Optimal. Leaf size=152 \[ \frac {5 b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 a^{13/4} \sqrt {a+\frac {b}{x^4}}}+\frac {5 x^3 \sqrt {a+\frac {b}{x^4}}}{4 a^3}-\frac {3 x^3}{4 a^2 \sqrt {a+\frac {b}{x^4}}}-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 152, 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 = {335, 290, 325, 220} \[ \frac {5 b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 a^{13/4} \sqrt {a+\frac {b}{x^4}}}+\frac {5 x^3 \sqrt {a+\frac {b}{x^4}}}{4 a^3}-\frac {3 x^3}{4 a^2 \sqrt {a+\frac {b}{x^4}}}-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right )^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {3 x^3}{4 a^2 \sqrt {a+\frac {b}{x^4}}}-\frac {15 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{4 a^2}\\ &=-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {3 x^3}{4 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {5 \sqrt {a+\frac {b}{x^4}} x^3}{4 a^3}+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{4 a^3}\\ &=-\frac {x^3}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {3 x^3}{4 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {5 \sqrt {a+\frac {b}{x^4}} x^3}{4 a^3}+\frac {5 b^{3/4} \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 )}{8 a^{13/4} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 94, normalized size = 0.62 \[ \frac {4 a^2 x^8-15 b \left (a x^4+b\right ) \sqrt {\frac {a x^4}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^4}{b}\right )+21 a b x^4+15 b^2}{12 a^3 x \sqrt {a+\frac {b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{14} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a^{3} x^{12} + 3 \, a^{2} b x^{8} + 3 \, a b^{2} x^{4} + b^{3}}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 304, normalized size = 2.00 \[ \frac {4 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{3} x^{13}+25 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} b \,x^{9}-15 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a^{2} b \,x^{8} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+36 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,b^{2} x^{5}-30 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a \,b^{2} x^{4} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+15 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{3} x -15 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, b^{3} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )}{12 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{3} x^{10}} \]
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 \[ \int \frac {x^{2}}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (a+\frac {b}{x^4}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.51, size = 42, normalized size = 0.28 \[ - \frac {x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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