Optimal. Leaf size=338 \[ -\frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1-(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac {16 b^3 e^4 (c+d x) \text {ArcSin}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {ArcSin}(c+d x))}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {ArcSin}(c+d x))}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {ArcSin}(c+d x))^3}{5 d} \]
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Rubi [A]
time = 0.34, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12,
4723, 4795, 4767, 4715, 267, 272, 45} \begin {gather*} -\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {ArcSin}(c+d x))}{125 d}-\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {ArcSin}(c+d x))}{75 d}+\frac {e^4 (c+d x)^5 (a+b \text {ArcSin}(c+d x))^3}{5 d}+\frac {3 b e^4 \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \text {ArcSin}(c+d x))^2}{25 d}+\frac {4 b e^4 \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2}{25 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2}{25 d}-\frac {16}{25} a b^2 e^4 x-\frac {16 b^3 e^4 (c+d x) \text {ArcSin}(c+d x)}{25 d}-\frac {6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}+\frac {76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac {298 b^3 e^4 \sqrt {1-(c+d x)^2}}{375 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 267
Rule 272
Rule 4715
Rule 4723
Rule 4767
Rule 4795
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^4 x^4 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int x^4 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (3 b e^4\right ) \text {Subst}\left (\int \frac {x^5 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d}\\ &=\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (12 b e^4\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d}-\frac {\left (6 b^2 e^4\right ) \text {Subst}\left (\int x^4 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d}-\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}+\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{125 d}\\ &=-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (16 b^2 e^4\right ) \text {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}+\frac {\left (3 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x}} \, dx,x,(c+d x)^2\right )}{125 d}+\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{75 d}\\ &=-\frac {16}{25} a b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (3 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-2 \sqrt {1-x}+(1-x)^{3/2}\right ) \, dx,x,(c+d x)^2\right )}{125 d}+\frac {\left (4 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,(c+d x)^2\right )}{75 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {16}{25} a b^2 e^4 x-\frac {6 b^3 e^4 \sqrt {1-(c+d x)^2}}{125 d}+\frac {4 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{125 d}-\frac {6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac {16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (4 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,(c+d x)^2\right )}{75 d}+\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1-(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac {16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )^3}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 307, normalized size = 0.91 \begin {gather*} \frac {e^4 \left ((c+d x)^5 (a+b \text {ArcSin}(c+d x))^3-\frac {1}{25} b \left (\frac {40}{9} b^2 \left (2+c^2+2 c d x+d^2 x^2\right ) \sqrt {1-(c+d x)^2}-\frac {2}{5} b^2 \sqrt {1-(c+d x)^2} \left (-15+10 \left (1-(c+d x)^2\right )-3 \left (-1+(c+d x)^2\right )^2\right )+\frac {40}{3} b (c+d x)^3 (a+b \text {ArcSin}(c+d x))+6 b (c+d x)^5 (a+b \text {ArcSin}(c+d x))-40 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2-20 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2-15 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^2+80 b \left (a d x+b \sqrt {1-(c+d x)^2}+b (c+d x) \text {ArcSin}(c+d x)\right )\right )\right )}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 383, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {e^{4} \left (d x +c \right )^{5} a^{3}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{3}}{5}+\frac {\arcsin \left (d x +c \right )^{2} \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {6 \left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{125}-\frac {2 \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{625}-\frac {8 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{75}-\frac {8 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{225}-\frac {16 \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {16 \left (d x +c \right ) \arcsin \left (d x +c \right )}{25}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(383\) |
default | \(\frac {\frac {e^{4} \left (d x +c \right )^{5} a^{3}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{3}}{5}+\frac {\arcsin \left (d x +c \right )^{2} \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {6 \left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{125}-\frac {2 \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{625}-\frac {8 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{75}-\frac {8 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{225}-\frac {16 \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {16 \left (d x +c \right ) \arcsin \left (d x +c \right )}{25}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(383\) |
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. 892 vs.
