3.214 \(\int (a+b \sin ^{-1}(c+d x))^5 \, dx\)

Optimal. Leaf size=164 \[ 120 a b^4 x-\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac {120 b^5 \sqrt {1-(c+d x)^2}}{d}+\frac {120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d} \]

[Out]

120*a*b^4*x+120*b^5*(d*x+c)*arcsin(d*x+c)/d-20*b^2*(d*x+c)*(a+b*arcsin(d*x+c))^3/d+(d*x+c)*(a+b*arcsin(d*x+c))
^5/d+120*b^5*(1-(d*x+c)^2)^(1/2)/d-60*b^3*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+5*b*(a+b*arcsin(d*x+c))^
4*(1-(d*x+c)^2)^(1/2)/d

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Rubi [A]  time = 0.21, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 261} \[ -\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+120 a b^4 x+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac {120 b^5 \sqrt {1-(c+d x)^2}}{d}+\frac {120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^5,x]

[Out]

120*a*b^4*x + (120*b^5*Sqrt[1 - (c + d*x)^2])/d + (120*b^5*(c + d*x)*ArcSin[c + d*x])/d - (60*b^3*Sqrt[1 - (c
+ d*x)^2]*(a + b*ArcSin[c + d*x])^2)/d - (20*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])^3)/d + (5*b*Sqrt[1 - (c + d
*x)^2]*(a + b*ArcSin[c + d*x])^4)/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^5)/d

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4619

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

Rule 4677

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

Rule 4803

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^5 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^5 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^4}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac {\left (20 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac {\left (60 b^3\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 )}{d}\\ &=-\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac {\left (120 b^4\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x-\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac {\left (120 b^5\right ) \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x+\frac {120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d}-\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac {\left (120 b^5\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x+\frac {120 b^5 \sqrt {1-(c+d x)^2}}{d}+\frac {120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d}-\frac {60 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 150, normalized size = 0.91 \[ \frac {-20 b^2 \left (-6 b^2 \left (a (c+d x)+b \sqrt {1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3+3 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5+5 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])^5,x]

[Out]

(5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^4 + (c + d*x)*(a + b*ArcSin[c + d*x])^5 - 20*b^2*(3*b*Sqrt[
1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + (c + d*x)*(a + b*ArcSin[c + d*x])^3 - 6*b^2*(a*(c + d*x) + b*Sqrt
[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/d

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fricas [B]  time = 0.44, size = 323, normalized size = 1.97 \[ \frac {{\left (b^{5} d x + b^{5} c\right )} \arcsin \left (d x + c\right )^{5} + 5 \, {\left (a b^{4} d x + a b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 10 \, {\left ({\left (a^{2} b^{3} - 2 \, b^{5}\right )} d x + {\left (a^{2} b^{3} - 2 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{5} - 20 \, a^{3} b^{2} + 120 \, a b^{4}\right )} d x + 10 \, {\left ({\left (a^{3} b^{2} - 6 \, a b^{4}\right )} d x + {\left (a^{3} b^{2} - 6 \, a b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 5 \, {\left ({\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} d x + {\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right ) + 5 \, {\left (b^{5} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{4} \arcsin \left (d x + c\right )^{3} + a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5} + 6 \, {\left (a^{2} b^{3} - 2 \, b^{5}\right )} \arcsin \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{2} - 6 \, a b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^5,x, algorithm="fricas")

[Out]

((b^5*d*x + b^5*c)*arcsin(d*x + c)^5 + 5*(a*b^4*d*x + a*b^4*c)*arcsin(d*x + c)^4 + 10*((a^2*b^3 - 2*b^5)*d*x +
(a^2*b^3 - 2*b^5)*c)*arcsin(d*x + c)^3 + (a^5 - 20*a^3*b^2 + 120*a*b^4)*d*x + 10*((a^3*b^2 - 6*a*b^4)*d*x + (
a^3*b^2 - 6*a*b^4)*c)*arcsin(d*x + c)^2 + 5*((a^4*b - 12*a^2*b^3 + 24*b^5)*d*x + (a^4*b - 12*a^2*b^3 + 24*b^5)
*c)*arcsin(d*x + c) + 5*(b^5*arcsin(d*x + c)^4 + 4*a*b^4*arcsin(d*x + c)^3 + a^4*b - 12*a^2*b^3 + 24*b^5 + 6*(
a^2*b^3 - 2*b^5)*arcsin(d*x + c)^2 + 4*(a^3*b^2 - 6*a*b^4)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)
)/d

