Optimal. Leaf size=66 \[ a^4 A \log (x)+4 a^3 A b x+3 a^2 A b^2 x^2+\frac {4}{3} a A b^3 x^3+\frac {B (a+b x)^5}{5 b}+\frac {1}{4} A b^4 x^4 \]
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 80
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*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 1.26 \begin {gather*} a^4 A \log (x)+a^4 B x+2 a^3 b x (2 A+B x)+a^2 b^2 x^2 (3 A+2 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) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 93, normalized size = 1.41 \begin {gather*} \frac {1}{5} \, B b^{4} x^{5} + A a^{4} \log \relax (x) + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 94, normalized size = 1.42 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 94, normalized size = 1.42 \begin {gather*} \frac {B \,b^{4} x^{5}}{5}+\frac {A \,b^{4} x^{4}}{4}+B a \,b^{3} x^{4}+\frac {4 A a \,b^{3} x^{3}}{3}+2 B \,a^{2} b^{2} x^{3}+3 A \,a^{2} b^{2} x^{2}+2 B \,a^{3} b \,x^{2}+A \,a^{4} \ln \relax (x )+4 A \,a^{3} b x +B \,a^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 93, normalized size = 1.41 \begin {gather*} \frac {1}{5} \, B b^{4} x^{5} + A a^{4} \log \relax (x) + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 84, normalized size = 1.27 \begin {gather*} x\,\left (B\,a^4+4\,A\,b\,a^3\right )+x^4\,\left (\frac {A\,b^4}{4}+B\,a\,b^3\right )+\frac {B\,b^4\,x^5}{5}+A\,a^4\,\ln \relax (x)+a^2\,b\,x^2\,\left (3\,A\,b+2\,B\,a\right )+\frac {2\,a\,b^2\,x^3\,\left (2\,A\,b+3\,B\,a\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 95, normalized size = 1.44 \begin {gather*} A a^{4} \log {\relax (x )} + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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