3.304 \(\int \sqrt {c e+d e x} (a+b \sin ^{-1}(c+d x))^4 \, dx\)
Optimal. Leaf size=84 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{\sqrt {1-(c+d x)^2}},x\right )}{3 e} \]
[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 [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 =
{} \[ \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
[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*} \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {e x} \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d e}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {(e x)^{3/2} \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}
________________________________________________________________________________________
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])^4,x]
[Out]
$Aborted
________________________________________________________________________________________
fricas [A] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}, x\right ) \]
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))^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)
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
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))^4,x, algorithm="giac")
[Out]
integrate(sqrt(d*e*x + c*e)*(b*arcsin(d*x + c) + a)^4, x)
________________________________________________________________________________________
maple [A] time = 0.36, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x +c e}\, \left (a +b \arcsin \left (d x +c \right )\right )^{4}\, dx \]
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))^4,x)
[Out]
int((d*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c))^4,x)
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
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))^4,x, algorithm="maxima")
[Out]
1/6*(4*(b^4*d*x + b^4*c)*sqrt(d*x + c)*sqrt(e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + (72*
a*b^3*d^2*sqrt(e)*integrate(1/3*sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(
d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 108*a^2*b^2*d^2*sqrt(e)*integrate(1/3*
sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))
)^2/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 144*a*b^3*c*d*sqrt(e)*integrate(1/3*sqrt(d*x + c)*x*arctan(d*x/(sqrt(d
*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x
) + 72*a^3*b*d^2*sqrt(e)*integrate(1/3*sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c
/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 216*a^2*b^2*c*d*sqrt(e)*integrate
(1/3*sqrt(d*x + c)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)))^2/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 72*a*b^3*c^2*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d
*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x
) + 144*a^3*b*c*d*sqrt(e)*integrate(1/3*sqrt(d*x + c)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/
(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 108*a^2*b^2*c^2*sqrt(e)*integrate(
1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))
)^2/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 72*a^3*b*c^2*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x
+ c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 3
*a^4*c^2*sqrt(e)*(2*arctan(sqrt(d*x + c)) - log(sqrt(d*x + c) + 1) + log(sqrt(d*x + c) - 1))/d + 48*b^4*d*sqrt
(e)*integrate(1/3*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 48*b^4*c*sqrt(e)*
integrate(1/3*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) - 6*(2*(c + 1)*arctan(sqr
t(d*x + c)) - (c - 1)*log(sqrt(d*x + c) + 1) + (c - 1)*log(sqrt(d*x + c) - 1) - 4*sqrt(d*x + c))*a^4*c*sqrt(e)
/d - 72*a*b^3*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(
d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) - 108*a^2*b^2*sqrt(e)*integrate(1/3*sqrt
(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^2/(d^2
*x^2 + 2*c*d*x + c^2 - 1), x) - 72*a^3*b*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqr
t(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + (6*(c^2 + 2*c
+ 1)*arctan(sqrt(d*x + c)) - 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) + 1) + 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) -
1) + 4*(d*x + c)^(3/2) - 24*sqrt(d*x + c)*c)*a^4*sqrt(e)/d - 3*a^4*sqrt(e)*(2*arctan(sqrt(d*x + c)) - log(sqr
t(d*x + c) + 1) + log(sqrt(d*x + c) - 1))/d)*d)/d
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]
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))^4,x)
[Out]
int((c*e + d*e*x)^(1/2)*(a + b*asin(c + d*x))^4, x)
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}\, dx \]
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))**4,x)
[Out]
Integral(sqrt(e*(c + d*x))*(a + b*asin(c + d*x))**4, x)
________________________________________________________________________________________