Optimal. Leaf size=45 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6715, 4619, 261} \[ \frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 4619
Rule 6715
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \sin ^{-1}(c x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 43, normalized size = 0.96 \[ \frac {a x^2}{2}+\frac {1}{2} b \left (\frac {\sqrt {1-c^2 x^4}}{c}+x^2 \sin ^{-1}\left (c x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 38, normalized size = 0.84 \[ \frac {b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + \sqrt {-c^{2} x^{4} + 1} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 38, normalized size = 0.84 \[ \frac {a c x^{2} + {\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 39, normalized size = 0.87 \[ \frac {a \,x^{2} c +b \left (x^{2} c \arcsin \left (c \,x^{2}\right )+\sqrt {-c^{2} x^{4}+1}\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, a x^{2} + \frac {{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 37, normalized size = 0.82 \[ \frac {a\,x^2}{2}+\frac {b\,\sqrt {1-c^2\,x^4}}{2\,c}+\frac {b\,x^2\,\mathrm {asin}\left (c\,x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.19, size = 42, normalized size = 0.93 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {asin}{\left (c x^{2} \right )}}{2} + \frac {b \sqrt {- c^{2} x^{4} + 1}}{2 c} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________