Optimal. Leaf size=128 \[ \frac {a^{3/2} c^2 \sqrt {c x} \sqrt [4]{1-\frac {a}{b x^2}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a-b x^2}}-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 128, 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 = {321, 315, 317, 335, 228} \[ \frac {a^{3/2} c^2 \sqrt {c x} \sqrt [4]{1-\frac {a}{b x^2}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a-b x^2}}-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b} \]
Antiderivative was successfully verified.
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Rule 228
Rule 315
Rule 317
Rule 321
Rule 335
Rubi steps
\begin {align*} \int \frac {(c x)^{5/2}}{\sqrt [4]{a-b x^2}} \, dx &=-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b}+\frac {\left (a c^2\right ) \int \frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}} \, dx}{2 b}\\ &=-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b}-\frac {\left (a^2 c^4\right ) \int \frac {1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx}{4 b^2}\\ &=-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b}-\frac {\left (a^2 c^2 \sqrt [4]{1-\frac {a}{b x^2}} \sqrt {c x}\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^2}} x^2} \, dx}{4 b^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b}+\frac {\left (a^2 c^2 \sqrt [4]{1-\frac {a}{b x^2}} \sqrt {c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^2}{b}}} \, dx,x,\frac {1}{x}\right )}{4 b^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {a c^3 \left (a-b x^2\right )^{3/4}}{2 b^2 \sqrt {c x}}-\frac {c (c x)^{3/2} \left (a-b x^2\right )^{3/4}}{3 b}+\frac {a^{3/2} c^2 \sqrt [4]{1-\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 71, normalized size = 0.55 \[ \frac {c (c x)^{3/2} \left (a \sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {b x^2}{a}\right )-a+b x^2\right )}{3 b \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} c^{2} x^{2}}{b x^{2} - a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {5}{2}}}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{5/2}}{{\left (a-b\,x^2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.78, size = 46, normalized size = 0.36 \[ \frac {c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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