Optimal. Leaf size=41 \[ \frac {a B-A b}{3 b^2 \left (a+b x^3\right )}+\frac {B \log \left (a+b x^3\right )}{3 b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 41, 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 {B \log \left (a+b x^3\right )}{3 b^2}-\frac {A b-a B}{3 b^2 \left (a+b x^3\right )} \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 )}{\left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^2}+\frac {B}{b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A b-a B}{3 b^2 \left (a+b x^3\right )}+\frac {B \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} \frac {a B-A b}{3 b^2 \left (a+b x^3\right )}+\frac {B \log \left (a+b x^3\right )}{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 )}{\left (a+b x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.56, size = 44, normalized size = 1.07 \begin {gather*} \frac {B a - A b + {\left (B b x^{3} + B a\right )} \log \left (b x^{3} + a\right )}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 65, normalized size = 1.59 \begin {gather*} -\frac {B {\left (\frac {\log \left (\frac {{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x^{3} + a\right )} b}\right )}}{3 \, b} - \frac {A}{3 \, {\left (b x^{3} + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 47, normalized size = 1.15 \begin {gather*} -\frac {A}{3 \left (b \,x^{3}+a \right ) b}+\frac {B a}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {B \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 = 0.58, size = 40, normalized size = 0.98 \begin {gather*} \frac {B a - A b}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {B \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 = 2.35, size = 37, normalized size = 0.90 \begin {gather*} \frac {B\,\ln \left (b\,x^3+a\right )}{3\,b^2}-\frac {A\,b-B\,a}{3\,b^2\,\left (b\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.23, size = 36, normalized size = 0.88 \begin {gather*} \frac {B \log {\left (a + b x^{3} \right )}}{3 b^{2}} + \frac {- A b + B a}{3 a b^{2} + 3 b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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