Optimal. Leaf size=66 \[ 3 a^2 A b^2 x^2+4 a^3 A b x+a^4 A \log (x)+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]
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Rubi [A] time = 0.0226127, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 80, 43} \[ 3 a^2 A b^2 x^2+4 a^3 A b x+a^4 A \log (x)+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]
Antiderivative was successfully verified.
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Rule 27
Rule 80
Rule 43
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{x} \, dx\\ &=\frac{B (a+b x)^5}{5 b}+A \int \frac{(a+b x)^4}{x} \, dx\\ &=\frac{B (a+b x)^5}{5 b}+A \int \left (4 a^3 b+\frac{a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx\\ &=4 a^3 A b x+3 a^2 A b^2 x^2+\frac{4}{3} a A b^3 x^3+\frac{1}{4} A b^4 x^4+\frac{B (a+b x)^5}{5 b}+a^4 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0219544, size = 83, normalized size = 1.26 \[ a^2 b^2 x^2 (3 A+2 B x)+2 a^3 b x (2 A+B x)+a^4 A \log (x)+a^4 B x+\frac{1}{3} a b^3 x^3 (4 A+3 B x)+\frac{1}{20} b^4 x^4 (5 A+4 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 94, normalized size = 1.4 \begin{align*}{\frac{{b}^{4}B{x}^{5}}{5}}+{\frac{A{b}^{4}{x}^{4}}{4}}+B{x}^{4}a{b}^{3}+{\frac{4\,aA{b}^{3}{x}^{3}}{3}}+2\,B{x}^{3}{a}^{2}{b}^{2}+3\,{a}^{2}A{b}^{2}{x}^{2}+2\,B{x}^{2}{a}^{3}b+4\,{a}^{3}Abx+{a}^{4}Bx+{a}^{4}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992208, size = 126, normalized size = 1.91 \begin{align*} \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3206, size = 205, normalized size = 3.11 \begin{align*} \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376294, size = 95, normalized size = 1.44 \begin{align*} A a^{4} \log{\left (x \right )} + \frac{B b^{4} x^{5}}{5} + x^{4} \left (\frac{A b^{4}}{4} + B a b^{3}\right ) + x^{3} \left (\frac{4 A a b^{3}}{3} + 2 B a^{2} b^{2}\right ) + x^{2} \left (3 A a^{2} b^{2} + 2 B a^{3} b\right ) + x \left (4 A a^{3} b + B a^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11463, size = 127, normalized size = 1.92 \begin{align*} \frac{1}{5} \, B b^{4} x^{5} + B a b^{3} x^{4} + \frac{1}{4} \, A b^{4} x^{4} + 2 \, B a^{2} b^{2} x^{3} + \frac{4}{3} \, A a b^{3} x^{3} + 2 \, B a^{3} b x^{2} + 3 \, A a^{2} b^{2} x^{2} + B a^{4} x + 4 \, A a^{3} b x + A a^{4} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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