Optimal. Leaf size=107 \[ -\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}+\frac {2 \sqrt [4]{2} \sqrt {c x} \sqrt {3-2 x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 107, 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 = {331, 326, 325,
324, 435} \begin {gather*} \frac {2 \sqrt [4]{2} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 324
Rule 325
Rule 326
Rule 331
Rule 435
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx &=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}-\frac {2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx}{3 c^2}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}-\frac {\left (2 \sqrt {c x}\right ) \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}} \, dx}{3 c^2 \sqrt {x}}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}-\frac {\left (2 \sqrt {c x} \sqrt {1-\frac {2 x^2}{3}}\right ) \int \frac {\sqrt {x}}{\sqrt {1-\frac {2 x^2}{3}}} \, dx}{3 c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}+\frac {\left (2 \sqrt [4]{\frac {2}{3}} \sqrt {c x} \sqrt {1-\frac {2 x^2}{3}}\right ) \text {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 )}{c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}+\frac {2 \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} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 51, normalized size = 0.48 \begin {gather*} -\frac {2 x \sqrt {3-2 x^2} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\frac {2 x^2}{3}\right )}{(c x)^{3/2} \sqrt {a \left (9-6 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(227\) vs.
\(2(85)=170\).
time = 0.08, size = 228, normalized size = 2.13
method | result | size |
risch | \(\frac {\frac {4 x^{2}}{3}-2}{c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}-\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) \sqrt {-c x a \left (2 x^{2}-3\right )}}{81 \sqrt {-2 a c \,x^{3}+3 a c x}\, c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(182\) |
elliptic | \(\frac {\sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (-\frac {2 \left (-2 c \,x^{2} a +3 a c \right )}{3 a \,c^{2} \sqrt {x \left (-2 c \,x^{2} a +3 a c \right )}}-\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{81 c \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(190\) |
default | \(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}-\sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}+24 x^{2}-36\right )}{18 c \sqrt {c x}\, a \left (2 x^{2}-3\right )}\) | \(228\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.33, size = 48, normalized size = 0.45 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} \sqrt {-a c} x {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, a c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.76, size = 54, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{3/2}\,\sqrt {3\,a-2\,a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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