Optimal. Leaf size=113 \[ \frac{2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac{2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac{4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]
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Rubi [A] time = 0.0669988, antiderivative size = 113, 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} \[ \frac{2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac{2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac{4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 368
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx &=\frac{x^9 \operatorname{Subst}\left (\int x^8 \sqrt{a+b x^3} \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{9/2}}\\ &=\frac{x^9 \operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 \left (c x^2\right )^{9/2}}\\ &=\frac{x^9 \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 \left (c x^2\right )^{9/2}}\\ &=\frac{2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}-\frac{4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}}+\frac{2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0389263, size = 67, normalized size = 0.59 \[ \frac{2 x \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2} \left (8 a^2-12 a b \left (c x^2\right )^{3/2}+15 b^2 c^3 x^6\right )}{315 b^3 c^4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{x}^{8}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34476, size = 169, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (15 \, b^{3} c^{5} x^{10} - 4 \, a^{2} b c^{2} x^{4} +{\left (3 \, a b^{2} c^{3} x^{6} + 8 \, a^{3}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{315 \, b^{3} c^{5} x} \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^{8} \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 = 1.18691, size = 74, normalized size = 0.65 \begin{align*} \frac{2 \,{\left (15 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3} c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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