Optimal. Leaf size=135 \[ -\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}-\frac {x^3 \sqrt {a-b x^4}}{5 b} \]
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Rubi [A] time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {321, 307, 224, 221, 1200, 1199, 424} \[ -\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}-\frac {x^3 \sqrt {a-b x^4}}{5 b} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 307
Rule 321
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {a-b x^4}} \, dx &=-\frac {x^3 \sqrt {a-b x^4}}{5 b}+\frac {(3 a) \int \frac {x^2}{\sqrt {a-b x^4}} \, dx}{5 b}\\ &=-\frac {x^3 \sqrt {a-b x^4}}{5 b}-\frac {\left (3 a^{3/2}\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{5 b^{3/2}}+\frac {\left (3 a^{3/2}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a-b x^4}} \, dx}{5 b^{3/2}}\\ &=-\frac {x^3 \sqrt {a-b x^4}}{5 b}-\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}}+\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}}\\ &=-\frac {x^3 \sqrt {a-b x^4}}{5 b}-\frac {3 a^{7/4} \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 )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}}\\ &=-\frac {x^3 \sqrt {a-b x^4}}{5 b}+\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}-\frac {3 a^{7/4} \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 )}{5 b^{7/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 66, normalized size = 0.49 \[ \frac {x^3 \left (a \sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {b x^4}{a}\right )-a+b x^4\right )}{5 b \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a} x^{6}}{b x^{4} - a}, x\right ) \]
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 \[ \int \frac {x^{6}}{\sqrt {-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 107, normalized size = 0.79 \[ -\frac {\sqrt {-b \,x^{4}+a}\, x^{3}}{5 b}-\frac {3 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {3}{2}}}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, b^{\frac {3}{2}}} \]
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 \[ \int \frac {x^{6}}{\sqrt {-b x^{4} + a}}\,{d x} \]
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 \[ \int \frac {x^6}{\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.03, size = 39, normalized size = 0.29 \[ \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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