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

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e}-\frac {2 b \text {Int}\left (\frac {(e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{\sqrt {1-(c+d x)^2}},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.20, 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 = {} \[ \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(1/2)*(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)*sqrt(d*e*x + c*e)
, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x +c e}\, \left (a +b \arcsin \left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^3,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(1/2)*(a + b*asin(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^(1/2)*(a + b*asin(c + d*x))^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(1/2)*(a+b*asin(d*x+c))**3,x)

[Out]

Integral(sqrt(e*(c + d*x))*(a + b*asin(c + d*x))**3, x)

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