Optimal. Leaf size=371 \[ -\frac{6 a^2 \sqrt{1-(a+b x)^2}}{b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}+\frac{3 a \sqrt{1-(a+b x)^2} (a+b x)}{4 b^3}+\frac{2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac{14 \sqrt{1-(a+b x)^2}}{9 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{\sqrt{1-(a+b x)^2} (a+b x)^2 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{3 a \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^2}{2 b^3}-\frac{4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}-\frac{3 a \sin ^{-1}(a+b x)}{4 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.45086, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {4805, 4743, 4773, 3317, 3296, 2638, 3311, 30, 2635, 8, 2633} \[ -\frac{6 a^2 \sqrt{1-(a+b x)^2}}{b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}+\frac{3 a \sqrt{1-(a+b x)^2} (a+b x)}{4 b^3}+\frac{2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac{14 \sqrt{1-(a+b x)^2}}{9 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{\sqrt{1-(a+b x)^2} (a+b x)^2 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{3 a \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^2}{2 b^3}-\frac{4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}-\frac{3 a \sin ^{-1}(a+b x)}{4 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4743
Rule 4773
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 30
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int x^2 \sin ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^2 \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3-\operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^3 \sin ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3-\operatorname{Subst}\left (\int x^2 \left (-\frac{a}{b}+\frac{\sin (x)}{b}\right )^3 \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3-\operatorname{Subst}\left (\int \left (-\frac{a^3 x^2}{b^3}+\frac{3 a^2 x^2 \sin (x)}{b^3}-\frac{3 a x^2 \sin ^2(x)}{b^3}+\frac{x^2 \sin ^3(x)}{b^3}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3-\frac{\operatorname{Subst}\left (\int x^2 \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}+\frac{(3 a) \operatorname{Subst}\left (\int x^2 \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac{3 a (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac{(a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3+\frac{2 \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{9 b^3}-\frac{2 \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}+\frac{(3 a) \operatorname{Subst}\left (\int x^2 \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac{(3 a) \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{3 a (a+b x) \sqrt{1-(a+b x)^2}}{4 b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac{3 a (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac{(a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3-\frac{2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt{1-(a+b x)^2}\right )}{9 b^3}-\frac{4 \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}-\frac{(3 a) \operatorname{Subst}\left (\int 1 \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac{2 \sqrt{1-(a+b x)^2}}{9 b^3}-\frac{6 a^2 \sqrt{1-(a+b x)^2}}{b^3}+\frac{3 a (a+b x) \sqrt{1-(a+b x)^2}}{4 b^3}+\frac{2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac{3 a \sin ^{-1}(a+b x)}{4 b^3}-\frac{4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac{3 a (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac{(a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3+\frac{4 \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}\\ &=-\frac{14 \sqrt{1-(a+b x)^2}}{9 b^3}-\frac{6 a^2 \sqrt{1-(a+b x)^2}}{b^3}+\frac{3 a (a+b x) \sqrt{1-(a+b x)^2}}{4 b^3}+\frac{2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac{3 a \sin ^{-1}(a+b x)}{4 b^3}-\frac{4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac{3 a (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac{(a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3\\ \end{align*}
