Optimal. Leaf size=32 \[ \frac{c \left (a+b x^2\right )^4 \sqrt{c \left (a+b x^2\right )^3}}{11 b} \]
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Rubi [A] time = 0.0265127, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ \frac{c \left (a+b x^2\right )^4 \sqrt{c \left (a+b x^2\right )^3}}{11 b} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^3\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int x^{9/2} \, dx,x,a+b x^2\right )}{2 b \left (a+b x^2\right )^{3/2}}\\ &=\frac{c \left (a+b x^2\right )^4 \sqrt{c \left (a+b x^2\right )^3}}{11 b}\\ \end{align*}
Mathematica [A] time = 0.0176661, size = 29, normalized size = 0.91 \[ \frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^3\right )^{3/2}}{11 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.8 \begin{align*}{\frac{b{x}^{2}+a}{11\,b} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0748, size = 95, normalized size = 2.97 \begin{align*} \frac{{\left (b^{4} c^{\frac{3}{2}} x^{8} + 4 \, a b^{3} c^{\frac{3}{2}} x^{6} + 6 \, a^{2} b^{2} c^{\frac{3}{2}} x^{4} + 4 \, a^{3} b c^{\frac{3}{2}} x^{2} + a^{4} c^{\frac{3}{2}}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{11 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62584, size = 181, normalized size = 5.66 \begin{align*} \frac{{\left (b^{4} c x^{8} + 4 \, a b^{3} c x^{6} + 6 \, a^{2} b^{2} c x^{4} + 4 \, a^{3} b c x^{2} + a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{11 \, b} \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 \left (c \left (a + b x^{2}\right )^{3}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19807, size = 450, normalized size = 14.06 \begin{align*} \frac{1155 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{4} \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{924 \,{\left (5 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}}\right )} a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{c} + \frac{198 \,{\left (35 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{2} - 42 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a c + 15 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}}\right )} a^{2} \mathrm{sgn}\left (b x^{2} + a\right )}{c^{2}} - \frac{44 \,{\left (105 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}}\right )} a \mathrm{sgn}\left (b x^{2} + a\right )}{c^{3}} + \frac{{\left (1155 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{4} c^{4} - 2772 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{3} c^{3} + 2970 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a^{2} c^{2} - 1540 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}} a c + 315 \,{\left (b c x^{2} + a c\right )}^{\frac{11}{2}}\right )} \mathrm{sgn}\left (b x^{2} + a\right )}{c^{4}}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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