Optimal. Leaf size=86 \[ 2 a^2 b x (2 a B+3 A b)+a^3 \log (x) (a B+4 A b)-\frac{a^4 A}{x}+a b^2 x^2 (3 a B+2 A b)+\frac{1}{3} b^3 x^3 (4 a B+A b)+\frac{1}{4} b^4 B x^4 \]
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Rubi [A] time = 0.0506458, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ 2 a^2 b x (2 a B+3 A b)+a^3 \log (x) (a B+4 A b)-\frac{a^4 A}{x}+a b^2 x^2 (3 a B+2 A b)+\frac{1}{3} b^3 x^3 (4 a B+A b)+\frac{1}{4} b^4 B x^4 \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^2} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{x^2} \, dx\\ &=\int \left (2 a^2 b (3 A b+2 a B)+\frac{a^4 A}{x^2}+\frac{a^3 (4 A b+a B)}{x}+2 a b^2 (2 A b+3 a B) x+b^3 (A b+4 a B) x^2+b^4 B x^3\right ) \, dx\\ &=-\frac{a^4 A}{x}+2 a^2 b (3 A b+2 a B) x+a b^2 (2 A b+3 a B) x^2+\frac{1}{3} b^3 (A b+4 a B) x^3+\frac{1}{4} b^4 B x^4+a^3 (4 A b+a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0356784, size = 85, normalized size = 0.99 \[ 3 a^2 b^2 x (2 A+B x)+a^3 \log (x) (a B+4 A b)-\frac{a^4 A}{x}+4 a^3 b B x+\frac{2}{3} a b^3 x^2 (3 A+2 B x)+\frac{1}{12} b^4 x^3 (4 A+3 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 95, normalized size = 1.1 \begin{align*}{\frac{{b}^{4}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{4}}{3}}+{\frac{4\,B{x}^{3}a{b}^{3}}{3}}+2\,A{x}^{2}a{b}^{3}+3\,B{x}^{2}{a}^{2}{b}^{2}+6\,A{a}^{2}{b}^{2}x+4\,B{a}^{3}bx+4\,A\ln \left ( x \right ){a}^{3}b+{a}^{4}B\ln \left ( x \right ) -{\frac{A{a}^{4}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00816, size = 127, normalized size = 1.48 \begin{align*} \frac{1}{4} \, B b^{4} x^{4} - \frac{A a^{4}}{x} + \frac{1}{3} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x +{\left (B a^{4} + 4 \, A a^{3} b\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27581, size = 223, normalized size = 2.59 \begin{align*} \frac{3 \, B b^{4} x^{5} - 12 \, A a^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 24 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x \log \left (x\right )}{12 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.425191, size = 94, normalized size = 1.09 \begin{align*} - \frac{A a^{4}}{x} + \frac{B b^{4} x^{4}}{4} + a^{3} \left (4 A b + B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{4}}{3} + \frac{4 B a b^{3}}{3}\right ) + x^{2} \left (2 A a b^{3} + 3 B a^{2} b^{2}\right ) + x \left (6 A a^{2} b^{2} + 4 B a^{3} b\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16837, size = 128, normalized size = 1.49 \begin{align*} \frac{1}{4} \, B b^{4} x^{4} + \frac{4}{3} \, B a b^{3} x^{3} + \frac{1}{3} \, A b^{4} x^{3} + 3 \, B a^{2} b^{2} x^{2} + 2 \, A a b^{3} x^{2} + 4 \, B a^{3} b x + 6 \, A a^{2} b^{2} x - \frac{A a^{4}}{x} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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