Optimal. Leaf size=59 \[ -\frac {4 \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {1}{8},\frac {7}{4};\frac {7}{8};-\frac {b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {365, 364} \[ -\frac {4 \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {1}{8},\frac {7}{4};\frac {7}{8};-\frac {b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/4} \left (a+b x^2\right )^{7/4}} \, dx &=\frac {\left (1+\frac {b x^2}{a}\right )^{3/4} \int \frac {1}{(c x)^{5/4} \left (1+\frac {b x^2}{a}\right )^{7/4}} \, dx}{a \left (a+b x^2\right )^{3/4}}\\ &=-\frac {4 \left (1+\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac {1}{8},\frac {7}{4};\frac {7}{8};-\frac {b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 0.97 \[ -\frac {4 x \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {1}{8},\frac {7}{4};\frac {7}{8};-\frac {b x^2}{a}\right )}{a (c x)^{5/4} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{4}}}{b^{2} c^{2} x^{6} + 2 \, a b c^{2} x^{4} + a^{2} c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (c x\right )^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {5}{4}} \left (b \,x^{2}+a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (c x\right )^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{5/4}\,{\left (b\,x^2+a\right )}^{7/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.45, size = 48, normalized size = 0.81 \[ \frac {\Gamma \left (- \frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{8}, \frac {7}{4} \\ \frac {7}{8} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} c^{\frac {5}{4}} \sqrt [4]{x} \Gamma \left (\frac {7}{8}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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