3.4.3 \(\int \frac {(a+b \text {ArcSin}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx\) [303]
Optimal. Leaf size=82 \[ -\frac {2 (a+b \text {ArcSin}(c+d x))^3}{3 d e (e (c+d x))^{3/2}}+\frac {2 b \text {Int}\left (\frac {(a+b \text {ArcSin}(c+d x))^2}{(e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}},x\right )}{e} \]
[Out]
-2/3*(a+b*arcsin(d*x+c))^3/d/e/(e*(d*x+c))^(3/2)+2*b*Unintegrable((a+b*arcsin(d*x+c))^2/(e*(d*x+c))^(3/2)/(1-(
d*x+c)^2)^(1/2),x)/e
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Rubi [A]
time = 0.14, 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 = {}
\begin {gather*} \int \frac {(a+b \text {ArcSin}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2),x]
[Out]
(-2*(a + b*ArcSin[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2)) + (2*b*Defer[Subst][Defer[Int][(a + b*ArcSin[x])^2/
((e*x)^(3/2)*Sqrt[1 - x^2]), x], x, c + d*x])/(d*e)
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 93.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcSin}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2),x]
[Out]
Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2), x]
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Maple [A]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x)
[Out]
int((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x)
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="maxima")
[Out]
-1/6*(4*b^3*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3*e^(1/2) - (18*a*b^2*d^2*e^(1/2)*integrate
(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*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 +
c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) + 18*a^2*b*d^2*e^(1/2)*integrate(sqrt(d*x + c)*x^2*ar
ctan(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*x^5*e^3 + 5*c
*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e
^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) + 36*a*b^2*c*d*e^(1/2)*integrate(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*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d
^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^
3*e^3), x) + 36*a^2*b*c*d*e^(1/2)*integrate(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*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x
^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) + 18*a*b^2*c^2
*e^(1/2)*integrate(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*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 -
d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) + 18*a^2*b*c^2*e^(1/2)*integrate(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*x^5
*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c
*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) - (6*arctan(sqrt(d*x + c))*e^(-3) + 3*e^(-3)*log(sqrt(d*x + c) + 1
) - 3*e^(-3)*log(sqrt(d*x + c) - 1) - 4*e^(-3)/(d*x + c)^(3/2))*a^3*c^2*e^(1/2)/d - 12*b^3*d*e^(1/2)*integrate
(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/(s
qrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^2/(d^5*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e
^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) - 12*b^3*c*e^(1/2)
*integrate(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)))^2/(d^5*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d
^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) + 2*(6*(c
+ 1)*arctan(sqrt(d*x + c))*e^(-3) + 3*(c - 1)*e^(-3)*log(sqrt(d*x + c) + 1) - 3*(c - 1)*e^(-3)*log(sqrt(d*x +
c) - 1) + 4*(3*d*x + 2*c)*e^(-3)/(d*x + c)^(3/2))*a^3*c*e^(1/2)/d - 18*a*b^2*e^(1/2)*integrate(sqrt(d*x + c)*a
rctan(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*x^5*e^3 +
5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^
2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3), x) - 18*a^2*b*e^(1/2)*integrate(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*x^5*e^3 + 5*c*d^4*x^4*e^3 + 10*c^2*d^3*x^
3*e^3 + 10*c^3*d^2*x^2*e^3 + 5*c^4*d*x*e^3 - d^3*x^3*e^3 + c^5*e^3 - 3*c*d^2*x^2*e^3 - 3*c^2*d*x*e^3 - c^3*e^3
), x) - (6*(c^2 + 2*c + 1)*arctan(sqrt(d*x + c))*e^(-3) + 3*(c^2 - 2*c + 1)*e^(-3)*log(sqrt(d*x + c) + 1) - 3*
(c^2 - 2*c + 1)*e^(-3)*log(sqrt(d*x + c) - 1) + 4*(6*(d*x + c)*c - c^2)*e^(-3)/(d*x + c)^(3/2))*a^3*e^(1/2)/d
+ (6*arctan(sqrt(d*x + c))*e^(-3) + 3*e^(-3)*log(sqrt(d*x + c) + 1) - 3*e^(-3)*log(sqrt(d*x + c) - 1) - 4*e^(-
3)/(d*x + c)^(3/2))*a^3*e^(1/2)/d)*(d^2*x*e^3 + c*d*e^3)*sqrt(d*x + c))/((d^2*x*e^3 + c*d*e^3)*sqrt(d*x + c))
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),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*x + c)*e^(
-5/2)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asin(d*x+c))**3/(d*e*x+c*e)**(5/2),x)
[Out]
Integral((a + b*asin(c + d*x))**3/(e*(c + d*x))**(5/2), x)
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="giac")
[Out]
integrate((b*arcsin(d*x + c) + a)^3/(d*e*x + c*e)^(5/2), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c + d*x))^3/(c*e + d*e*x)^(5/2),x)
[Out]
int((a + b*asin(c + d*x))^3/(c*e + d*e*x)^(5/2), x)
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