Optimal. Leaf size=54 \[ \log (x) (a C+A b)-\frac {a A}{2 x^2}+x (a D+b B)-\frac {a B}{x}+\frac {1}{2} b C x^2+\frac {1}{3} b D x^3 \]
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Rubi [A] time = 0.05, antiderivative size = 54, 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*} \log (x) (a C+A b)-\frac {a A}{2 x^2}+x (a D+b B)-\frac {a B}{x}+\frac {1}{2} b C x^2+\frac {1}{3} b D x^3 \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^3} \, dx &=\int \left (b B \left (1+\frac {a D}{b B}\right )+\frac {a A}{x^3}+\frac {a B}{x^2}+\frac {A b+a C}{x}+b C x+b D x^2\right ) \, dx\\ &=-\frac {a A}{2 x^2}-\frac {a B}{x}+(b B+a D) x+\frac {1}{2} b C x^2+\frac {1}{3} b D x^3+(A b+a C) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.94 \begin {gather*} \log (x) (a C+A b)-\frac {a \left (A+2 B x-2 D x^3\right )}{2 x^2}+\frac {1}{6} b x \left (6 B+3 C x+2 D x^2\right ) \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^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.62, size = 55, normalized size = 1.02 \begin {gather*} \frac {2 \, D b x^{5} + 3 \, C b x^{4} + 6 \, {\left (D a + B b\right )} x^{3} + 6 \, {\left (C a + A b\right )} x^{2} \log \relax (x) - 6 \, B a x - 3 \, A a}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 48, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, D b x^{3} + \frac {1}{2} \, C b x^{2} + D a x + B b x + {\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.89 \begin {gather*} \frac {D b \,x^{3}}{3}+\frac {C b \,x^{2}}{2}+A b \ln \relax (x )+B b x +C a \ln \relax (x )+D a x -\frac {B a}{x}-\frac {A a}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 48, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, D b x^{3} + \frac {1}{2} \, C b x^{2} + {\left (D a + B b\right )} x + {\left (C a + A b\right )} \log \relax (x) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 47, normalized size = 0.87 \begin {gather*} \frac {b\,x^3\,D}{3}+B\,b\,x-\frac {A\,a}{2\,x^2}-\frac {B\,a}{x}+\frac {C\,b\,x^2}{2}+A\,b\,\ln \relax (x)+C\,a\,\ln \relax (x)+a\,x\,D \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 51, normalized size = 0.94 \begin {gather*} \frac {C b x^{2}}{2} + \frac {D b x^{3}}{3} + x \left (B b + D a\right ) + \left (A b + C a\right ) \log {\relax (x )} + \frac {- A a - 2 B a x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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