Optimal. Leaf size=83 \[ \frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {320, 290, 342,
202} \begin {gather*} \frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 290
Rule 320
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}} \, dx &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}-\frac {1}{2} a \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}-\frac {\left (a \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\left (a \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 56, normalized size = 0.67 \begin {gather*} \frac {2 x \sqrt {c x} \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 )}{3 \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 44, normalized size = 0.53 \begin {gather*} \frac {\sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c\,x}}{{\left (b\,x^2+a\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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