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giac [B]  time = 0.29, size = 482, normalized size = 2.94 \[ \frac {{\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{5}}{d} + \frac {5 \, {\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{4}}{d} + \frac {10 \, {\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )^{3}}{d} - \frac {20 \, {\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{3}}{d} + \frac {20 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac {10 \, {\left (d x + c\right )} a^{3} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {60 \, {\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac {30 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3} \arcsin \left (d x + c\right )^{2}}{d} - \frac {60 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{2}}{d} + \frac {5 \, {\left (d x + c\right )} a^{4} b \arcsin \left (d x + c\right )}{d} - \frac {60 \, {\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )}{d} + \frac {120 \, {\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )}{d} + \frac {20 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b^{2} \arcsin \left (d x + c\right )}{d} - \frac {120 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{5}}{d} - \frac {20 \, {\left (d x + c\right )} a^{3} b^{2}}{d} + \frac {120 \, {\left (d x + c\right )} a b^{4}}{d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{4} b}{d} - \frac {60 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3}}{d} + \frac {120 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^5,x, algorithm="giac")

[Out]

(d*x + c)*b^5*arcsin(d*x + c)^5/d + 5*(d*x + c)*a*b^4*arcsin(d*x + c)^4/d + 5*sqrt(-(d*x + c)^2 + 1)*b^5*arcsi
n(d*x + c)^4/d + 10*(d*x + c)*a^2*b^3*arcsin(d*x + c)^3/d - 20*(d*x + c)*b^5*arcsin(d*x + c)^3/d + 20*sqrt(-(d
*x + c)^2 + 1)*a*b^4*arcsin(d*x + c)^3/d + 10*(d*x + c)*a^3*b^2*arcsin(d*x + c)^2/d - 60*(d*x + c)*a*b^4*arcsi
n(d*x + c)^2/d + 30*sqrt(-(d*x + c)^2 + 1)*a^2*b^3*arcsin(d*x + c)^2/d - 60*sqrt(-(d*x + c)^2 + 1)*b^5*arcsin(
d*x + c)^2/d + 5*(d*x + c)*a^4*b*arcsin(d*x + c)/d - 60*(d*x + c)*a^2*b^3*arcsin(d*x + c)/d + 120*(d*x + c)*b^
5*arcsin(d*x + c)/d + 20*sqrt(-(d*x + c)^2 + 1)*a^3*b^2*arcsin(d*x + c)/d - 120*sqrt(-(d*x + c)^2 + 1)*a*b^4*a
rcsin(d*x + c)/d + (d*x + c)*a^5/d - 20*(d*x + c)*a^3*b^2/d + 120*(d*x + c)*a*b^4/d + 5*sqrt(-(d*x + c)^2 + 1)
*a^4*b/d - 60*sqrt(-(d*x + c)^2 + 1)*a^2*b^3/d + 120*sqrt(-(d*x + c)^2 + 1)*b^5/d

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maple [B]  time = 0.07, size = 367, normalized size = 2.24 \[ \frac {a^{5} \left (d x +c \right )+b^{5} \left (\arcsin \left (d x +c \right )^{5} \left (d x +c \right )+5 \arcsin \left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}-20 \left (d x +c \right ) \arcsin \left (d x +c \right )^{3}-60 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}+120 \sqrt {1-\left (d x +c \right )^{2}}+120 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+5 a \,b^{4} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{4}+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+10 a^{2} b^{3} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{3}+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+10 a^{3} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+5 a^{4} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^5,x)

[Out]

