Optimal. Leaf size=94 \[ \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{2 \sqrt [4]{6} a \sqrt {a \left (3-2 x^2\right )}} \]
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Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {288, 329, 224, 221} \[ \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^{3/2} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{2 \sqrt [4]{6} a \sqrt {a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 288
Rule 329
Rubi steps
\begin {align*} \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^2 \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx}{4 a}\\ &=\frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{2 a}\\ &=\frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {\left (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 )}{2 \sqrt {3} a \sqrt {a \left (3-2 x^2\right )}}\\ &=\frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {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 )}{2 \sqrt [4]{6} a \sqrt {a \left (3-2 x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.63 \[ -\frac {c \sqrt {c x} \left (\sqrt {9-6 x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {2 x^2}{3}\right )-3\right )}{6 a \sqrt {a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c x}{4 \, a^{2} x^{4} - 12 \, a^{2} x^{2} + 9 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 126, normalized size = 1.34 \[ \frac {\sqrt {c x}\, \sqrt {-\left (2 x^{2}-3\right ) a}\, \left (-12 x +\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right ) c}{24 \left (2 x^{2}-3\right ) a^{2} x} \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{3/2}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 51, normalized size = 0.54 \[ \frac {\sqrt {3} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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