Optimal. Leaf size=35 \[ \frac {(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac {B x^3}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \begin {gather*} \frac {(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac {B x^3}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {B}{b}+\frac {A b-a B}{b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {B x^3}{3 b}+\frac {(A b-a B) \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.89 \begin {gather*} \frac {(A b-a B) \log \left (a+b x^3\right )+b B x^3}{3 b^2} \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 {x^2 \left (A+B x^3\right )}{a+b x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.58, size = 30, normalized size = 0.86 \begin {gather*} \frac {B b x^{3} - {\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 32, normalized size = 0.91 \begin {gather*} \frac {B x^{3}}{3 \, b} - \frac {{\left (B a - A b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.14 \begin {gather*} \frac {B \,x^{3}}{3 b}+\frac {A \ln \left (b \,x^{3}+a \right )}{3 b}-\frac {B a \ln \left (b \,x^{3}+a \right )}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 31, normalized size = 0.89 \begin {gather*} \frac {B x^{3}}{3 \, b} - \frac {{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 31, normalized size = 0.89 \begin {gather*} \frac {B\,x^3}{3\,b}+\frac {\ln \left (b\,x^3+a\right )\,\left (A\,b-B\,a\right )}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 27, normalized size = 0.77 \begin {gather*} \frac {B x^{3}}{3 b} - \frac {\left (- A b + B a\right ) \log {\left (a + b x^{3} \right )}}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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