1/d*(a^5*(d*x+c)+b^5*(arcsin(d*x+c)^5*(d*x+c)+5*arcsin(d*x+c)^4*(1-(d*x+c)^2)^(1/2)-20*(d*x+c)*arcsin(d*x+c)^3
-60*arcsin(d*x+c)^2*(1-(d*x+c)^2)^(1/2)+120*(1-(d*x+c)^2)^(1/2)+120*(d*x+c)*arcsin(d*x+c))+5*a*b^4*((d*x+c)*ar
csin(d*x+c)^4+4*arcsin(d*x+c)^3*(1-(d*x+c)^2)^(1/2)-12*arcsin(d*x+c)^2*(d*x+c)+24*d*x+24*c-24*arcsin(d*x+c)*(1
-(d*x+c)^2)^(1/2))+10*a^2*b^3*((d*x+c)*arcsin(d*x+c)^3+3*arcsin(d*x+c)^2*(1-(d*x+c)^2)^(1/2)-6*(1-(d*x+c)^2)^(
1/2)-6*(d*x+c)*arcsin(d*x+c))+10*a^3*b^2*(arcsin(d*x+c)^2*(d*x+c)-2*d*x-2*c+2*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2
))+5*a^4*b*((d*x+c)*arcsin(d*x+c)+(1-(d*x+c)^2)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{5} x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{5} + a^{5} x + \frac {5 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a^{4} b}{d} + \int \frac {5 \, {\left (\sqrt {d x + c + 1} \sqrt {-d x - c + 1} b^{5} d x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{4} + {\left (a b^{4} d^{2} x^{2} + 2 \, a b^{4} c d x + a b^{4} c^{2} - a b^{4}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{4} + 2 \, {\left (a^{2} b^{3} d^{2} x^{2} + 2 \, a^{2} b^{3} c d x + a^{2} b^{3} c^{2} - a^{2} b^{3}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 2 \, {\left (a^{3} b^{2} d^{2} x^{2} + 2 \, a^{3} b^{2} c d x + a^{3} b^{2} c^{2} - a^{3} b^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^5,x, algorithm="maxima")

[Out]

b^5*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^5 + a^5*x + 5*((d*x + c)*arcsin(d*x + c) + sqrt(-
(d*x + c)^2 + 1))*a^4*b/d + integrate(5*(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b^5*d*x*arctan2(d*x + c, sqrt(d*
x + c + 1)*sqrt(-d*x - c + 1))^4 + (a*b^4*d^2*x^2 + 2*a*b^4*c*d*x + a*b^4*c^2 - a*b^4)*arctan2(d*x + c, sqrt(d
*x + c + 1)*sqrt(-d*x - c + 1))^4 + 2*(a^2*b^3*d^2*x^2 + 2*a^2*b^3*c*d*x + a^2*b^3*c^2 - a^2*b^3)*arctan2(d*x
+ c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 2*(a^3*b^2*d^2*x^2 + 2*a^3*b^2*c*d*x + a^3*b^2*c^2 - a^3*b^2)*a
rctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

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mupad [B]  time = 0.67, size = 317, normalized size = 1.93 \[ a^5\,x+\frac {10\,a^3\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {5\,a^4\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {b^5\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^5-20\,{\mathrm {asin}\left (c+d\,x\right )}^3+120\,\mathrm {asin}\left (c+d\,x\right )\right )}{d}+\frac {b^5\,\sqrt {1-{\left (c+d\,x\right )}^2}\,\left (5\,{\mathrm {asin}\left (c+d\,x\right )}^4-60\,{\mathrm {asin}\left (c+d\,x\right )}^2+120\right )}{d}+\frac {5\,a\,b^4\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^4-12\,{\mathrm {asin}\left (c+d\,x\right )}^2+24\right )}{d}+\frac {10\,a^2\,b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}-\frac {10\,a^2\,b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d}-\frac {5\,a\,b^4\,\left (24\,\mathrm {asin}\left (c+d\,x\right )-4\,{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^5,x)

