Optimal. Leaf size=68 \[ -\frac {2 \sqrt {b} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\sqrt {a} c^2 \left (a-b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {329, 237, 335, 275, 232} \[ -\frac {2 \sqrt {b} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 232
Rule 237
Rule 275
Rule 329
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x} \left (a-b x^2\right )^{3/4}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{c}\\ &=\frac {\left (2 \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {c x}\right )}{c \left (a-b x^2\right )^{3/4}}\\ &=-\frac {\left (2 \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{c \left (a-b x^2\right )^{3/4}}\\ &=-\frac {\left (\left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{c x}\right )}{c \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt {b} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \left (a-b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 0.81 \[ \frac {2 x \left (1-\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {b x^2}{a}\right )}{\sqrt {c x} \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{b c x^{3} - a c x}, x\right ) \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x}\, \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c\,x}\,{\left (a-b\,x^2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.89, size = 32, normalized size = 0.47 \[ \frac {i e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{b^{\frac {3}{4}} \sqrt {c} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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