3.189 \(\int (c e+d e x)^3 (a+b \sin ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=176 \[ \frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac {b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{8 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{32 d}-\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b^2 e^3 (c+d x)^2}{32 d} \]

[Out]

-3/32*b^2*e^3*(d*x+c)^2/d-1/32*b^2*e^3*(d*x+c)^4/d-3/32*e^3*(a+b*arcsin(d*x+c))^2/d+1/4*e^3*(d*x+c)^4*(a+b*arc
sin(d*x+c))^2/d+3/16*b*e^3*(d*x+c)*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2)^(1/2)/d+1/8*b*e^3*(d*x+c)^3*(a+b*arcsin(d
*x+c))*(1-(d*x+c)^2)^(1/2)/d

________________________________________________________________________________________

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.

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^2,x]

[Out]

(-3*b^2*e^3*(c + d*x)^2)/(32*d) - (b^2*e^3*(c + d*x)^4)/(32*d) + (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a +
b*ArcSin[c + d*x]))/(16*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(8*d) - (3*e^3
*(a + b*ArcSin[c + d*x])^2)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/(4*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

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*}

________________________________________________________________________________________

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.

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^2,x]

[Out]

(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^2 + (-(b^2*(c + d*x)^4) + 4*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a +
b*ArcSin[c + d*x]) - 3*(b^2*(c + d*x)^2 - 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (a +
b*ArcSin[c + d*x])^2))/8))/(4*d)

________________________________________________________________________________________

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.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/32*((8*a^2 - b^2)*d^4*e^3*x^4 + 4*(8*a^2 - b^2)*c*d^3*e^3*x^3 + 3*(2*(8*a^2 - b^2)*c^2 - b^2)*d^2*e^3*x^2 +
2*(2*(8*a^2 - b^2)*c^3 - 3*b^2*c)*d*e^3*x + (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 + (8*b^2*c^4 - 3*b^2)*e^3)*arcsin(d*x + c)^2 + 2*(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 + (8*a*b*c^4 - 3*a*b)*e^3)*arcsin(d*x + c) + 2*(2*a*b*d^3*e^3
*x^3 + 6*a*b*c*d^2*e^3*x^2 + 3*(2*a*b*c^2 + a*b)*d*e^3*x + (2*a*b*c^3 + 3*a*b*c)*e^3 + (2*b^2*d^3*e^3*x^3 + 6*
b^2*c*d^2*e^3*x^2 + 3*(2*b^2*c^2 + b^2)*d*e^3*x + (2*b^2*c^3 + 3*b^2*c)*e^3)*arcsin(d*x + c))*sqrt(-d^2*x^2 -
2*c*d*x - c^2 + 1))/d

________________________________________________________________________________________

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.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="giac")

[Out]

1/4*(d*x + c)^4*a^2*e^3/d + 1/4*((d*x + c)^2 - 1)^2*b^2*arcsin(d*x + c)^2*e^3/d - 1/8*(-(d*x + c)^2 + 1)^(3/2)
*(d*x + c)*b^2*arcsin(d*x + c)*e^3/d + 1/2*((d*x + c)^2 - 1)^2*a*b*arcsin(d*x + c)*e^3/d + 1/2*((d*x + c)^2 -
1)*b^2*arcsin(d*x + c)^2*e^3/d - 1/8*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b*e^3/d + 5/16*sqrt(-(d*x + c)^2 + 1
)*(d*x + c)*b^2*arcsin(d*x + c)*e^3/d - 1/32*((d*x + c)^2 - 1)^2*b^2*e^3/d + ((d*x + c)^2 - 1)*a*b*arcsin(d*x
+ c)*e^3/d + 5/32*b^2*arcsin(d*x + c)^2*e^3/d + 5/16*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b*e^3/d - 5/32*((d*x +
c)^2 - 1)*b^2*e^3/d + 5/16*a*b*arcsin(d*x + c)*e^3/d - 17/256*b^2*e^3/d

________________________________________________________________________________________

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.

