Optimal. Leaf size=261 \[ \frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{11/4} \sqrt {a+b x^4}}-\frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^4}}+\frac {7 a^2 x \sqrt {a+b x^4}}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {7 a x^3 \sqrt {a+b x^4}}{45 b^2}+\frac {x^7 \sqrt {a+b x^4}}{9 b} \]
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Rubi [A] time = 0.09, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {321, 305, 220, 1196} \[ \frac {7 a^2 x \sqrt {a+b x^4}}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{11/4} \sqrt {a+b x^4}}-\frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^4}}-\frac {7 a x^3 \sqrt {a+b x^4}}{45 b^2}+\frac {x^7 \sqrt {a+b x^4}}{9 b} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 321
Rule 1196
Rubi steps
\begin {align*} \int \frac {x^{10}}{\sqrt {a+b x^4}} \, dx &=\frac {x^7 \sqrt {a+b x^4}}{9 b}-\frac {(7 a) \int \frac {x^6}{\sqrt {a+b x^4}} \, dx}{9 b}\\ &=-\frac {7 a x^3 \sqrt {a+b x^4}}{45 b^2}+\frac {x^7 \sqrt {a+b x^4}}{9 b}+\frac {\left (7 a^2\right ) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{15 b^2}\\ &=-\frac {7 a x^3 \sqrt {a+b x^4}}{45 b^2}+\frac {x^7 \sqrt {a+b x^4}}{9 b}+\frac {\left (7 a^{5/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{15 b^{5/2}}-\frac {\left (7 a^{5/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 b^{5/2}}\\ &=-\frac {7 a x^3 \sqrt {a+b x^4}}{45 b^2}+\frac {x^7 \sqrt {a+b x^4}}{9 b}+\frac {7 a^2 x \sqrt {a+b x^4}}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^4}}+\frac {7 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{11/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 0.31 \[ \frac {x^3 \left (7 a^2 \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )-7 a^2-2 a b x^4+5 b^2 x^8\right )}{45 b^2 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{10}}{\sqrt {b x^{4} + a}}, x\right ) \]
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 \[ \int \frac {x^{10}}{\sqrt {b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 133, normalized size = 0.51 \[ \frac {\sqrt {b \,x^{4}+a}\, x^{7}}{9 b}-\frac {7 \sqrt {b \,x^{4}+a}\, a \,x^{3}}{45 b^{2}}+\frac {7 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {5}{2}}}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{10}}{\sqrt {b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{10}}{\sqrt {b\,x^4+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.38, size = 37, normalized size = 0.14 \[ \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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