Optimal. Leaf size=100 \[ \frac {5 a^{9/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt {a-b x^4}}-\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b} \]
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Rubi [A] time = 0.03, antiderivative size = 100, 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 = {321, 224, 221} \[ \frac {5 a^{9/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt {a-b x^4}}-\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 321
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {a-b x^4}} \, dx &=-\frac {x^5 \sqrt {a-b x^4}}{7 b}+\frac {(5 a) \int \frac {x^4}{\sqrt {a-b x^4}} \, dx}{7 b}\\ &=-\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{21 b^2}\\ &=-\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b}+\frac {\left (5 a^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 b^2 \sqrt {a-b x^4}}\\ &=-\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b}+\frac {5 a^{9/4} \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 )}{21 b^{9/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 80, normalized size = 0.80 \[ \frac {5 a^2 x \sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b x^4}{a}\right )-5 a^2 x+2 a b x^5+3 b^2 x^9}{21 b^2 \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a} x^{8}}{b x^{4} - a}, x\right ) \]
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 \[ \int \frac {x^{8}}{\sqrt {-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 107, normalized size = 1.07 \[ -\frac {\sqrt {-b \,x^{4}+a}\, x^{5}}{7 b}+\frac {5 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, b^{2}}-\frac {5 \sqrt {-b \,x^{4}+a}\, a x}{21 b^{2}} \]
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 \[ \int \frac {x^{8}}{\sqrt {-b x^{4} + a}}\,{d x} \]
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 \[ \int \frac {x^8}{\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 39, normalized size = 0.39 \[ \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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