Optimal. Leaf size=49 \[ \frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5, 321, 217, 206} \[ \frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5
Rule 206
Rule 217
Rule 321
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {x^2}{\sqrt {a+b x^2}} \, dx\\ &=\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b}\\ &=\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b}\\ &=\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ \frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 93, normalized size = 1.90 \[ \left [\frac {2 \, \sqrt {b x^{2} + a} b x + a \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, b^{2}}, \frac {\sqrt {b x^{2} + a} b x + a \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{2 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 40, normalized size = 0.82 \[ \frac {\sqrt {b x^{2} + a} x}{2 \, b} + \frac {a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.80 \[ -\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, x}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 31, normalized size = 0.63 \[ \frac {\sqrt {b x^{2} + a} x}{2 \, b} - \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 56, normalized size = 1.14 \[ \left \{\begin {array}{cl} \frac {x^3}{3\,\sqrt {a}} & \text {\ if\ \ }b=0\\ \frac {x\,\sqrt {b\,x^2+a}}{2\,b}-\frac {a\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.91, size = 42, normalized size = 0.86 \[ \frac {\sqrt {a} x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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