Optimal. Leaf size=104 \[ -6 a b^2 x+\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \sin ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 261} \[ -6 a b^2 x+\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \sin ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=-6 a b^2 x+\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=-6 a b^2 x-\frac {6 b^3 (c+d x) \sin ^{-1}(c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=-6 a b^2 x-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \sin ^{-1}(c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 96, normalized size = 0.92 \[ \frac {-6 b^2 \left (a (c+d x)+b \sqrt {1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3+3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 158, normalized size = 1.52 \[ \frac {{\left (b^{3} d x + b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{3} - 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{2} b - 2 \, b^{3}\right )} d x + {\left (a^{2} b - 2 \, b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 3 \, {\left (b^{3} \arcsin \left (d x + c\right )^{2} + 2 \, a b^{2} \arcsin \left (d x + c\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 208, normalized size = 2.00 \[ \frac {{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {3 \, {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, {\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right )}{d} - \frac {6 \, {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )}{d} + \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{3}}{d} - \frac {6 \, {\left (d x + c\right )} a b^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b}{d} - \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 166, normalized size = 1.60 \[ \frac {\left (d x +c \right ) a^{3}+b^{3} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{3}+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d} \]
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 \[ b^{3} x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + a^{3} x + \frac {3 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a^{2} b}{d} + \int \frac {3 \, {\left (\sqrt {d x + c + 1} \sqrt {-d x - c + 1} b^{3} d x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} + {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - a b^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 152, normalized size = 1.46 \[ a^3\,x-\frac {b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d}+\frac {3\,a\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {3\,a^2\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 282, normalized size = 2.71 \[ \begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {asin}{\left (c + d x \right )} + \frac {3 a^{2} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {3 a b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 6 a b^{2} x + \frac {6 a b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {6 b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 6 b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {3 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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