\(\int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx\) [226]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \arcsin (c+d x))^2},x\right )}{e} \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*arcsin(d*x+c))^2,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 = {} \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx \]

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2),x]

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcSin[x])^2), 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))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx \]

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsin(d*x+c))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.86 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\frac {\int \frac {1}{a^{2} c + a^{2} d x + 2 a b c \operatorname {asin}{\left (c + d x \right )} + 2 a b d x \operatorname {asin}{\left (c + d x \right )} + b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )} + b^{2} d x \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx}{e} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 4.16 (sec) , antiderivative size = 357, normalized size of antiderivative = 15.52 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsin(d*x+c))^2,x, algorithm="maxima")

[Out]

((a*b*d^2*e*x + a*b*c*d*e + (b^2*d^2*e*x + b^2*c*d*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))*
integrate(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)/(a*b*d^4*e*x^4 + 4*a*b*c*d^3*e*x^3 + (6*a*b*c^2 - a*b)*d^2*e*x^
2 + 2*(2*a*b*c^3 - a*b*c)*d*e*x + (a*b*c^4 - a*b*c^2)*e + (b^2*d^4*e*x^4 + 4*b^2*c*d^3*e*x^3 + (6*b^2*c^2 - b^
2)*d^2*e*x^2 + 2*(2*b^2*c^3 - b^2*c)*d*e*x + (b^2*c^4 - b^2*c^2)*e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d
*x - c + 1))), x) - sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))/(a*b*d^2*e*x + a*b*c*d*e + (b^2*d^2*e*x + b^2*c*d*e)
*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))

Giac [N/A]

Not integrable

Time = 2.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsin(d*x+c))^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \arcsin (c+d x))^2} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int(1/((c*e + d*e*x)*(a + b*asin(c + d*x))^2),x)

[Out]

int(1/((c*e + d*e*x)*(a + b*asin(c + d*x))^2), x)