Mathematica [A] time = 0.221958, size = 181, normalized size = 0.49 \[ \frac{-\sqrt{-a^2-2 a b x-b^2 x^2+1} \left (575 a^2-65 a b x+8 b^2 x^2+160\right )+18 \left (2 a^3+3 a+2 b^3 x^3\right ) \sin ^{-1}(a+b x)^3+18 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (11 a^2-5 a b x+2 b^2 x^2+4\right ) \sin ^{-1}(a+b x)^2-3 \left (132 a^2 b x+170 a^3+a \left (75-30 b^2 x^2\right )+8 b x \left (b^2 x^2+6\right )\right ) \sin ^{-1}(a+b x)}{108 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 344, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ( -{\frac{a}{4} \left ( 4\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3} \left ( bx+a \right ) ^{2}+6\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{1- \left ( bx+a \right ) ^{2}} \left ( bx+a \right ) -2\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}-6\,\arcsin \left ( bx+a \right ) \left ( bx+a \right ) ^{2}-3\, \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}}+3\,\arcsin \left ( bx+a \right ) \right ) }+{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{3} \left ( \left ( bx+a \right ) ^{2}-3 \right ) \left ( bx+a \right ) }{3}}+ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{1- \left ( bx+a \right ) ^{2}}-{\frac{14}{9}\sqrt{1- \left ( bx+a \right ) ^{2}}}-2\, \left ( bx+a \right ) \arcsin \left ( bx+a \right ) +{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( -1+ \left ( bx+a \right ) ^{2} \right ) }{3}\sqrt{1- \left ( bx+a \right ) ^{2}}}-{\frac{2\,\arcsin \left ( bx+a \right ) \left ( \left ( bx+a \right ) ^{2}-3 \right ) \left ( bx+a \right ) }{9}}-{\frac{-2+2\, \left ( bx+a \right ) ^{2}}{27}\sqrt{1- \left ( bx+a \right ) ^{2}}}+{a}^{2} \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 ) + \left ( \arcsin \left ( bx+a \right ) \right ) ^{3} \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.88483, size = 378, normalized size = 1.02 \begin{align*} \frac{18 \,{\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \,{\left (8 \, b^{3} x^{3} - 30 \, a b^{2} x^{2} + 170 \, a^{3} + 12 \,{\left (11 \, a^{2} + 4\right )} b x + 75 \, a\right )} \arcsin \left (b x + a\right ) -{\left (8 \, b^{2} x^{2} - 65 \, a b x - 18 \,{\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \arcsin \left (b x + a\right )^{2} + 575 \, a^{2} + 160\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{108 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.00306, size = 432, normalized size = 1.16 \begin{align*} \begin{cases} \frac{a^{3} \operatorname{asin}^{3}{\left (a + b x \right )}}{3 b^{3}} - \frac{85 a^{3} \operatorname{asin}{\left (a + b x \right )}}{18 b^{3}} - \frac{11 a^{2} x \operatorname{asin}{\left (a + b x \right )}}{3 b^{2}} + \frac{11 a^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{6 b^{3}} - \frac{575 a^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{3}} + \frac{5 a x^{2} \operatorname{asin}{\left (a + b x \right )}}{6 b} - \frac{5 a x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{6 b^{2}} + \frac{65 a x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{2}} + \frac{a \operatorname{asin}^{3}{\left (a + b x \right )}}{2 b^{3}} - \frac{25 a \operatorname{asin}{\left (a + b x \right )}}{12 b^{3}} + \frac{x^{3} \operatorname{asin}^{3}{\left (a + b x \right )}}{3} - \frac{2 x^{3} \operatorname{asin}{\left (a + b x \right )}}{9} + \frac{x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{3 b} - \frac{2 x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b} - \frac{4 x \operatorname{asin}{\left (a + b x \right )}}{3 b^{2}} + \frac{2 \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac{40 \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \operatorname{asin}^{3}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21234, size = 525, normalized size = 1.42 \begin{align*} \frac{{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{3}}{b^{3}} + \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )}{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{3}}{b^{3}} - \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} + \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac{6 \,{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{b^{3}} + \frac{{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac{a \arcsin \left (b x + a\right )^{3}}{2 \, b^{3}} - \frac{{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac{2 \,{\left ({\left (b x + a\right )}^{2} - 1\right )}{\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac{3 \,{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )}{2 \, b^{3}} + \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )} a}{4 \, b^{3}} - \frac{6 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac{14 \,{\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac{3 \, a \arcsin \left (b x + a\right )}{4 \, b^{3}} + \frac{2 \,{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}}}{27 \, b^{3}} - \frac{14 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}}{9 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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