Optimal. Leaf size=176 \[ \frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac {b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}-\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b^2 e^3 (c+d x)^2}{32 d} \]
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Rubi [A] time = 0.26, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4805, 12, 4627, 4707, 4641, 30} \[ \frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac {b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}-\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b^2 e^3 (c+d x)^2}{32 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 4627
Rule 4641
Rule 4707
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{16 d}+\frac {b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{16 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{16 d}\\ &=-\frac {3 b^2 e^3 (c+d x)^2}{32 d}-\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{16 d}+\frac {b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 142, normalized size = 0.81 \[ \frac {e^3 \left (\frac {1}{8} \left (-3 \left (-2 b \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )+\left (a+b \sin ^{-1}(c+d x)\right )^2+b^2 (c+d x)^2\right )+4 b \sqrt {1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )-b^2 (c+d x)^4\right )+(c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 442, normalized size = 2.51 \[ \frac {{\left (8 \, a^{2} - b^{2}\right )} d^{4} e^{3} x^{4} + 4 \, {\left (8 \, a^{2} - b^{2}\right )} c d^{3} e^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} - b^{2}\right )} c^{2} - b^{2}\right )} d^{2} e^{3} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} - b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d e^{3} x + {\left (8 \, b^{2} d^{4} e^{3} x^{4} + 32 \, b^{2} c d^{3} e^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{2} c^{3} d e^{3} x + {\left (8 \, b^{2} c^{4} - 3 \, b^{2}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (8 \, a b d^{4} e^{3} x^{4} + 32 \, a b c d^{3} e^{3} x^{3} + 48 \, a b c^{2} d^{2} e^{3} x^{2} + 32 \, a b c^{3} d e^{3} x + {\left (8 \, a b c^{4} - 3 \, a b\right )} e^{3}\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (2 \, a b d^{3} e^{3} x^{3} + 6 \, a b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b c^{2} + a b\right )} d e^{3} x + {\left (2 \, a b c^{3} + 3 \, a b c\right )} e^{3} + {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} + b^{2}\right )} d e^{3} x + {\left (2 \, b^{2} c^{3} + 3 \, b^{2} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 328, normalized size = 1.86 \[ \frac {{\left (d x + c\right )}^{4} a^{2} e^{3}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{4 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right ) e^{3}}{8 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b \arcsin \left (d x + c\right ) e^{3}}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{2 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b e^{3}}{8 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right ) e^{3}}{16 \, d} - \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{2} e^{3}}{32 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a b \arcsin \left (d x + c\right ) e^{3}}{d} + \frac {5 \, b^{2} \arcsin \left (d x + c\right )^{2} e^{3}}{32 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b e^{3}}{16 \, d} - \frac {5 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e^{3}}{32 \, d} + \frac {5 \, a b \arcsin \left (d x + c\right ) e^{3}}{16 \, d} - \frac {17 \, b^{2} e^{3}}{256 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 206, normalized size = 1.17 \[ \frac {\frac {e^{3} \left (d x +c \right )^{4} a^{2}}{4}+e^{3} b^{2} \left (\frac {\arcsin \left (d x +c \right )^{2} \left (d x +c \right )^{4}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d} \]
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}{4} \, a^{2} d^{3} e^{3} x^{4} + a^{2} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{2} c^{2} d e^{3} 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 b c^{2} d e^{3} + \frac {1}{3} \, {\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 b c d^{2} e^{3} + \frac {1}{48} \, {\left (24 \, x^{4} \arcsin \left (d x + c\right ) + {\left (\frac {6 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x^{3}}{d^{2}} - \frac {14 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c x^{2}}{d^{3}} + \frac {105 \, c^{4} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{5}} + \frac {35 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c^{2} x}{d^{4}} - \frac {90 \, {\left (c^{2} - 1\right )} c^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{5}} - \frac {105 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c^{3}}{d^{5}} - \frac {9 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (c^{2} - 1\right )} x}{d^{4}} + \frac {9 \, {\left (c^{2} - 1\right )}^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{5}} + \frac {55 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (c^{2} - 1\right )} c}{d^{5}}\right )} d\right )} a b d^{3} e^{3} + a^{2} c^{3} e^{3} x + \frac {2 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a b c^{3} e^{3}}{d} + \frac {1}{4} \, {\left (b^{2} d^{3} e^{3} x^{4} + 4 \, b^{2} c d^{2} e^{3} x^{3} + 6 \, b^{2} c^{2} d e^{3} x^{2} + 4 \, b^{2} c^{3} e^{3} x\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} + \int \frac {{\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} 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 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{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 = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.17, size = 916, normalized size = 5.20 \[ \begin {cases} a^{2} c^{3} e^{3} x + \frac {3 a^{2} c^{2} d e^{3} x^{2}}{2} + a^{2} c d^{2} e^{3} x^{3} + \frac {a^{2} d^{3} e^{3} x^{4}}{4} + \frac {a b c^{4} e^{3} \operatorname {asin}{\left (c + d x \right )}}{2 d} + 2 a b c^{3} e^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {a b c^{3} e^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{8 d} + 3 a b c^{2} d e^{3} x^{2} \operatorname {asin}{\left (c + d x \right )} + \frac {3 a b c^{2} e^{3} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{8} + 2 a b c d^{2} e^{3} x^{3} \operatorname {asin}{\left (c + d x \right )} + \frac {3 a b c d e^{3} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{8} + \frac {3 a b c e^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16 d} + \frac {a b d^{3} e^{3} x^{4} \operatorname {asin}{\left (c + d x \right )}}{2} + \frac {a b d^{2} e^{3} x^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{8} + \frac {3 a b e^{3} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16} - \frac {3 a b e^{3} \operatorname {asin}{\left (c + d x \right )}}{16 d} + \frac {b^{2} c^{4} e^{3} \operatorname {asin}^{2}{\left (c + d x \right )}}{4 d} + b^{2} c^{3} e^{3} x \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {b^{2} c^{3} e^{3} x}{8} + \frac {b^{2} c^{3} e^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{8 d} + \frac {3 b^{2} c^{2} d e^{3} x^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{2} - \frac {3 b^{2} c^{2} d e^{3} x^{2}}{16} + \frac {3 b^{2} c^{2} e^{3} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{8} + b^{2} c d^{2} e^{3} x^{3} \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {b^{2} c d^{2} e^{3} x^{3}}{8} + \frac {3 b^{2} c d e^{3} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{8} - \frac {3 b^{2} c e^{3} x}{16} + \frac {3 b^{2} c e^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{16 d} + \frac {b^{2} d^{3} e^{3} x^{4} \operatorname {asin}^{2}{\left (c + d x \right )}}{4} - \frac {b^{2} d^{3} e^{3} x^{4}}{32} + \frac {b^{2} d^{2} e^{3} x^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{8} - \frac {3 b^{2} d e^{3} x^{2}}{32} + \frac {3 b^{2} e^{3} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{16} - \frac {3 b^{2} e^{3} \operatorname {asin}^{2}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {asin}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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