Optimal. Leaf size=69 \[ \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 b}{2 a^2 \sqrt {a+b x^2}}-\frac {1}{2 a x^2 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 0.99, 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, 51, 63, 208} \[ -\frac {3 \sqrt {a+b x^2}}{2 a^2 x^2}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {1}{a x^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{a x^2 \sqrt {a+b x^2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {1}{a x^2 \sqrt {a+b x^2}}-\frac {3 \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {1}{a x^2 \sqrt {a+b x^2}}-\frac {3 \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^2}\\ &=\frac {1}{a x^2 \sqrt {a+b x^2}}-\frac {3 \sqrt {a+b x^2}}{2 a^2 x^2}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.51 \[ -\frac {b \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x^2}{a}+1\right )}{a^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 171, normalized size = 2.48 \[ \left [\frac {3 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (3 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {3 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 72, normalized size = 1.04 \[ -\frac {3 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (b x^{2} + a\right )} b - 2 \, a b}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{2} + a} a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.91 \[ \frac {3 b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {3 b}{2 \sqrt {b \,x^{2}+a}\, a^{2}}-\frac {1}{2 \sqrt {b \,x^{2}+a}\, a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 51, normalized size = 0.74 \[ \frac {3 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {3 \, b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {1}{2 \, \sqrt {b x^{2} + a} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 53, normalized size = 0.77 \[ \frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {1}{2\,a\,x^2\,\sqrt {b\,x^2+a}}-\frac {3\,b}{2\,a^2\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.39, size = 73, normalized size = 1.06 \[ - \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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