Optimal. Leaf size=64 \[ -\frac {(b c-a d)^3 \log (a+b x)}{a b^3}+\frac {d^2 x (3 b c-a d)}{b^2}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^2}{2 b} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {72} \[ \frac {d^2 x (3 b c-a d)}{b^2}-\frac {(b c-a d)^3 \log (a+b x)}{a b^3}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x (a+b x)} \, dx &=\int \left (\frac {d^2 (3 b c-a d)}{b^2}+\frac {c^3}{a x}+\frac {d^3 x}{b}+\frac {(-b c+a d)^3}{a b^2 (a+b x)}\right ) \, dx\\ &=\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^2}{2 b}+\frac {c^3 \log (x)}{a}-\frac {(b c-a d)^3 \log (a+b x)}{a b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.92 \[ \frac {a b d^2 x (-2 a d+6 b c+b d x)-2 (b c-a d)^3 \log (a+b x)+2 b^3 c^3 \log (x)}{2 a b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 97, normalized size = 1.52 \[ \frac {a b^{2} d^{3} x^{2} + 2 \, b^{3} c^{3} \log \relax (x) + 2 \, {\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 91, normalized size = 1.42 \[ \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {b d^{3} x^{2} + 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 103, normalized size = 1.61 \[ \frac {d^{3} x^{2}}{2 b}+\frac {a^{2} d^{3} \ln \left (b x +a \right )}{b^{3}}-\frac {3 a c \,d^{2} \ln \left (b x +a \right )}{b^{2}}-\frac {a \,d^{3} x}{b^{2}}+\frac {c^{3} \ln \relax (x )}{a}-\frac {c^{3} \ln \left (b x +a \right )}{a}+\frac {3 c^{2} d \ln \left (b x +a \right )}{b}+\frac {3 c \,d^{2} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 91, normalized size = 1.42 \[ \frac {c^{3} \log \relax (x)}{a} + \frac {b d^{3} x^{2} + 2 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 65, normalized size = 1.02 \[ \frac {d^3\,x^2}{2\,b}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {c^3\,\ln \relax (x)}{a}+\frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.48, size = 112, normalized size = 1.75 \[ x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) + \frac {d^{3} x^{2}}{2 b} + \frac {c^{3} \log {\relax (x )}}{a} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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