Optimal. Leaf size=110 \[ \frac {3 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 110, 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 = {288, 320, 319, 318, 424} \[ \frac {3 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 318
Rule 319
Rule 320
Rule 424
Rubi steps
\begin {align*} \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {\left (3 c^2\right ) \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx}{4 a}\\ &=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {\left (3 c^2 \sqrt {c x}\right ) \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}} \, dx}{4 a \sqrt {x}}\\ &=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {\left (3 c^2 \sqrt {c x} \sqrt {1-\frac {2 x^2}{3}}\right ) \int \frac {\sqrt {x}}{\sqrt {1-\frac {2 x^2}{3}}} \, dx}{4 a \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}+\frac {\left (3 \left (\frac {3}{2}\right )^{3/4} c^2 \sqrt {c x} \sqrt {1-\frac {2 x^2}{3}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {\frac {2}{3}} x}}{\sqrt {2}}\right )}{2 a \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}+\frac {3 \sqrt [4]{3} c^2 \sqrt {c x} \sqrt {3-2 x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.54 \[ \frac {c (c x)^{3/2} \left (\sqrt {9-6 x^2} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};\frac {2 x^2}{3}\right )-3\right )}{3 a \sqrt {a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x^{2}}{4 \, a^{2} x^{4} - 12 \, a^{2} x^{2} + 9 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 230, normalized size = 2.09 \[ -\frac {\sqrt {c x}\, \sqrt {-\left (2 x^{2}-3\right ) a}\, \left (8 x^{2}+2 \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right ) c^{2}}{16 \left (2 x^{2}-3\right ) a^{2} x} \]
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 \[ \int \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{5/2}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.34, size = 51, normalized size = 0.46 \[ \frac {\sqrt {3} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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