3.4.8 \(\int (c e+d e x)^m (a+b \text {ArcSin}(c+d x))^4 \, dx\) [308]

Optimal. Leaf size=89 \[ \frac {(e (c+d x))^{1+m} (a+b \text {ArcSin}(c+d x))^4}{d e (1+m)}-\frac {4 b \text {Int}\left (\frac {(e (c+d x))^{1+m} (a+b \text {ArcSin}(c+d x))^3}{\sqrt {1-(c+d x)^2}},x\right )}{e (1+m)} \]

[Out]

(e*(d*x+c))^(1+m)*(a+b*arcsin(d*x+c))^4/d/e/(1+m)-4*b*Unintegrable((e*(d*x+c))^(1+m)*(a+b*arcsin(d*x+c))^3/(1-
(d*x+c)^2)^(1/2),x)/e/(1+m)

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (c e+d e x)^m (a+b \text {ArcSin}(c+d x))^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^4,x]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^4)/(d*e*(1 + m)) - (4*b*Defer[Subst][Defer[Int][((e*x)^(1 + m)*
(a + b*ArcSin[x])^3)/Sqrt[1 - x^2], x], x, c + d*x])/(d*e*(1 + m))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 2.64, size = 0, normalized size = 0.00 \begin {gather*} \int (c e+d e x)^m (a+b \text {ArcSin}(c+d x))^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^4,x]

[Out]

Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^4, x]

________________________________________________________________________________________

Maple [A]
time = 1.12, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{m} \left (a +b \arcsin \left (d x +c \right )\right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^4,x)

________________________________________________________________________________________

Maxima [A]
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.

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x
+ c) + a^4)*((d*x + c)*e)^m, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**m*(a+b*asin(d*x+c))**4,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*asin(c + d*x))**4, x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^4*(d*e*x + c*e)^m, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^m*(a + b*asin(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^m*(a + b*asin(c + d*x))^4, x)

________________________________________________________________________________________