Optimal. Leaf size=56 \[ \frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac{2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \]
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Rubi [A] time = 0.0448986, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {368, 266, 43} \[ \frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac{2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \]
Antiderivative was successfully verified.
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Rule 368
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^5 \sqrt{a+b x^3} \, dx,x,\sqrt{c x^2}\right )}{c^3}\\ &=\frac{\operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 c^3}\\ &=-\frac{2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3}+\frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0213297, size = 43, normalized size = 0.77 \[ \frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2} \left (3 b \left (c x^2\right )^{3/2}-2 a\right )}{45 b^2 c^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{x}^{5}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941693, size = 58, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\left (c x^{2}\right )^{\frac{3}{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\left (c x^{2}\right )^{\frac{3}{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{45 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32915, size = 127, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (3 \, b^{2} c^{3} x^{6} + \sqrt{c x^{2}} a b c x^{2} - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{45 \, b^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15588, size = 51, normalized size = 0.91 \begin{align*} \frac{2 \,{\left (3 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}} a\right )}}{45 \, b^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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