Optimal. Leaf size=68 \[ -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {3 b \sqrt {a+b x^2}}{8 x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 68, 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 = {266, 47, 63, 208} \begin {gather*} -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {3 b \sqrt {a+b x^2}}{8 x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 b \sqrt {a+b x^2}}{8 x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac {1}{16} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {3 b \sqrt {a+b x^2}}{8 x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {3 b \sqrt {a+b x^2}}{8 x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 x^4}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 1.12 \begin {gather*} -\frac {2 a^2+3 b^2 x^4 \sqrt {\frac {b x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )+7 a b x^2+5 b^2 x^4}{8 x^4 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 59, normalized size = 0.87 \begin {gather*} \frac {\left (-2 a-5 b x^2\right ) \sqrt {a+b x^2}}{8 x^4}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 136, normalized size = 2.00 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (5 \, a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{16 \, a x^{4}}, \frac {3 \, \sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (5 \, a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{8 \, a x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 70, normalized size = 1.03 \begin {gather*} \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} - 3 \, \sqrt {b x^{2} + a} a b^{3}}{b^{2} x^{4}}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 102, normalized size = 1.50 \begin {gather*} -\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+\frac {3 \sqrt {b \,x^{2}+a}\, b^{2}}{8 a}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}{8 a^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 90, normalized size = 1.32 \begin {gather*} -\frac {3 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 52, normalized size = 0.76 \begin {gather*} \frac {3\,a\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {5\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.98, size = 71, normalized size = 1.04 \begin {gather*} - \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{4 x^{3}} - \frac {5 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{8 x} - \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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