Optimal. Leaf size=92 \[ -\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{32 \sqrt {b}}-\frac {5 a^2 \sqrt {a+\frac {b}{x^4}}}{32 x^2}-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {335, 275, 195, 217, 206} \[ -\frac {5 a^2 \sqrt {a+\frac {b}{x^4}}}{32 x^2}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{32 \sqrt {b}}-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 275
Rule 335
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x^3} \, dx &=-\operatorname {Subst}\left (\int x \left (a+b x^4\right )^{5/2} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b x^2\right )^{5/2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2}-\frac {1}{12} (5 a) \operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2}-\frac {1}{16} \left (5 a^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x^4}}}{32 x^2}-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2}-\frac {1}{32} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x^4}}}{32 x^2}-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2}-\frac {1}{32} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^4}} x^2}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x^4}}}{32 x^2}-\frac {5 a \left (a+\frac {b}{x^4}\right )^{3/2}}{48 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{12 x^2}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right )}{32 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 96, normalized size = 1.04 \[ -\frac {\sqrt {a+\frac {b}{x^4}} \left (15 a^3 x^{12} \sqrt {\frac {a x^4}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {a x^4}{b}+1}\right )+33 a^3 x^{12}+59 a^2 b x^8+34 a b^2 x^4+8 b^3\right )}{96 x^{10} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 182, normalized size = 1.98 \[ \left [\frac {15 \, a^{3} \sqrt {b} x^{10} \log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) - 2 \, {\left (33 \, a^{2} b x^{8} + 26 \, a b^{2} x^{4} + 8 \, b^{3}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{192 \, b x^{10}}, \frac {15 \, a^{3} \sqrt {-b} x^{10} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{b}\right ) - {\left (33 \, a^{2} b x^{8} + 26 \, a b^{2} x^{4} + 8 \, b^{3}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{96 \, b x^{10}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 0.95 \[ \frac {\frac {15 \, a^{4} \arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {33 \, {\left (a x^{4} + b\right )}^{\frac {5}{2}} a^{4} - 40 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}} a^{4} b + 15 \, \sqrt {a x^{4} + b} a^{4} b^{2}}{a^{3} x^{12}}}{96 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 113, normalized size = 1.23 \[ -\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} \left (15 a^{3} x^{12} \ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )+33 \sqrt {a \,x^{4}+b}\, a^{2} \sqrt {b}\, x^{8}+26 \sqrt {a \,x^{4}+b}\, a \,b^{\frac {3}{2}} x^{4}+8 \sqrt {a \,x^{4}+b}\, b^{\frac {5}{2}}\right )}{96 \left (a \,x^{4}+b \right )^{\frac {5}{2}} \sqrt {b}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.99, size = 158, normalized size = 1.72 \[ \frac {5 \, a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right )}{64 \, \sqrt {b}} - \frac {33 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} a^{3} x^{10} - 40 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{3} b x^{6} + 15 \, \sqrt {a + \frac {b}{x^{4}}} a^{3} b^{2} x^{2}}{96 \, {\left ({\left (a + \frac {b}{x^{4}}\right )}^{3} x^{12} - 3 \, {\left (a + \frac {b}{x^{4}}\right )}^{2} b x^{8} + 3 \, {\left (a + \frac {b}{x^{4}}\right )} b^{2} x^{4} - b^{3}\right )}} \]
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 \[ \int \frac {{\left (a+\frac {b}{x^4}\right )}^{5/2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.23, size = 102, normalized size = 1.11 \[ - \frac {11 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{32 x^{2}} - \frac {13 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x^{4}}}}{48 x^{6}} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x^{4}}}}{12 x^{10}} - \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{32 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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