Optimal. Leaf size=38 \[ \frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \]
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Rubi [A] time = 0.0238234, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \]
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 )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^{3/2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}\\ \end{align*}
Mathematica [A] time = 0.0143225, size = 28, normalized size = 0.74 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (5 b x^3-2 a\right )}{105 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-10\,b{x}^{3}+4\,a}{105\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26559, size = 41, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{2}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41838, size = 99, normalized size = 2.61 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{9} + 8 \, a b^{2} x^{6} + a^{2} b x^{3} - 2 \, a^{3}\right )} \sqrt{b x^{3} + a}}{105 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.3229, size = 88, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{4 a^{3} \sqrt{a + b x^{3}}}{105 b^{2}} + \frac{2 a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b} + \frac{16 a x^{6} \sqrt{a + b x^{3}}}{105} + \frac{2 b x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13056, size = 105, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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