Optimal. Leaf size=117 \[ \frac {6^{3/4} a c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{7 \sqrt {a \left (3-2 x^2\right )}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac {2}{7} c \sqrt {3 a-2 a x^2} \sqrt {c x} \]
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Rubi [A] time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {279, 321, 329, 224, 221} \[ \frac {6^{3/4} a c^{3/2} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{7 \sqrt {a \left (3-2 x^2\right )}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac {2}{7} c \sqrt {3 a-2 a x^2} \sqrt {c x} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 279
Rule 321
Rule 329
Rubi steps
\begin {align*} \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx &=\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} (6 a) \int \frac {(c x)^{3/2}}{\sqrt {3 a-2 a x^2}} \, dx\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} \left (3 a c^2\right ) \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} (6 a c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {\left (2 \sqrt {3} a c \sqrt {3-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {2 x^4}{3 c^2}}} \, dx,x,\sqrt {c x}\right )}{7 \sqrt {a \left (3-2 x^2\right )}}\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {6^{3/4} a c^{3/2} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{7 \sqrt {a \left (3-2 x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 74, normalized size = 0.63 \[ \frac {c \sqrt {a \left (3-2 x^2\right )} \sqrt {c x} \left (3 \sqrt {3} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {2 x^2}{3}\right )-\left (3-2 x^2\right )^{3/2}\right )}{7 \sqrt {3-2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 133, normalized size = 1.14 \[ -\frac {\sqrt {c x}\, \sqrt {-\left (2 x^{2}-3\right ) a}\, \left (-8 x^{5}+20 x^{3}-12 x +\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right ) c}{14 \left (2 x^{2}-3\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,x\right )}^{3/2}\,\sqrt {3\,a-2\,a\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.73, size = 53, normalized size = 0.45 \[ \frac {\sqrt {3} \sqrt {a} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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