Optimal. Leaf size=83 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {329, 240, 212, 208, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 240
Rule 329
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x} \sqrt [4]{a+b x^2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{c}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )}{c}\\ &=\operatorname {Subst}\left (\int \frac {1}{c-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{c+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{b} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 65, normalized size = 0.78 \[ \frac {\sqrt {x} \left (\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{\sqrt [4]{b} \sqrt {c x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 241, normalized size = 2.90 \[ -2 \, \left (\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} b c \left (\frac {1}{b c^{2}}\right )^{\frac {3}{4}} - {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {\sqrt {b x^{2} + a} c x + {\left (b c^{2} x^{2} + a c^{2}\right )} \sqrt {\frac {1}{b c^{2}}}}{b x^{2} + a}} \left (\frac {1}{b c^{2}}\right )^{\frac {3}{4}}}{b x^{2} + a}\right ) + \frac {1}{2} \, \left (\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} + {\left (b c x^{2} + a c\right )} \left (\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - \frac {1}{2} \, \left (\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} - {\left (b c x^{2} + a c\right )} \left (\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c x}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c\,x}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.51, size = 44, normalized size = 0.53 \[ \frac {\sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \sqrt {c} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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