Optimal. Leaf size=77 \[ -\frac {x \sqrt {a-b x^4}}{3 b}+\frac {a^{5/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 b^{5/4} \sqrt {a-b x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {327, 230, 227}
\begin {gather*} \frac {a^{5/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 b^{5/4} \sqrt {a-b x^4}}-\frac {x \sqrt {a-b x^4}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a-b x^4}} \, dx &=-\frac {x \sqrt {a-b x^4}}{3 b}+\frac {a \int \frac {1}{\sqrt {a-b x^4}} \, dx}{3 b}\\ &=-\frac {x \sqrt {a-b x^4}}{3 b}+\frac {\left (a \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{3 b \sqrt {a-b x^4}}\\ &=-\frac {x \sqrt {a-b x^4}}{3 b}+\frac {a^{5/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 b^{5/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 64, normalized size = 0.83 \begin {gather*} \frac {x \left (-a+b x^4+a \sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b x^4}{a}\right )\right )}{3 b \sqrt {a-b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 86, normalized size = 1.12
method | result | size |
default | \(-\frac {x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {a \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(86\) |
risch | \(-\frac {x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {a \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(86\) |
elliptic | \(-\frac {x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {a \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(86\) |
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 [A]
time = 0.08, size = 46, normalized size = 0.60 \begin {gather*} \frac {\sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {-b x^{4} + a} x}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 39, normalized size = 0.51 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{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 {x^4}{\sqrt {a-b\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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