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

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \arcsin (c+d x))^3},x\right )}{e} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx \]

[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 [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{3}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.96 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.87 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\frac {\int \frac {1}{a^{3} c + a^{3} d x + 3 a^{2} b c \operatorname {asin}{\left (c + d x \right )} + 3 a^{2} b d x \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )} + 3 a b^{2} d x \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )} + b^{3} d x \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx}{e} \]

[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

Maxima [N/A]

Not integrable

Time = 196.97 (sec) , antiderivative size = 520, normalized size of antiderivative = 22.61 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

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

Giac [N/A]

Not integrable

Time = 7.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[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 [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^3} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]

[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)