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

Optimal. Leaf size=88 \[ \frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d e (m+1)}-\frac{3 b \text{Unintegrable}\left (\frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^2}{\sqrt{1-(c+d x)^2}},x\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.178789, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^3)/(d*e*(1 + m)) - (3*b*Defer[Subst][Defer[Int][((e*x)^(1 + m)*
(a + b*ArcSin[x])^2)/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 )^3 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^m \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d e (1+m)}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(e x)^{1+m} \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ \end{align*}

Mathematica [A]  time = 1.8278, size = 0, normalized size = 0. \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.286, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{m} \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}\right )}{\left (d e x + c e\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}{\left (d e x + c e\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsin(d*x + c) + a)^3*(d*e*x + c*e)^m, x)