3.308 \(\int (c e+d e x)^m (a+b \sin ^{-1}(c+d x))^4 \, dx\)

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

[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)

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Rubi [A]  time = 0.19, 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 = {} \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx \]

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 {\operatorname {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) \operatorname {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*}

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Mathematica [A]  time = 3.22, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx \]

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]

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]

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*e*x + c*e)^m, x)

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

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)

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maple [A]  time = 2.20, 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)

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

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*e*x + c*e)^(m + 1)*a^4/(d*e*(m + 1)) + ((b^4*d*e^m*x + 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*e^m*x + 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*c^2 - a*b^3)*e^m*m +
(a*b^3*d^2*e^m*m + a*b^3*d^2*e^m)*x^2 + (a*b^3*c^2 - a*b^3)*e^m + 2*(a*b^3*c*d*e^m*m + a*b^3*c*d*e^m)*x)*(d*x
+ c)^m*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*((a^2*b^2*c^2 - a^2*b^2)*e^m*m + (a^2*b^2*
d^2*e^m*m + a^2*b^2*d^2*e^m)*x^2 + (a^2*b^2*c^2 - a^2*b^2)*e^m + 2*(a^2*b^2*c*d*e^m*m + a^2*b^2*c*d*e^m)*x)*(d
*x + c)^m*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*((a^3*b*c^2 - a^3*b)*e^m*m + (a^3*b*d^2
*e^m*m + a^3*b*d^2*e^m)*x^2 + (a^3*b*c^2 - a^3*b)*e^m + 2*(a^3*b*c*d*e^m*m + a^3*b*c*d*e^m)*x)*(d*x + c)^m*arc
tan2(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)

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

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)

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

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)

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