[In]

int((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x)

[Out]

1/d*(1/4*e^3*(d*x+c)^4*a^2+e^3*b^2*(1/4*arcsin(d*x+c)^2*(d*x+c)^4-1/16*arcsin(d*x+c)*(-2*(d*x+c)^3*(1-(d*x+c)^
2)^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/2)+3*arcsin(d*x+c))+3/32*arcsin(d*x+c)^2-1/32*(d*x+c)^4-3/32*(d*x+c)^2)+2*
e^3*a*b*(1/4*(d*x+c)^4*arcsin(d*x+c)+1/16*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1-(d*x+c)^2)^(1/2)-3/32*
arcsin(d*x+c)))

________________________________________________________________________________________

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.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^2*c^2*d*e^3*x^2 + 3/2*(2*x^2*arcsin(d*x + c) + d*(3*c^2*arcsin
(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 + sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x/d^2 - (c^2 - 1)*arcs
in(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 - 3*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c/d^3))*a*b*c^2*d*
e^3 + 1/3*(6*x^3*arcsin(d*x + c) + d*(2*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^2/d^2 - 15*c^3*arcsin(-(d^2*x + c
*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^4 - 5*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c*x/d^3 + 9*(c^2 - 1)*c*arcsin(-
(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^4 + 15*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2/d^4 - 4*sqrt(-d^2
*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)/d^4))*a*b*c*d^2*e^3 + 1/48*(24*x^4*arcsin(d*x + c) + (6*sqrt(-d^2*x^2 - 2*
c*d*x - c^2 + 1)*x^3/d^2 - 14*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsin(-(d^2*x + c*d)/sqr
t(c^2*d^2 - (c^2 - 1)*d^2))/d^5 + 35*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*arcsin(-(
d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 - 105*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^3/d^5 - 9*sqrt(-d^2
*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))
/d^5 + 55*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)*c/d^5)*d)*a*b*d^3*e^3 + a^2*c^3*e^3*x + 2*((d*x + c)*ar
csin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a*b*c^3*e^3/d + 1/4*(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)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + integrate(1/2*(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)*sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

________________________________________________________________________________________

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.

[In]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^2, x)

________________________________________________________________________________________

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.

[In]

integrate((d*e*x+c*e)**3*(a+b*asin(d*x+c))**2,x)

[Out]

Piecewise((a**2*c**3*e**3*x + 3*a**2*c**2*d*e**3*x**2/2 + a**2*c*d**2*e**3*x**3 + a**2*d**3*e**3*x**4/4 + a*b*
c**4*e**3*asin(c + d*x)/(2*d) + 2*a*b*c**3*e**3*x*asin(c + d*x) + 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*asin(c + d*x) + 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*asin(c + d*x) + 3*a*b*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/8 + 3*a*
b*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(16*d) + a*b*d**3*e**3*x**4*asin(c + d*x)/2 + a*b*d**2*e**3*x**
3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/8 + 3*a*b*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 - 3*a*b*e**3
*asin(c + d*x)/(16*d) + b**2*c**4*e**3*asin(c + d*x)**2/(4*d) + b**2*c**3*e**3*x*asin(c + d*x)**2 - b**2*c**3*
e**3*x/8 + b**2*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(8*d) + 3*b**2*c**2*d*e**3*x**2*
asin(c + d*x)**2/2 - 3*b**2*c**2*d*e**3*x**2/16 + 3*b**2*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asi
n(c + d*x)/8 + b**2*c*d**2*e**3*x**3*asin(c + d*x)**2 - b**2*c*d**2*e**3*x**3/8 + 3*b**2*c*d*e**3*x**2*sqrt(-c
**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 3*b**2*c*e**3*x/16 + 3*b**2*c*e**3*sqrt(-c**2 - 2*c*d*x - d**
2*x**2 + 1)*asin(c + d*x)/(16*d) + b**2*d**3*e**3*x**4*asin(c + d*x)**2/4 - b**2*d**3*e**3*x**4/32 + b**2*d**2
*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 3*b**2*d*e**3*x**2/32 + 3*b**2*e**3*x*sqrt(
-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/16 - 3*b**2*e**3*asin(c + d*x)**2/(32*d), Ne(d, 0)), (c**3*e**3
*x*(a + b*asin(c))**2, True))

________________________________________________________________________________________