Optimal. Leaf size=54 \[ -\frac {a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac {x^3 (A b-a B)}{3 b^2}+\frac {B x^6}{6 b} \]
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Rubi [A] time = 0.06, antiderivative size = 54, 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 = {446, 77} \begin {gather*} \frac {x^3 (A b-a B)}{3 b^2}-\frac {a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac {B x^6}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x (A+B x)}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-a B}{b^2}+\frac {B x}{b}+\frac {a (-A b+a B)}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^6}{6 b}-\frac {a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.87 \begin {gather*} \frac {b x^3 \left (-2 a B+2 A b+b B x^3\right )+2 a (a B-A b) \log \left (a+b x^3\right )}{6 b^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 {x^5 \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 = 0.56, size = 51, normalized size = 0.94 \begin {gather*} \frac {B b^{2} x^{6} - 2 \, {\left (B a b - A b^{2}\right )} x^{3} + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 52, normalized size = 0.96 \begin {gather*} \frac {B b x^{6} - 2 \, B a x^{3} + 2 \, A b x^{3}}{6 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 1.15 \begin {gather*} \frac {B \,x^{6}}{6 b}+\frac {A \,x^{3}}{3 b}-\frac {B a \,x^{3}}{3 b^{2}}-\frac {A a \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {B \,a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 50, normalized size = 0.93 \begin {gather*} \frac {B b x^{6} - 2 \, {\left (B a - A b\right )} x^{3}}{6 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 52, normalized size = 0.96 \begin {gather*} x^3\,\left (\frac {A}{3\,b}-\frac {B\,a}{3\,b^2}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (B\,a^2-A\,a\,b\right )}{3\,b^3}+\frac {B\,x^6}{6\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 46, normalized size = 0.85 \begin {gather*} \frac {B x^{6}}{6 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x^{3} \right )}}{3 b^{3}} + x^{3} \left (\frac {A}{3 b} - \frac {B a}{3 b^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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