Optimal. Leaf size=64 \[ -\frac{a^2 (a B+3 A b)}{2 x^2}-\frac{a^3 A}{3 x^3}+b^2 \log (x) (3 a B+A b)-\frac{3 a b (a B+A b)}{x}+b^3 B x \]
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Rubi [A] time = 0.0360875, antiderivative size = 64, normalized size of antiderivative = 1., 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 = {76} \[ -\frac{a^2 (a B+3 A b)}{2 x^2}-\frac{a^3 A}{3 x^3}+b^2 \log (x) (3 a B+A b)-\frac{3 a b (a B+A b)}{x}+b^3 B x \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^4} \, dx &=\int \left (b^3 B+\frac{a^3 A}{x^4}+\frac{a^2 (3 A b+a B)}{x^3}+\frac{3 a b (A b+a B)}{x^2}+\frac{b^2 (A b+3 a B)}{x}\right ) \, dx\\ &=-\frac{a^3 A}{3 x^3}-\frac{a^2 (3 A b+a B)}{2 x^2}-\frac{3 a b (A b+a B)}{x}+b^3 B x+b^2 (A b+3 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0316233, size = 67, normalized size = 1.05 \[ b^2 \log (x) (3 a B+A b)-\frac{9 a^2 b x (A+2 B x)+a^3 (2 A+3 B x)+18 a A b^2 x^2-6 b^3 B x^4}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 72, normalized size = 1.1 \begin{align*}{b}^{3}Bx+A\ln \left ( x \right ){b}^{3}+3\,B\ln \left ( x \right ) a{b}^{2}-{\frac{A{a}^{3}}{3\,{x}^{3}}}-{\frac{3\,{a}^{2}bA}{2\,{x}^{2}}}-{\frac{{a}^{3}B}{2\,{x}^{2}}}-3\,{\frac{a{b}^{2}A}{x}}-3\,{\frac{{a}^{2}bB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00529, size = 93, normalized size = 1.45 \begin{align*} B b^{3} x +{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70937, size = 166, normalized size = 2.59 \begin{align*} \frac{6 \, B b^{3} x^{4} + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} \log \left (x\right ) - 2 \, A a^{3} - 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.907381, size = 70, normalized size = 1.09 \begin{align*} B b^{3} x + b^{2} \left (A b + 3 B a\right ) \log{\left (x \right )} - \frac{2 A a^{3} + x^{2} \left (18 A a b^{2} + 18 B a^{2} b\right ) + x \left (9 A a^{2} b + 3 B a^{3}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18388, size = 95, normalized size = 1.48 \begin{align*} B b^{3} x +{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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