Optimal. Leaf size=65 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}+\frac {3}{8} a x \sqrt {a+b x^2}+\frac {1}{4} x \left (a+b x^2\right )^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}+\frac {3}{8} a x \sqrt {a+b x^2}+\frac {1}{4} x \left (a+b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{3/2} \, dx &=\frac {1}{4} x \left (a+b x^2\right )^{3/2}+\frac {1}{4} (3 a) \int \sqrt {a+b x^2} \, dx\\ &=\frac {3}{8} a x \sqrt {a+b x^2}+\frac {1}{4} x \left (a+b x^2\right )^{3/2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {3}{8} a x \sqrt {a+b x^2}+\frac {1}{4} x \left (a+b x^2\right )^{3/2}+\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {3}{8} a x \sqrt {a+b x^2}+\frac {1}{4} x \left (a+b x^2\right )^{3/2}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 65, normalized size = 1.00 \[ \frac {1}{8} \sqrt {a+b x^2} \left (\frac {3 a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\frac {b x^2}{a}+1}}+5 a x+2 b x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 124, normalized size = 1.91 \[ \left [\frac {3 \, a^{2} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, b^{2} x^{3} + 5 \, a b x\right )} \sqrt {b x^{2} + a}}{16 \, b}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b^{2} x^{3} + 5 \, a b x\right )} \sqrt {b x^{2} + a}}{8 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 49, normalized size = 0.75 \[ \frac {1}{8} \, {\left (2 \, b x^{2} + 5 \, a\right )} \sqrt {b x^{2} + a} x - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 51, normalized size = 0.78 \[ \frac {3 a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}+\frac {3 \sqrt {b \,x^{2}+a}\, a x}{8}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} x}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 43, normalized size = 0.66 \[ \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {b x^{2} + a} a x + \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.61, size = 37, normalized size = 0.57 \[ \frac {x\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 70, normalized size = 1.08 \[ \frac {5 a^{\frac {3}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{8} + \frac {\sqrt {a} b x^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{4} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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