Optimal. Leaf size=66 \[ \frac{2 \left (a+b x^2\right )^2 \left (c \sqrt{a+b x^2}\right )^{3/2}}{11 b^2}-\frac{2 a \left (a+b x^2\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{7 b^2} \]
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Rubi [A] time = 0.136821, antiderivative size = 74, normalized size of antiderivative = 1.12, 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 = {6720, 266, 43} \[ \frac{2 c \left (a+b x^2\right )^{5/2} \sqrt{c \sqrt{a+b x^2}}}{11 b^2}-\frac{2 a c \left (a+b x^2\right )^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (c \sqrt{a+b x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \int x^3 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int x (a+b x)^{3/4} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/4}}{b}+\frac{(a+b x)^{7/4}}{b}\right ) \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 a c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b^2}+\frac{2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{5/2}}{11 b^2}\\ \end{align*}
Mathematica [A] time = 0.0202312, size = 41, normalized size = 0.62 \[ \frac{2 \left (a+b x^2\right ) \left (7 b x^2-4 a\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{77 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,b{x}^{2}+2\,a \right ) \left ( -7\,b{x}^{2}+4\,a \right ) }{77\,{b}^{2}} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997918, size = 58, normalized size = 0.88 \begin{align*} -\frac{2 \,{\left (11 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{7}{2}} a c^{2} - 7 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{11}{2}}\right )}}{77 \, b^{2} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68915, size = 119, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (7 \, b^{2} c x^{4} + 3 \, a b c x^{2} - 4 \, a^{2} c\right )} \sqrt{b x^{2} + a} \sqrt{\sqrt{b x^{2} + a} c}}{77 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 52.6148, size = 87, normalized size = 1.32 \begin{align*} \begin{cases} - \frac{8 a^{2} c^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}}{77 b^{2}} + \frac{6 a c^{\frac{3}{2}} x^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}}{77 b} + \frac{2 c^{\frac{3}{2}} x^{4} \left (a + b x^{2}\right )^{\frac{3}{4}}}{11} & \text{for}\: b \neq 0 \\\frac{x^{4} \left (\sqrt{a} c\right )^{\frac{3}{2}}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23085, size = 43, normalized size = 0.65 \begin{align*} \frac{2 \,{\left (7 \,{\left (b x^{2} + a\right )}^{\frac{11}{4}} - 11 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} a\right )} c^{\frac{3}{2}}}{77 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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