Optimal. Leaf size=272 \[ -\frac {4 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}+\frac {8 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}-\frac {8 a^2 \sqrt {b} \sqrt {a+\frac {b}{x^4}}}{3 x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}+x \left (a+\frac {b}{x^4}\right )^{5/2}-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {4 a b \sqrt {a+\frac {b}{x^4}}}{3 x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {242, 277, 279, 305, 220, 1196} \[ -\frac {8 a^2 \sqrt {b} \sqrt {a+\frac {b}{x^4}}}{3 x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}-\frac {4 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}+\frac {8 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}+x \left (a+\frac {b}{x^4}\right )^{5/2}-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {4 a b \sqrt {a+\frac {b}{x^4}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 242
Rule 277
Rule 279
Rule 305
Rule 1196
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^4}\right )^{5/2} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b x^4\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {b}{x^4}\right )^{5/2} x-(10 b) \operatorname {Subst}\left (\int x^2 \left (a+b x^4\right )^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}+\left (a+\frac {b}{x^4}\right )^{5/2} x-\frac {1}{3} (20 a b) \operatorname {Subst}\left (\int x^2 \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 a b \sqrt {a+\frac {b}{x^4}}}{3 x^3}-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}+\left (a+\frac {b}{x^4}\right )^{5/2} x-\frac {1}{3} \left (8 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 a b \sqrt {a+\frac {b}{x^4}}}{3 x^3}-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}+\left (a+\frac {b}{x^4}\right )^{5/2} x-\frac {1}{3} \left (8 a^{5/2} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \left (8 a^{5/2} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 a b \sqrt {a+\frac {b}{x^4}}}{3 x^3}-\frac {10 b \left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {8 a^2 \sqrt {b} \sqrt {a+\frac {b}{x^4}}}{3 \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}+\left (a+\frac {b}{x^4}\right )^{5/2} x+\frac {8 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}-\frac {4 a^{9/4} \sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.20 \[ -\frac {b^2 \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {5}{2},-\frac {9}{4};-\frac {5}{4};-\frac {a x^4}{b}\right )}{9 x^7 \sqrt {\frac {a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{x^{8}}, x\right ) \]
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 \[ \int {\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 251, normalized size = 0.92 \[ \frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} \left (-15 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{3} x^{12}-24 i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a^{\frac {5}{2}} \sqrt {b}\, x^{9} \EllipticE \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+24 i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a^{\frac {5}{2}} \sqrt {b}\, x^{9} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )-19 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} b \,x^{8}-5 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,b^{2} x^{4}-\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{3}\right ) x}{9 \left (a \,x^{4}+b \right )^{3} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}} \]
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 \[ \int {\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 38, normalized size = 0.14 \[ -\frac {x\,{\left (a+\frac {b}{x^4}\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {9}{4};\ -\frac {5}{4};\ -\frac {a\,x^4}{b}\right )}{9\,{\left (\frac {a\,x^4}{b}+1\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.98, size = 42, normalized size = 0.15 \[ - \frac {a^{\frac {5}{2}} x \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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