Optimal. Leaf size=134 \[ \frac {7 a^{5/2} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 b^{5/2} \sqrt [4]{a-b x^4}}-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b} \]
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Rubi [A] time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {321, 311, 313, 335, 275, 228} \[ -\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}+\frac {7 a^{5/2} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 b^{5/2} \sqrt [4]{a-b x^4}}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b} \]
Antiderivative was successfully verified.
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Rule 228
Rule 275
Rule 311
Rule 313
Rule 321
Rule 335
Rubi steps
\begin {align*} \int \frac {x^{10}}{\sqrt [4]{a-b x^4}} \, dx &=-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}+\frac {(7 a) \int \frac {x^6}{\sqrt [4]{a-b x^4}} \, dx}{10 b}\\ &=-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}+\frac {\left (7 a^2\right ) \int \frac {x^2}{\sqrt [4]{a-b x^4}} \, dx}{20 b^2}\\ &=-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}-\frac {\left (7 a^3\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{40 b^3}\\ &=-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}-\frac {\left (7 a^3 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3} \, dx}{40 b^3 \sqrt [4]{a-b x^4}}\\ &=-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}+\frac {\left (7 a^3 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,\frac {1}{x}\right )}{40 b^3 \sqrt [4]{a-b x^4}}\\ &=-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}+\frac {\left (7 a^3 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^2}{b}}} \, dx,x,\frac {1}{x^2}\right )}{80 b^3 \sqrt [4]{a-b x^4}}\\ &=-\frac {7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac {7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac {x^7 \left (a-b x^4\right )^{3/4}}{10 b}+\frac {7 a^{5/2} \sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 b^{5/2} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 0.60 \[ \frac {x^3 \left (7 a^2 \sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {b x^4}{a}\right )-7 a^2+a b x^4+6 b^2 x^8\right )}{60 b^2 \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{10}}{b x^{4} - a}, x\right ) \]
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 \[ \int \frac {x^{10}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {x^{10}}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {x^{10}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {x^{10}}{{\left (a-b\,x^4\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.46, size = 39, normalized size = 0.29 \[ \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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