Optimal. Leaf size=104 \[ -\frac {4 a x \left (a-b x^2\right )^{3/4}}{15 b^2}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}+\frac {8 a^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {327, 235, 234}
\begin {gather*} \frac {8 a^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a-b x^2}}-\frac {4 a x \left (a-b x^2\right )^{3/4}}{15 b^2}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 235
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{a-b x^2}} \, dx &=-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}+\frac {(2 a) \int \frac {x^2}{\sqrt [4]{a-b x^2}} \, dx}{3 b}\\ &=-\frac {4 a x \left (a-b x^2\right )^{3/4}}{15 b^2}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}+\frac {\left (4 a^2\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{15 b^2}\\ &=-\frac {4 a x \left (a-b x^2\right )^{3/4}}{15 b^2}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}+\frac {\left (4 a^2 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{15 b^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {4 a x \left (a-b x^2\right )^{3/4}}{15 b^2}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}+\frac {8 a^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{5/2} \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 = 6.68, size = 79, normalized size = 0.76 \begin {gather*} \frac {2 \left (-6 a^2 x+a b x^3+5 b^2 x^5+6 a^2 x \sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )\right )}{45 b^2 \sqrt [4]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]
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 \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 = 0.48, size = 29, normalized size = 0.28 \begin {gather*} \frac {x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 \sqrt [4]{a}} \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 {x^4}{{\left (a-b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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