Optimal. Leaf size=282 \[ \frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}}-\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}-\frac {x^{11}}{2 b \sqrt {a+b x^4}} \]
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Rubi [A] time = 0.11, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {288, 321, 305, 220, 1196} \[ \frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}}-\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}-\frac {x^{11}}{2 b \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 288
Rule 305
Rule 321
Rule 1196
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}+\frac {11 \int \frac {x^{10}}{\sqrt {a+b x^4}} \, dx}{2 b}\\ &=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}-\frac {(77 a) \int \frac {x^6}{\sqrt {a+b x^4}} \, dx}{18 b^2}\\ &=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {\left (77 a^2\right ) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{30 b^3}\\ &=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {\left (77 a^{5/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{30 b^{7/2}}-\frac {\left (77 a^{5/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{30 b^{7/2}}\\ &=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {77 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 )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {77 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 )}{60 b^{15/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 0.28 \[ \frac {x^3 \left (-77 a^2 \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {b x^4}{a}\right )+77 a^2-11 a b x^4+5 b^2 x^8\right )}{45 b^3 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} x^{14}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{14}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 157, normalized size = 0.56 \[ \frac {\sqrt {b \,x^{4}+a}\, x^{7}}{9 b^{2}}-\frac {a^{2} x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, b^{3}}-\frac {16 \sqrt {b \,x^{4}+a}\, a \,x^{3}}{45 b^{3}}+\frac {77 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {5}{2}}}{30 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{14}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{14}}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.88, size = 37, normalized size = 0.13 \[ \frac {x^{15} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {19}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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