Optimal. Leaf size=211 \[ -\frac {a^2 \sin ^{-1}(a+b x)^3}{2 b^2}-\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {6 a \sqrt {1-(a+b x)^2}}{b^2}-\frac {\sin ^{-1}(a+b x)^3}{4 b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}-\frac {3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}+\frac {6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}+\frac {3 \sin ^{-1}(a+b x)}{8 b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.31, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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.
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Rule 8
Rule 30
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rule 4743
Rule 4773
Rule 4805
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*}
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Mathematica [A] time = 0.17, 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.
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fricas [A] time = 0.42, size = 108, normalized size = 0.51 \[ \frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (2 \, {\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - b x + 15 \, a\right )}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 203, normalized size = 0.96 \[ -\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{3}}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{3}}{2 \, b^{2}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac {6 \, {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {\arcsin \left (b x + a\right )^{3}}{4 \, b^{2}} - \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{8 \, b^{2}} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac {3 \, \arcsin \left (b x + a\right )}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 185, normalized size = 0.88 \[ \frac {\frac {\arcsin \left (b x +a \right )^{3} \left (-1+\left (b x +a \right )^{2}\right )}{2}+\frac {3 \arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{4}-\frac {3 \arcsin \left (b x +a \right ) \left (-1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsin \left (b x +a \right )}{8}-\frac {\arcsin \left (b x +a \right )^{3}}{2}-a \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arcsin \left (b x +a \right )\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{3} + 3 \, b \int \frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\mathrm {asin}\left (a+b\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 248, normalized size = 1.18 \[ \begin {cases} - \frac {a^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2 b^{2}} + \frac {21 a^{2} \operatorname {asin}{\left (a + b x \right )}}{4 b^{2}} + \frac {9 a x \operatorname {asin}{\left (a + b x \right )}}{2 b} - \frac {9 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b} - \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asin}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}^{3}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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