Optimal. Leaf size=48 \[ \frac{\left (a+b x^3\right )^{p+2}}{3 b^2 (p+2)}-\frac{a \left (a+b x^3\right )^{p+1}}{3 b^2 (p+1)} \]
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Rubi [A] time = 0.028962, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{\left (a+b x^3\right )^{p+2}}{3 b^2 (p+2)}-\frac{a \left (a+b x^3\right )^{p+1}}{3 b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^3\right )^p \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^{1+p}}{3 b^2 (1+p)}+\frac{\left (a+b x^3\right )^{2+p}}{3 b^2 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0210368, size = 40, normalized size = 0.83 \[ \frac{\left (a+b x^3\right )^{p+1} \left (b (p+1) x^3-a\right )}{3 b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 42, normalized size = 0.9 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) ^{1+p} \left ( -{x}^{3}pb-b{x}^{3}+a \right ) }{3\,{b}^{2} \left ({p}^{2}+3\,p+2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988165, size = 63, normalized size = 1.31 \begin{align*} \frac{{\left (b^{2}{\left (p + 1\right )} x^{6} + a b p x^{3} - a^{2}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51117, size = 115, normalized size = 2.4 \begin{align*} \frac{{\left ({\left (b^{2} p + b^{2}\right )} x^{6} + a b p x^{3} - a^{2}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.42768, size = 524, normalized size = 10.92 \begin{align*} \begin{cases} \frac{a^{p} x^{6}}{6} & \text{for}\: b = 0 \\\frac{a \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{a \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac{2 a \log{\left (2 \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{a}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{b x^{3} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{b x^{3} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac{2 b x^{3} \log{\left (2 \right )}}{3 a b^{2} + 3 b^{3} x^{3}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 b^{2}} - \frac{a \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 b^{2}} + \frac{x^{3}}{3 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{a b p x^{3} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{b^{2} p x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{b^{2} x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10728, size = 127, normalized size = 2.65 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} p -{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a p +{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} - 2 \,{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a}{3 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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