Optimal. Leaf size=235 \[ -\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac {4}{3} a b^2 e^2 x+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d}-\frac {4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d} \]
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Rubi [A] time = 0.32, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4805, 12, 4627, 4707, 4677, 4619, 261, 266, 43} \[ -\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac {4}{3} a b^2 e^2 x+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d}-\frac {4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4707
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (2 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (4 b^2 e^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {4}{3} a b^2 e^2 x-\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac {\left (b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,(c+d x)^2\right )}{9 d}-\frac {\left (4 b^3 e^2\right ) \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d}-\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac {\left (b^3 e^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,(c+d x)^2\right )}{9 d}+\frac {\left (4 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4}{3} a b^2 e^2 x-\frac {14 b^3 e^2 \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d}-\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 199, normalized size = 0.85 \[ \frac {e^2 \left ((c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3-b \left (\frac {2}{3} b (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )-\sqrt {1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2-2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2+4 b \left (a d x+b \sqrt {1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+\frac {2}{9} b^2 \left (c^2+2 c d x+d^2 x^2+2\right ) \sqrt {1-(c+d x)^2}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 530, normalized size = 2.26 \[ \frac {3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} - 9 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \, {\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + 27 \, {\left (a b^{2} d^{3} e^{2} x^{3} + 3 \, a b^{2} c d^{2} e^{2} x^{2} + 3 \, a b^{2} c^{2} d e^{2} x + a b^{2} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{3} c - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right ) + {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d e^{2} x + {\left (18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} e^{2} + 9 \, {\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + {\left (b^{3} c^{2} + 2 \, b^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 18 \, {\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + {\left (a b^{2} c^{2} + 2 \, a b^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{27 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 485, normalized size = 2.06 \[ \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} + \frac {{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{3 \, d} + \frac {{\left (d x + c\right )}^{3} a^{3} e^{2}}{3 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right ) e^{2}}{d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} + \frac {{\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{2} \arcsin \left (d x + c\right ) e^{2}}{3 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac {{\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right ) e^{2}}{d} - \frac {14 \, {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{2} b e^{2}}{3 \, d} + \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} e^{2}}{27 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} \arcsin \left (d x + c\right ) e^{2}}{d} - \frac {14 \, {\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b e^{2}}{d} - \frac {14 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{2}}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 280, normalized size = 1.19 \[ \frac {\frac {e^{2} \left (d x +c \right )^{3} a^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \arcsin \left (d x +c \right ) \left (d x +c \right )^{3}}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\arcsin \left (d x +c \right )^{2} \left (d x +c \right )^{3}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+3 e^{2} a^{2} b \left (\frac {\arcsin \left (d x +c \right ) \left (d x +c \right )^{3}}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d} \]
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 \[ \frac {1}{3} \, a^{3} d^{2} e^{2} x^{3} + a^{3} c d e^{2} x^{2} + \frac {3}{2} \, {\left (2 \, x^{2} \arcsin \left (d x + c\right ) + d {\left (\frac {3 \, c^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c}{d^{3}}\right )}\right )} a^{2} b c d e^{2} + \frac {1}{6} \, {\left (6 \, x^{3} \arcsin \left (d x + c\right ) + d {\left (\frac {2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} - \frac {5 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} + \frac {15 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} a^{2} b d^{2} e^{2} + a^{3} c^{2} e^{2} 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 c^{2} e^{2}}{d} + \frac {1}{3} \, {\left (b^{3} d^{2} e^{2} x^{3} + 3 \, b^{3} c d e^{2} x^{2} + 3 \, b^{3} c^{2} e^{2} x\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + \int \frac {{\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x\right )} \sqrt {d x + c + 1} \sqrt {-d x - c + 1} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} + 3 \, {\left (a b^{2} d^{4} e^{2} x^{4} + 4 \, a b^{2} c d^{3} e^{2} x^{3} + {\left (6 \, a b^{2} c^{2} - a b^{2}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (2 \, a b^{2} c^{3} - a b^{2} c\right )} d e^{2} x + {\left (a b^{2} c^{4} - a b^{2} c^{2}\right )} e^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}}{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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.61, size = 1173, normalized size = 4.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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