{(-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {c}{2 d (b c-a d) (c+d x)^2}-\frac {a}{(b c-a d)^2 (c+d x)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 85, normalized size = 1.00 \[ -\frac {a}{(c+d x) (b c-a d)^2}+\frac {c}{2 d (c+d x)^2 (a d-b c)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 244, normalized size = 2.87 \[ -\frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x + 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (b x + a\right ) - 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (d x + c\right )}{2 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 165, normalized size = 1.94 \[ -\frac {a b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {a b d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.99 \[ \frac {a b \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}-\frac {a b \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {a}{\left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.11, size = 208, normalized size = 2.45 \[ -\frac {a b \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {a b \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, a d^{2} x + b c^{2} + a c d}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 185, normalized size = 2.18 \[ \frac {2\,a\,b\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3}-\frac {\frac {b\,c^2+a\,d\,c}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,d\,x}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{c^2+2\,c\,d\,x+d^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.88, size = 401, normalized size = 4.72 \[ - \frac {a b \log {\left (x + \frac {- \frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {a b \log {\left (x + \frac {\frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a c d - 2 a d^{2} x - b c^{2}}{2 a^{2} c^{2} d^{3} - 4 a b c^{3} d^{2} + 2 b^{2} c^{4} d + x^{2} \left (2 a^{2} d^{5} - 4 a b c d^{4} + 2 b^{2} c^{2} d^{3}\right ) + x \left (4 a^{2} c d^{4} - 8 a b c^{2} d^{3} + 4 b^{2} c^{3} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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