Optimal. Leaf size=140 \[ -\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}+\frac {\sqrt [4]{2} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{3^{3/4} a c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {1}{3 a c \sqrt {3 a-2 a x^2} \sqrt {c x}} \]
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Rubi [A] time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {290, 325, 320, 319, 318, 424} \[ -\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}+\frac {\sqrt [4]{2} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{3^{3/4} a c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {1}{3 a c \sqrt {3 a-2 a x^2} \sqrt {c x}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 318
Rule 319
Rule 320
Rule 325
Rule 424
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}+\frac {\int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx}{2 a}\\ &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}-\frac {\int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx}{3 a c^2}\\ &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}-\frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}} \, dx}{3 a c^2 \sqrt {x}}\\ &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}-\frac {\left (\sqrt {c x} \sqrt {1-\frac {2 x^2}{3}}\right ) \int \frac {\sqrt {x}}{\sqrt {1-\frac {2 x^2}{3}}} \, dx}{3 a c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}+\frac {\left (\sqrt [4]{\frac {2}{3}} \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 )}{a c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=\frac {1}{3 a c \sqrt {c x} \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3 a-2 a x^2}}{3 a^2 c \sqrt {c x}}+\frac {\sqrt [4]{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 )}{3^{3/4} a c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 58, normalized size = 0.41 \[ -\frac {2 x \left (3-2 x^2\right )^{3/2} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};\frac {2 x^2}{3}\right )}{3 \sqrt {3} \left (a \left (3-2 x^2\right )\right )^{3/2} (c x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{4 \, a^{2} c^{2} x^{6} - 12 \, a^{2} c^{2} x^{4} + 9 \, a^{2} c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 228, normalized size = 1.63 \[ -\frac {\sqrt {-\left (2 x^{2}-3\right ) a}\, \left (24 x^{2}+2 \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-24\right )}{36 \sqrt {c x}\, \left (2 x^{2}-3\right ) a^{2} c} \]
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 \[ \int \frac {1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.16, size = 54, normalized size = 0.39 \[ \frac {\sqrt {3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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