\(\int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx\) [304]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\frac {2 (e (c+d x))^{3/2} (a+b \arcsin (c+d x))^4}{3 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{3/2} (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}},x\right )}{3 e} \]

[Out]

2/3*(e*(d*x+c))^(3/2)*(a+b*arcsin(d*x+c))^4/d/e-8/3*b*Unintegrable((e*(d*x+c))^(3/2)*(a+b*arcsin(d*x+c))^3/(1-
(d*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx \]

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^4)/(3*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(3/2)*(a + b*A
rcSin[x])^3)/Sqrt[1 - x^2], x], x, c + d*x])/(3*d*e)

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {e x} (a+b \arcsin (x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{3/2} (a+b \arcsin (c+d x))^4}{3 d e}-\frac {(8 b) \text {Subst}\left (\int \frac {(e x)^{3/2} (a+b \arcsin (x))^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 164.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \sqrt {d e x +c e}\, \left (a +b \arcsin \left (d x +c \right )\right )^{4}d x\]

[In]

int((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^4,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int { \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 73.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}\, dx \]

[In]

integrate((d*e*x+c*e)**(1/2)*(a+b*asin(d*x+c))**4,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 1.75 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int { \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \arcsin (c+d x))^4 \, dx=\int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((c*e + d*e*x)^(1/2)*(a + b*asin(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(1/2)*(a + b*asin(c + d*x))^4, x)