[Out]

a^5*x + (10*a^3*b^2*(2*asin(c + d*x)*(1 - (c + d*x)^2)^(1/2) + (asin(c + d*x)^2 - 2)*(c + d*x)))/d + (5*a^4*b*
((1 - (c + d*x)^2)^(1/2) + asin(c + d*x)*(c + d*x)))/d + (b^5*(c + d*x)*(120*asin(c + d*x) - 20*asin(c + d*x)^
3 + asin(c + d*x)^5))/d + (b^5*(1 - (c + d*x)^2)^(1/2)*(5*asin(c + d*x)^4 - 60*asin(c + d*x)^2 + 120))/d + (5*
a*b^4*(c + d*x)*(asin(c + d*x)^4 - 12*asin(c + d*x)^2 + 24))/d + (10*a^2*b^3*(3*asin(c + d*x)^2 - 6)*(1 - (c +
d*x)^2)^(1/2))/d - (10*a^2*b^3*(6*asin(c + d*x) - asin(c + d*x)^3)*(c + d*x))/d - (5*a*b^4*(24*asin(c + d*x)
- 4*asin(c + d*x)^3)*(1 - (c + d*x)^2)^(1/2))/d

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sympy [A]  time = 3.50, size = 663, normalized size = 4.04 \[ \begin {cases} a^{5} x + \frac {5 a^{4} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 5 a^{4} b x \operatorname {asin}{\left (c + d x \right )} + \frac {5 a^{4} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {10 a^{3} b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 20 a^{3} b^{2} x + \frac {20 a^{3} b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {10 a^{2} b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {60 a^{2} b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 60 a^{2} b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {30 a^{2} b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {60 a^{2} b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {5 a b^{4} c \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {60 a b^{4} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 5 a b^{4} x \operatorname {asin}^{4}{\left (c + d x \right )} - 60 a b^{4} x \operatorname {asin}^{2}{\left (c + d x \right )} + 120 a b^{4} x + \frac {20 a b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {120 a b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {b^{5} c \operatorname {asin}^{5}{\left (c + d x \right )}}{d} - \frac {20 b^{5} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} + \frac {120 b^{5} c \operatorname {asin}{\left (c + d x \right )}}{d} + b^{5} x \operatorname {asin}^{5}{\left (c + d x \right )} - 20 b^{5} x \operatorname {asin}^{3}{\left (c + d x \right )} + 120 b^{5} x \operatorname {asin}{\left (c + d x \right )} + \frac {5 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {60 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + \frac {120 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\relax (c )}\right )^{5} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))**5,x)

[Out]

Piecewise((a**5*x + 5*a**4*b*c*asin(c + d*x)/d + 5*a**4*b*x*asin(c + d*x) + 5*a**4*b*sqrt(-c**2 - 2*c*d*x - d*
*2*x**2 + 1)/d + 10*a**3*b**2*c*asin(c + d*x)**2/d + 10*a**3*b**2*x*asin(c + d*x)**2 - 20*a**3*b**2*x + 20*a**
3*b**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d + 10*a**2*b**3*c*asin(c + d*x)**3/d - 60*a**2*b**
3*c*asin(c + d*x)/d + 10*a**2*b**3*x*asin(c + d*x)**3 - 60*a**2*b**3*x*asin(c + d*x) + 30*a**2*b**3*sqrt(-c**2
- 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d - 60*a**2*b**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + 5*a*b*
*4*c*asin(c + d*x)**4/d - 60*a*b**4*c*asin(c + d*x)**2/d + 5*a*b**4*x*asin(c + d*x)**4 - 60*a*b**4*x*asin(c +
d*x)**2 + 120*a*b**4*x + 20*a*b**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 120*a*b**4*sqrt(
-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d + b**5*c*asin(c + d*x)**5/d - 20*b**5*c*asin(c + d*x)**3/d +
120*b**5*c*asin(c + d*x)/d + b**5*x*asin(c + d*x)**5 - 20*b**5*x*asin(c + d*x)**3 + 120*b**5*x*asin(c + d*x) +
5*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**4/d - 60*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 +
1)*asin(c + d*x)**2/d + 120*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d, Ne(d, 0)), (x*(a + b*asin(c))**5, Tr
ue))

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