Optimal. Leaf size=178 \[ -\frac {4 a b^2 x}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac {4 b^3 x \sin ^{-1}(c x)}{3 c^2}+\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4627, 4707, 4677, 4619, 261, 266, 43} \[ -\frac {4 a b^2 x}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3+\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}-\frac {4 b^3 x \sin ^{-1}(c x)}{3 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}(c x)\right )^3 \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-(b c) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac {1}{3} \left (2 b^2\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {(2 b) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{3 c}\\ &=-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac {\left (4 b^2\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^2}+\frac {1}{9} \left (2 b^3 c\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {4 a b^2 x}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac {\left (4 b^3\right ) \int \sin ^{-1}(c x) \, dx}{3 c^2}+\frac {1}{9} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {4 a b^2 x}{3 c^2}-\frac {4 b^3 x \sin ^{-1}(c x)}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3+\frac {\left (4 b^3\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c}+\frac {1}{9} \left (b^3 c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {4 a b^2 x}{3 c^2}-\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}+\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {4 b^3 x \sin ^{-1}(c x)}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.38, size = 163, normalized size = 0.92 \[ \frac {1}{27} \left (\frac {b \left (9 c^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+18 \left (\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )-2 b \left (3 c^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )\right )\right )}{c^3}+9 x^3 \left (a+b \sin ^{-1}(c x)\right )^3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.77, size = 194, normalized size = 1.09 \[ \frac {9 \, b^{3} c^{3} x^{3} \arcsin \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arcsin \left (c x\right )^{2} + 3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arcsin \left (c x\right ) + {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.59, size = 368, normalized size = 2.07 \[ \frac {1}{3} \, a^{3} x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} + \frac {a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} \arcsin \left (c x\right )^{2}}{3 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b^{2} x}{9 \, c^{2}} + \frac {a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac {14 \, b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b^{2} \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} \arcsin \left (c x\right )^{2}}{c^{3}} - \frac {14 \, a b^{2} x}{9 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} b}{3 \, c^{3}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3}}{27 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arcsin \left (c x\right )}{c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b}{c^{3}} - \frac {14 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 235, normalized size = 1.32 \[ \frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arcsin \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{9}-\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 273, normalized size = 1.53 \[ \frac {1}{3} \, b^{3} x^{3} \arcsin \left (c x\right )^{3} + a b^{2} x^{3} \arcsin \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b + \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} + \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.49, size = 328, normalized size = 1.84 \[ \begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {asin}{\left (c x \right )} + \frac {a^{2} b x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )} - \frac {2 a b^{2} x^{3}}{9} + \frac {2 a b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c} - \frac {4 a b^{2} x}{3 c^{2}} + \frac {4 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {asin}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3 c} - \frac {2 b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{27 c} - \frac {4 b^{3} x \operatorname {asin}{\left (c x \right )}}{3 c^{2}} + \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3 c^{3}} - \frac {40 b^{3} \sqrt {- c^{2} x^{2} + 1}}{27 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________