3.307 \(\int \frac {(a+b \sin ^{-1}(c+d x))^4}{(c e+d e x)^{5/2}} \, dx\)
Optimal. Leaf size=84 \[ \frac {8 b \text {Int}\left (\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{\sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}},x\right )}{3 e}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d e (e (c+d x))^{3/2}} \]
[Out]
-2/3*(a+b*arcsin(d*x+c))^4/d/e/(e*(d*x+c))^(3/2)+8/3*b*Unintegrable((a+b*arcsin(d*x+c))^3/(e*(d*x+c))^(3/2)/(1
-(d*x+c)^2)^(1/2),x)/e
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Rubi [A] time = 0.21, 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 =
{} \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{(c e+d e x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2),x]
[Out]
(-2*(a + b*ArcSin[c + d*x])^4)/(3*d*e*(e*(c + d*x))^(3/2)) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcSin[x])^3/
((e*x)^(3/2)*Sqrt[1 - x^2]), x], x, c + d*x])/(3*d*e)
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{(c e+d e x)^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^4}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{3 d e (e (c+d x))^{3/2}}+\frac {(8 b) \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}
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Mathematica [A] time = 46.91, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{(c e+d e x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2),x]
[Out]
Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2), x]
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fricas [A] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),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)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x, algorithm="giac")
[Out]
integrate((b*arcsin(d*x + c) + a)^4/(d*e*x + c*e)^(5/2), x)
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maple [A] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x)
[Out]
int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x)
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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((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x, algorithm="maxima")
[Out]
-1/6*(4*b^4*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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^
3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), 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^5*
e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 -
3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), 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^5*e^3*x^5 + 5*c*d^4*
e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 -
3*c^2*d*e^3*x - c^3*e^3), 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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^
3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3
*e^3), 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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c
^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), 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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^
4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), x) + 144*a^3*b*c*d*sqrt(e)*int
egrate(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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*
x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), 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^5*e^
3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*
c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), x) + 72*a^3*b*c^2*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sq
rt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^5*e^3*x^5 + 5*c*d^4*e^3*x^4
+ 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d
*e^3*x - c^3*e^3), x) - a^4*c^2*sqrt(e)*(6*arctan(sqrt(d*x + c))/e^3 + 3*log(sqrt(d*x + c) + 1)/e^3 - 3*log(sq
rt(d*x + c) - 1)/e^3 - 4/((d*x + c)^(3/2)*e^3))/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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3
*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), 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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3
*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), x) + 2*a^4*c*sqrt(e)*(6*(c + 1)*arcta
n(sqrt(d*x + c))/e^3 + 3*(c - 1)*log(sqrt(d*x + c) + 1)/e^3 - 3*(c - 1)*log(sqrt(d*x + c) - 1)/e^3 + 4*(3*d*x
+ 2*c)/((d*x + c)^(3/2)*e^3))/d - 72*a*b^3*sqrt(e)*integrate(1/3*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*s
qrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e
^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^
3), 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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^
3*x^2 + 5*c^4*d*e^3*x - d^3*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), x) - 72*a^3*b*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^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + 10*c^2*d^3*e^3*x^3 + 10*c^3*d^2*e^3*x^2 + 5*c^4*d*e^3*x - d^3
*e^3*x^3 + c^5*e^3 - 3*c*d^2*e^3*x^2 - 3*c^2*d*e^3*x - c^3*e^3), x) - a^4*sqrt(e)*(6*(c^2 + 2*c + 1)*arctan(sq
rt(d*x + c))/e^3 + 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) + 1)/e^3 - 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) - 1)/e^3
+ 4*(6*(d*x + c)*c - c^2)/((d*x + c)^(3/2)*e^3))/d + a^4*sqrt(e)*(6*arctan(sqrt(d*x + c))/e^3 + 3*log(sqrt(d*
x + c) + 1)/e^3 - 3*log(sqrt(d*x + c) - 1)/e^3 - 4/((d*x + c)^(3/2)*e^3))/d)*(d^2*e^3*x + c*d*e^3)*sqrt(d*x +
c))/((d^2*e^3*x + c*d*e^3)*sqrt(d*x + c))
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(5/2),x)
[Out]
int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(5/2), x)
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asin(d*x+c))**4/(d*e*x+c*e)**(5/2),x)
[Out]
Integral((a + b*asin(c + d*x))**4/(e*(c + d*x))**(5/2), x)
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