Optimal. Leaf size=287 \[ -\frac {45 b^3 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{128 d}+\frac {45 b^3 e^3 \text {ArcSin}(c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {ArcSin}(c+d x))}{32 d}-\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{16 d}-\frac {3 e^3 (a+b \text {ArcSin}(c+d x))^3}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^3}{4 d} \]
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Rubi [A]
time = 0.29, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4889, 12,
4723, 4795, 4737, 327, 222} \begin {gather*} -\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {ArcSin}(c+d x))}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^3}{4 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \text {ArcSin}(c+d x))^2}{16 d}+\frac {9 b e^3 \sqrt {1-(c+d x)^2} (c+d x) (a+b \text {ArcSin}(c+d x))^2}{32 d}-\frac {3 e^3 (a+b \text {ArcSin}(c+d x))^3}{32 d}+\frac {45 b^3 e^3 \text {ArcSin}(c+d x)}{256 d}-\frac {3 b^3 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{128 d}-\frac {45 b^3 e^3 \sqrt {1-(c+d x)^2} (c+d x)}{256 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4795
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{16 d}-\frac {\left (3 b^2 e^3\right ) \text {Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (9 b^2 e^3\right ) \text {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{16 d}+\frac {\left (3 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac {45 b^3 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{256 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {45 b^3 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{128 d}+\frac {45 b^3 e^3 \sin ^{-1}(c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}-\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 232, normalized size = 0.81 \begin {gather*} \frac {e^3 \left ((c+d x)^4 (a+b \text {ArcSin}(c+d x))^3-\frac {3}{8} \left (\frac {15}{8} b^3 (c+d x) \sqrt {1-(c+d x)^2}+\frac {1}{4} b^3 (c+d x)^3 \sqrt {1-(c+d x)^2}-\frac {15}{8} b^3 \text {ArcSin}(c+d x)+3 b^2 (c+d x)^2 (a+b \text {ArcSin}(c+d x))+b^2 (c+d x)^4 (a+b \text {ArcSin}(c+d x))-3 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2-2 b (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2+(a+b \text {ArcSin}(c+d x))^3\right )\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 394, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {\frac {e^{3} \left (d x +c \right )^{4} a^{3}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\) | \(394\) |
default | \(\frac {\frac {e^{3} \left (d x +c \right )^{4} a^{3}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\) | \(394\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 681 vs.
\(2 (252) = 504\).
time = 1.10, size = 681, normalized size = 2.37 \begin {gather*} \frac {8 \, {\left (8 \, b^{3} d^{4} x^{4} + 32 \, b^{3} c d^{3} x^{3} + 48 \, b^{3} c^{2} d^{2} x^{2} + 32 \, b^{3} c^{3} d x + 8 \, b^{3} c^{4} - 3 \, b^{3}\right )} \arcsin \left (d x + c\right )^{3} e^{3} + 24 \, {\left (8 \, a b^{2} d^{4} x^{4} + 32 \, a b^{2} c d^{3} x^{3} + 48 \, a b^{2} c^{2} d^{2} x^{2} + 32 \, a b^{2} c^{3} d x + 8 \, a b^{2} c^{4} - 3 \, a b^{2}\right )} \arcsin \left (d x + c\right )^{2} e^{3} + 3 \, {\left (8 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{4} x^{4} + 32 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{3} x^{3} - 24 \, b^{3} c^{2} + 8 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{4} - 24 \, {\left (b^{3} - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d^{2} x^{2} - 24 \, a^{2} b + 15 \, b^{3} - 16 \, {\left (3 \, b^{3} c - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3}\right )} d x\right )} \arcsin \left (d x + c\right ) e^{3} + 8 \, {\left ({\left (8 \, a^{3} - 3 \, a b^{2}\right )} d^{4} x^{4} + 4 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c d^{3} x^{3} - 3 \, {\left (3 \, a b^{2} - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{2}\right )} d^{2} x^{2} - 2 \, {\left (9 \, a b^{2} c - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{3}\right )} d x\right )} e^{3} + 3 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (8 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b^{3} c + 3 \, {\left (2 \, b^{3} c^{2} + b^{3}\right )} d x\right )} \arcsin \left (d x + c\right )^{2} e^{3} + 16 \, {\left (2 \, a b^{2} d^{3} x^{3} + 6 \, a b^{2} c d^{2} x^{2} + 2 \, a b^{2} c^{3} + 3 \, a b^{2} c + 3 \, {\left (2 \, a b^{2} c^{2} + a b^{2}\right )} d x\right )} \arcsin \left (d x + c\right ) e^{3} + {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{3} x^{3} + 6 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{2} x^{2} + 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3} + 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d x + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c\right )} e^{3}\right )}}{256 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1828 vs.
\(2 (260) = 520\).
time = 0.98, size = 1828, normalized size = 6.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 641 vs.
\(2 (261) = 522\).
time = 0.45, size = 641, normalized size = 2.23 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{4 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{16 \, d} + \frac {{\left (d x + c\right )}^{4} a^{3} e^{3}}{4 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{2 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )}{8 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{2} b e^{3} \arcsin \left (d x + c\right )}{4 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} e^{3} \arcsin \left (d x + c\right )}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{2 \, d} + \frac {5 \, b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{32 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a^{2} b e^{3}}{16 \, d} + \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} e^{3}}{128 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )}{16 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} e^{3}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b e^{3} \arcsin \left (d x + c\right )}{2 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} e^{3} \arcsin \left (d x + c\right )}{32 \, d} + \frac {15 \, a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{32 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b e^{3}}{32 \, d} - \frac {51 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} e^{3}}{256 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} e^{3}}{32 \, d} + \frac {15 \, a^{2} b e^{3} \arcsin \left (d x + c\right )}{32 \, d} - \frac {51 \, b^{3} e^{3} \arcsin \left (d x + c\right )}{256 \, d} - \frac {51 \, a b^{2} e^{3}}{256 \, d} \end {gather*}
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 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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