\(2 (293) = 586\).
time = 1.99, size = 892, normalized size = 2.64 \begin {gather*} \frac {1125 \, {\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \arcsin \left (d x + c\right )^{3} e^{4} + 3375 \, {\left (a b^{2} d^{5} x^{5} + 5 \, a b^{2} c d^{4} x^{4} + 10 \, a b^{2} c^{2} d^{3} x^{3} + 10 \, a b^{2} c^{3} d^{2} x^{2} + 5 \, a b^{2} c^{4} d x + a b^{2} c^{5}\right )} \arcsin \left (d x + c\right )^{2} e^{4} + 15 \, {\left (9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} d^{5} x^{5} + 45 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c d^{4} x^{4} - 10 \, {\left (4 \, b^{3} - 9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d^{3} x^{3} - 40 \, b^{3} c^{3} + 9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{5} - 30 \, {\left (4 \, b^{3} c - 3 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{3}\right )} d^{2} x^{2} - 240 \, b^{3} c - 15 \, {\left (8 \, b^{3} c^{2} - 3 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{4} + 16 \, b^{3}\right )} d x\right )} \arcsin \left (d x + c\right ) e^{4} + 15 \, {\left (3 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} d^{5} x^{5} + 15 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c d^{4} x^{4} - 10 \, {\left (4 \, a b^{2} - 3 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{2}\right )} d^{3} x^{3} - 30 \, {\left (4 \, a b^{2} c - {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{3}\right )} d^{2} x^{2} - 15 \, {\left (8 \, a b^{2} c^{2} - {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{4} + 16 \, a b^{2}\right )} d x\right )} e^{4} + \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (225 \, {\left (3 \, b^{3} d^{4} x^{4} + 12 \, b^{3} c d^{3} x^{3} + 3 \, b^{3} c^{4} + 4 \, b^{3} c^{2} + 2 \, {\left (9 \, b^{3} c^{2} + 2 \, b^{3}\right )} d^{2} x^{2} + 8 \, b^{3} + 4 \, {\left (3 \, b^{3} c^{3} + 2 \, b^{3} c\right )} d x\right )} \arcsin \left (d x + c\right )^{2} e^{4} + 450 \, {\left (3 \, a b^{2} d^{4} x^{4} + 12 \, a b^{2} c d^{3} x^{3} + 3 \, a b^{2} c^{4} + 4 \, a b^{2} c^{2} + 2 \, {\left (9 \, a b^{2} c^{2} + 2 \, a b^{2}\right )} d^{2} x^{2} + 8 \, a b^{2} + 4 \, {\left (3 \, a b^{2} c^{3} + 2 \, a b^{2} c\right )} d x\right )} \arcsin \left (d x + c\right ) e^{4} + {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} d^{4} x^{4} + 108 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c d^{3} x^{3} + 27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{4} + 2 \, {\left (450 \, a^{2} b - 136 \, b^{3} + 81 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d^{2} x^{2} + 1800 \, a^{2} b - 4144 \, b^{3} + 4 \, {\left (225 \, a^{2} b - 68 \, b^{3}\right )} c^{2} + 4 \, {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{3} + 2 \, {\left (225 \, a^{2} b - 68 \, b^{3}\right )} c\right )} d x\right )} e^{4}\right )}}{5625 \, 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. 2518 vs.
\(2 (306) = 612\).
time = 1.46, size = 2518, normalized size = 7.45 \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. 832 vs.
\(2 (304) = 608\).
time = 0.47, size = 832, normalized size = 2.46 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )^{3}}{5 \, d} + \frac {{\left (d x + c\right )}^{5} a^{3} e^{4}}{5 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} a b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )^{3}}{5 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{4} \arcsin \left (d x + c\right )^{2}}{25 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} a^{2} b e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )}{125 \, d} + \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {{\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )^{3}}{5 \, d} + \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} e^{4} \arcsin \left (d x + c\right )}{25 \, d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} - \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} a b^{2} e^{4}}{125 \, d} + \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {76 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )}{375 \, d} + \frac {3 \, {\left (d x + c\right )} a b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b e^{4}}{25 \, d} - \frac {6 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{4}}{625 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{2} e^{4} \arcsin \left (d x + c\right )}{5 \, d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} - \frac {76 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{2} e^{4}}{375 \, d} + \frac {3 \, {\left (d x + c\right )} a^{2} b e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {298 \, {\left (d x + c\right )} b^{3} e^{4} \arcsin \left (d x + c\right )}{375 \, d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{2} b e^{4}}{5 \, d} + \frac {76 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{3} e^{4}}{1125 \, d} + \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {298 \, {\left (d x + c\right )} a b^{2} e^{4}}{375 \, d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b e^{4}}{5 \, d} - \frac {298 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} e^{4}}{375 \, 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 )}^4\,{\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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