Optimal. Leaf size=61 \[ \frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{a x}+\frac {d^3 x}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 61, 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 = {88} \[ \frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{a x}+\frac {d^3 x}{b} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx &=\int \left (\frac {d^3}{b}+\frac {c^3}{a x^2}+\frac {c^2 (-b c+3 a d)}{a^2 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{a x}+\frac {d^3 x}{b}-\frac {c^2 (b c-3 a d) \log (x)}{a^2}+\frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 1.08 \[ \frac {b^2 c^2 x \log (x) (3 a d-b c)+a b \left (a d^3 x^2-b c^3\right )+x (b c-a d)^3 \log (a+b x)}{a^2 b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 98, normalized size = 1.61 \[ \frac {a^{2} b d^{3} x^{2} - a b^{2} c^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \log \left (b x + a\right ) - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x \log \relax (x)}{a^{2} b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 91, normalized size = 1.49 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 102, normalized size = 1.67 \[ -\frac {a \,d^{3} \ln \left (b x +a \right )}{b^{2}}+\frac {3 c^{2} d \ln \relax (x )}{a}-\frac {3 c^{2} d \ln \left (b x +a \right )}{a}-\frac {b \,c^{3} \ln \relax (x )}{a^{2}}+\frac {b \,c^{3} \ln \left (b x +a \right )}{a^{2}}+\frac {3 c \,d^{2} \ln \left (b x +a \right )}{b}+\frac {d^{3} x}{b}-\frac {c^{3}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 89, normalized size = 1.46 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \relax (x)}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 88, normalized size = 1.44 \[ \frac {d^3\,x}{b}-\frac {c^3}{a\,x}+\frac {c^2\,\ln \relax (x)\,\left (3\,a\,d-b\,c\right )}{a^2}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.91, size = 196, normalized size = 3.21 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} + \frac {c^{2} \left (3 a d - b c\right ) \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} - a b c^{2} \left (3 a d - b c\right )}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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