Optimal. Leaf size=86 \[ -\frac {15}{8} b^2 \sqrt {a+\frac {b}{x^2}}+\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )+\frac {5}{8} b x^2 \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{4} x^4 \left (a+\frac {b}{x^2}\right )^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 86, 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 = {266, 47, 50, 63, 208} \[ -\frac {15}{8} b^2 \sqrt {a+\frac {b}{x^2}}+\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )+\frac {1}{4} x^4 \left (a+\frac {b}{x^2}\right )^{5/2}+\frac {5}{8} b x^2 \left (a+\frac {b}{x^2}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^{5/2} x^3 \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{5/2} x^4-\frac {1}{8} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {5}{8} b \left (a+\frac {b}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{5/2} x^4-\frac {1}{16} \left (15 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {15}{8} b^2 \sqrt {a+\frac {b}{x^2}}+\frac {5}{8} b \left (a+\frac {b}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{5/2} x^4-\frac {1}{16} \left (15 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {15}{8} b^2 \sqrt {a+\frac {b}{x^2}}+\frac {5}{8} b \left (a+\frac {b}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{5/2} x^4-\frac {1}{8} (15 a b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=-\frac {15}{8} b^2 \sqrt {a+\frac {b}{x^2}}+\frac {5}{8} b \left (a+\frac {b}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{5/2} x^4+\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 49, normalized size = 0.57 \[ -\frac {b^2 \sqrt {a+\frac {b}{x^2}} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-\frac {a x^2}{b}\right )}{\sqrt {\frac {a x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 157, normalized size = 1.83 \[ \left [\frac {15}{16} \, \sqrt {a} b^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + \frac {1}{8} \, {\left (2 \, a^{2} x^{4} + 9 \, a b x^{2} - 8 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}, -\frac {15}{8} \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + \frac {1}{8} \, {\left (2 \, a^{2} x^{4} + 9 \, a b x^{2} - 8 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 95, normalized size = 1.10 \[ -\frac {15}{16} \, \sqrt {a} b^{2} \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, \sqrt {a} b^{3} \mathrm {sgn}\relax (x)}{{\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b} + \frac {1}{8} \, {\left (2 \, a^{2} x^{2} \mathrm {sgn}\relax (x) + 9 \, a b \mathrm {sgn}\relax (x)\right )} \sqrt {a x^{2} + b} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 127, normalized size = 1.48 \[ \frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (15 a \,b^{3} x \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )+15 \sqrt {a \,x^{2}+b}\, a^{\frac {3}{2}} b^{2} x^{2}+10 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,x^{2}+8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{\frac {3}{2}} x^{2}-8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} \sqrt {a}\right ) x^{4}}{8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {a}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.80, size = 115, normalized size = 1.34 \[ -\frac {15}{16} \, \sqrt {a} b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \sqrt {a + \frac {b}{x^{2}}} b^{2} + \frac {9 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a b^{2} - 7 \, \sqrt {a + \frac {b}{x^{2}}} a^{2} b^{2}}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} a + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 72, normalized size = 0.84 \[ \frac {9\,a\,x^4\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{8}-b^2\,\sqrt {a+\frac {b}{x^2}}-\frac {7\,a^2\,x^4\,\sqrt {a+\frac {b}{x^2}}}{8}-\frac {\sqrt {a}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x^2}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,15{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.67, size = 117, normalized size = 1.36 \[ \frac {15 \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{8} + \frac {a^{3} x^{5}}{4 \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {11 a^{2} \sqrt {b} x^{3}}{8 \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {a b^{\frac {3}{2}} x}{8 \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {b^{\frac {5}{2}}}{x \sqrt {\frac {a x^{2}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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