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 = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.89, size = 58, normalized size = 1.21 \[ \frac {{\left ({\left (b^{2} p + b^{2}\right )} x^{6} + a b p x^{3} - a^{2}\right )} {\left (b x^{3} + a\right )}^{p}}{3 \, {\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 94, normalized size = 1.96 \[ \frac {{\left (b x^{3} + a\right )}^{2} {\left (b x^{3} + a\right )}^{p} p - {\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p} a p + {\left (b x^{3} + a\right )}^{2} {\left (b x^{3} + a\right )}^{p} - 2 \, {\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p} a}{3 \, {\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 0.88 \[ -\frac {\left (-p b \,x^{3}-b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{p +1}}{3 \left (p^{2}+3 p +2\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 47, normalized size = 0.98 \[ \frac {{\left (b^{2} {\left (p + 1\right )} x^{6} + a b p x^{3} - a^{2}\right )} {\left (b x^{3} + a\right )}^{p}}{3 \, {\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 68, normalized size = 1.42 \[ {\left (b\,x^3+a\right )}^p\,\left (\frac {x^6\,\left (p+1\right )}{3\,\left (p^2+3\,p+2\right )}-\frac {a^2}{3\,b^2\,\left (p^2+3\,p+2\right )}+\frac {a\,p\,x^3}{3\,b\,\left (p^2+3\,p+2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.07, size = 524, normalized size = 10.92 \[ \begin {cases} \frac {a^{p} x^{6}}{6} & \text {for}\: b = 0 \\\frac {a \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {a \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac {2 a \log {\relax (2 )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {a}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {b x^{3} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {b x^{3} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac {2 b x^{3} \log {\relax (2 )}}{3 a b^{2} + 3 b^{3} x^{3}} & \text {for}\: p = -2 \\- \frac {a \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 b^{2}} - \frac {a \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 b^{2}} + \frac {x^{3}}{3 b} & \text {for}\: p = -1 \\- \frac {a^{2} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac {a b p x^{3} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac {b^{2} p x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac {b^{2} x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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