Optimal. Leaf size=68 \[ \frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}} \]
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Rubi [A]
time = 0.03, 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 = {272, 43, 65,
214} \begin {gather*} \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {3}{8} b x^2 \sqrt {a+\frac {b}{x^2}}+\frac {1}{4} x^4 \left (a+\frac {b}{x^2}\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=\frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 69, normalized size = 1.01 \begin {gather*} \frac {1}{8} \sqrt {a+\frac {b}{x^2}} x \left (5 b x+2 a x^3-\frac {3 b^2 \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{\sqrt {a} \sqrt {b+a x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 84, normalized size = 1.24
method | result | size |
risch | \(\frac {x^{2} \left (2 a \,x^{2}+5 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{8}+\frac {3 b^{2} \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{8 \sqrt {a}\, \sqrt {a \,x^{2}+b}}\) | \(77\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (2 x \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}+3 \sqrt {a}\, \sqrt {a \,x^{2}+b}\, b x +3 \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) b^{2}\right )}{8 \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 98, normalized size = 1.44 \begin {gather*} -\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x^{2}}} a b^{2}}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} a + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 157, normalized size = 2.31 \begin {gather*} \left [\frac {3 \, \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 ) + 2 \, {\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.73, size = 70, normalized size = 1.03 \begin {gather*} \frac {a \sqrt {b} x^{3} \sqrt {\frac {a x^{2}}{b} + 1}}{4} + \frac {5 b^{\frac {3}{2}} x \sqrt {\frac {a x^{2}}{b} + 1}}{8} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{8 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.67, size = 68, normalized size = 1.00 \begin {gather*} -\frac {3 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, \sqrt {a}} + \frac {1}{8} \, {\left (2 \, a x^{2} \mathrm {sgn}\left (x\right ) + 5 \, b \mathrm {sgn}\left (x\right )\right )} \sqrt {a x^{2} + b} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.65, size = 52, normalized size = 0.76 \begin {gather*} \frac {5\,x^4\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{8}+\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {3\,a\,x^4\,\sqrt {a+\frac {b}{x^2}}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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