Optimal. Leaf size=49 \[ \frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {327, 223, 212}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 327
Rubi steps
\begin {align*} \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 \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 51, normalized size = 1.04 \begin {gather*} \frac {x \sqrt {a+b x^2}}{2 b}+\frac {a \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 39, normalized size = 0.80
method | result | size |
default | \(\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\) | \(39\) |
risch | \(\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 31, normalized size = 0.63 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.06, size = 93, normalized size = 1.90 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.08, size = 42, normalized size = 0.86 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.48, size = 40, normalized size = 0.82 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.82, size = 56, normalized size = 1.14 \begin {gather*} \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 . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________