Optimal. Leaf size=56 \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{x^3} \, dx &=\int \left (2 a B c+\frac {a^2 A}{x^3}+\frac {a^2 B}{x^2}+\frac {2 a A c}{x}+A c^2 x+B c^2 x^2\right ) \, dx\\ &=-\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3+2 a A c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \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+c x^2\right )^2}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 55, normalized size = 0.98 \begin {gather*} \frac {2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 12 \, A a c x^{2} \log \relax (x) - 6 \, B a^{2} x - 3 \, A a^{2}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 0.91 \begin {gather*} \frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+2 A a c \ln \relax (x )+2 B a c x -\frac {B \,a^{2}}{x}-\frac {A \,a^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \relax (x) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 50, normalized size = 0.89 \begin {gather*} \frac {A\,c^2\,x^2}{2}-\frac {\frac {A\,a^2}{2}+B\,a^2\,x}{x^2}+\frac {B\,c^2\,x^3}{3}+2\,A\,a\,c\,\ln \relax (x)+2\,B\,a\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 58, normalized size = 1.04 \begin {gather*} 2 A a c \log {\relax (x )} + \frac {A c^{2} x^{2}}{2} + 2 B a c x + \frac {B c^{2} x^{3}}{3} + \frac {- A a^{2} - 2 B a^{2} x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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