Optimal. Leaf size=211 \[ -\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}-\frac{3 (a+b x) \sqrt{1-(a+b x)^2}}{8 b^2}+\frac{6 a \sqrt{1-(a+b x)^2}}{b^2}-\frac{\sin ^{-1}(a+b x)^3}{4 b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}+\frac{6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}+\frac{3 \sin ^{-1}(a+b x)}{8 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.308878, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4805, 4743, 4773, 3317, 3296, 2638, 3311, 30, 2635, 8} \[ -\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}-\frac{3 (a+b x) \sqrt{1-(a+b x)^2}}{8 b^2}+\frac{6 a \sqrt{1-(a+b x)^2}}{b^2}-\frac{\sin ^{-1}(a+b x)^3}{4 b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}+\frac{6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}+\frac{3 \sin ^{-1}(a+b x)}{8 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4805
Rule 4743
Rule 4773
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \sin ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right ) \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3-\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^2 \sin ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3-\frac{3}{2} \operatorname{Subst}\left (\int x^2 \left (-\frac{a}{b}+\frac{\sin (x)}{b}\right )^2 \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3-\frac{3}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 x^2}{b^2}-\frac{2 a x^2 \sin (x)}{b^2}+\frac{x^2 \sin ^2(x)}{b^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3-\frac{3 \operatorname{Subst}\left (\int x^2 \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}+\frac{(3 a) \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3-\frac{3 \operatorname{Subst}\left (\int x^2 \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}+\frac{3 \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}+\frac{(6 a) \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=-\frac{3 (a+b x) \sqrt{1-(a+b x)^2}}{8 b^2}+\frac{6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{\sin ^{-1}(a+b x)^3}{4 b^2}-\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3+\frac{3 \operatorname{Subst}\left (\int 1 \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^2}-\frac{(6 a) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{6 a \sqrt{1-(a+b x)^2}}{b^2}-\frac{3 (a+b x) \sqrt{1-(a+b x)^2}}{8 b^2}+\frac{3 \sin ^{-1}(a+b x)}{8 b^2}+\frac{6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{\sin ^{-1}(a+b x)^3}{4 b^2}-\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3\\ \end{align*}
Mathematica [A] time = 0.141906, size = 135, normalized size = 0.64 \[ \frac{3 (15 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+\left (-4 a^2+4 b^2 x^2-2\right ) \sin ^{-1}(a+b x)^3-6 (3 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)^2+\left (42 a^2+36 a b x-6 b^2 x^2+3\right ) \sin ^{-1}(a+b x)}{8 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 185, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{3} \left ( -1+ \left ( bx+a \right ) ^{2} \right ) }{2}}+{\frac{3\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}{4} \left ( \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}}+\arcsin \left ( bx+a \right ) \right ) }-{\frac{3\,\arcsin \left ( bx+a \right ) \left ( -1+ \left ( bx+a \right ) ^{2} \right ) }{4}}-{\frac{3\,bx+3\,a}{8}\sqrt{1- \left ( bx+a \right ) ^{2}}}-{\frac{3\,\arcsin \left ( bx+a \right ) }{8}}-{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}}{2}}-a \left ( \left ( \arcsin \left ( bx+a \right ) \right ) ^{3} \left ( bx+a \right ) +3\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{1- \left ( bx+a \right ) ^{2}}-6\,\sqrt{1- \left ( bx+a \right ) ^{2}}-6\, \left ( bx+a \right ) \arcsin \left ( bx+a \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.90174, size = 266, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{3} - 3 \,{\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (2 \,{\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - b x + 15 \, a\right )}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.59027, size = 248, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{asin}^{3}{\left (a + b x \right )}}{2 b^{2}} + \frac{21 a^{2} \operatorname{asin}{\left (a + b x \right )}}{4 b^{2}} + \frac{9 a x \operatorname{asin}{\left (a + b x \right )}}{2 b} - \frac{9 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac{45 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b^{2}} + \frac{x^{2} \operatorname{asin}^{3}{\left (a + b x \right )}}{2} - \frac{3 x^{2} \operatorname{asin}{\left (a + b x \right )}}{4} + \frac{3 x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4 b} - \frac{3 x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b} - \frac{\operatorname{asin}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac{3 \operatorname{asin}{\left (a + b x \right )}}{8 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{asin}^{3}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19873, size = 274, normalized size = 1.3 \begin{align*} -\frac{{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{3}}{b^{2}} + \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{3}}{2 \, b^{2}} + \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac{6 \,{\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{2}} + \frac{\arcsin \left (b x + a\right )^{3}}{4 \, b^{2}} - \frac{3 \,{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} - \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )}}{8 \, b^{2}} + \frac{6 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac{3 \, \arcsin \left (b x + a\right )}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]