3.3.32 \(\int \frac {1}{(c e+d e x) (a+b \text {ArcSin}(c+d x))^3} \, dx\) [232]

Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \text {ArcSin}(c+d x))^3},x\right )}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, 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 {1}{(c e+d e x) (a+b \text {ArcSin}(c+d x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3),x]

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) (a+b \text {ArcSin}(c+d x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3), x]

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

-1/2*((b*d*x + b*c)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1) - b*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c
+ 1)) - 2*(a^2*b^2*d^3*x^2*e + 2*a^2*b^2*c*d^2*x*e + a^2*b^2*c^2*d*e + (b^4*d^3*x^2*e + 2*b^4*c*d^2*x*e + b^4*
c^2*d*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*(a*b^3*d^3*x^2*e + 2*a*b^3*c*d^2*x*e + a
*b^3*c^2*d*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))*integrate(1/(a*b^2*d^3*x^3*e + 3*a*b^2*c
*d^2*x^2*e + 3*a*b^2*c^2*d*x*e + a*b^2*c^3*e + (b^3*d^3*x^3*e + 3*b^3*c*d^2*x^2*e + 3*b^3*c^2*d*x*e + b^3*c^3*
e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))), x) - a)/(a^2*b^2*d^3*x^2*e + 2*a^2*b^2*c*d^2*x*e +
a^2*b^2*c^2*d*e + (b^4*d^3*x^2*e + 2*b^4*c*d^2*x*e + b^4*c^2*d*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*
x - c + 1))^2 + 2*(a*b^3*d^3*x^2*e + 2*a*b^3*c*d^2*x*e + a*b^3*c^2*d*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqr
t(-d*x - c + 1)))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{3} c + a^{3} d x + 3 a^{2} b c \operatorname {asin}{\left (c + d x \right )} + 3 a^{2} b d x \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )} + 3 a b^{2} d x \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )} + b^{3} d x \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________