Optimal. Leaf size=56 \[ \frac {4 \sqrt [4]{c x} \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 56, 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} \[ \frac {4 \sqrt [4]{c x} \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/4} \sqrt [4]{a+b x^2}} \, dx &=\frac {\sqrt [4]{1+\frac {b x^2}{a}} \int \frac {1}{(c x)^{3/4} \sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {4 \sqrt [4]{c x} \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 0.96 \[ \frac {4 x \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {b x^2}{a}\right )}{(c x)^{3/4} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {1}{4}}}{b c x^{3} + a c x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {1}{\left (c x \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{3/4}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.09, size = 44, normalized size = 0.79 \[ \frac {\sqrt [4]{x} \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, \frac {1}{4} \\ \frac {9}{8} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} c^{\frac {3}{4}} \Gamma \left (\frac {9}{8}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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