Optimal. Leaf size=56 \[ \frac {1}{2} x^2 (a C+A b)+a A \log (x)+\frac {1}{3} x^3 (a D+b B)+a B x+\frac {1}{4} b C x^4+\frac {1}{5} b D x^5 \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1802} \begin {gather*} \frac {1}{2} x^2 (a C+A b)+a A \log (x)+\frac {1}{3} x^3 (a D+b B)+a B x+\frac {1}{4} b C x^4+\frac {1}{5} b D x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 1802
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{x} \, dx &=\int \left (a B+\frac {a A}{x}+(A b+a C) x+(b B+a D) x^2+b C x^3+b D x^4\right ) \, dx\\ &=a B x+\frac {1}{2} (A b+a C) x^2+\frac {1}{3} (b B+a D) x^3+\frac {1}{4} b C x^4+\frac {1}{5} b D x^5+a A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (a C+A b)+a A \log (x)+\frac {1}{3} x^3 (a D+b B)+a B x+\frac {1}{4} b C x^4+\frac {1}{5} b D x^5 \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 {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.82, size = 48, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, D b x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{3} \, {\left (D a + B b\right )} x^{3} + B a x + \frac {1}{2} \, {\left (C a + A b\right )} x^{2} + A a \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 53, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, D b x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{3} \, D a x^{3} + \frac {1}{3} \, B b x^{3} + \frac {1}{2} \, C a x^{2} + \frac {1}{2} \, A b x^{2} + B a x + A a \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 = 0.00, size = 53, normalized size = 0.95 \begin {gather*} \frac {D b \,x^{5}}{5}+\frac {C b \,x^{4}}{4}+\frac {B b \,x^{3}}{3}+\frac {D a \,x^{3}}{3}+\frac {A b \,x^{2}}{2}+\frac {C a \,x^{2}}{2}+A a \ln \relax (x )+B a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 48, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, D b x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{3} \, {\left (D a + B b\right )} x^{3} + B a x + \frac {1}{2} \, {\left (C a + A b\right )} x^{2} + A a \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 52, normalized size = 0.93 \begin {gather*} \frac {a\,x^3\,D}{3}+\frac {b\,x^5\,D}{5}+B\,a\,x+\frac {A\,b\,x^2}{2}+\frac {C\,a\,x^2}{2}+\frac {B\,b\,x^3}{3}+\frac {C\,b\,x^4}{4}+A\,a\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 54, normalized size = 0.96 \begin {gather*} A a \log {\relax (x )} + B a x + \frac {C b x^{4}}{4} + \frac {D b x^{5}}{5} + x^{3} \left (\frac {B b}{3} + \frac {D a}{3}\right ) + x^{2} \left (\frac {A b}{2} + \frac {C a}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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