\(\int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx\) [219]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c x))^3}{3 d (d x)^{3/2}}+\frac {2 b c \text {Int}\left (\frac {(a+b \arcsin (c x))^2}{(d x)^{3/2} \sqrt {1-c^2 x^2}},x\right )}{d} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx \]

[In]

Int[(a + b*ArcSin[c*x])^3/(d*x)^(5/2),x]

[Out]

(-2*(a + b*ArcSin[c*x])^3)/(3*d*(d*x)^(3/2)) + (2*b*c*Defer[Int][(a + b*ArcSin[c*x])^2/((d*x)^(3/2)*Sqrt[1 - c
^2*x^2]), x])/d

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b \arcsin (c x))^3}{3 d (d x)^{3/2}}+\frac {(2 b c) \int \frac {(a+b \arcsin (c x))^2}{(d x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 41.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx \]

[In]

Integrate[(a + b*ArcSin[c*x])^3/(d*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcSin[c*x])^3/(d*x)^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{3}}{\left (d x \right )^{\frac {5}{2}}}d x\]

[In]

int((a+b*arcsin(c*x))^3/(d*x)^(5/2),x)

[Out]

int((a+b*arcsin(c*x))^3/(d*x)^(5/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*asin(c*x))**3/(d*x)**(5/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [N/A]

Not integrable

Time = 3.59 (sec) , antiderivative size = 471, normalized size of antiderivative = 26.17 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(5/2),x, algorithm="maxima")

[Out]

-1/6*(4*b^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3 + (3*a^3*c^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/(sqrt
(c)*d^3) - log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(sqrt(c)*d^3)) - 18*a*b^2*c^2*sqrt(d)*integrate(x^
(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^2/(c^2*d^3*x^5 - d^3*x^3), x) - 18*a^2*b*c^2*sqrt(d)*integrat
e(x^(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d^3*x^5 - d^3*x^3), x) + 12*b^3*c*sqrt(d)*integrate(
sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^2/(c^2*d^3*x^5 - d^3*x^3), x)
- a^3*sqrt(d)*(6*c^(3/2)*arctan(sqrt(c)*sqrt(x))/d^3 - 3*c^(3/2)*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c
)))/d^3 - 4/(d^3*x^(3/2))) + 18*a*b^2*sqrt(d)*integrate(sqrt(x)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^2/(
c^2*d^3*x^5 - d^3*x^3), x) + 18*a^2*b*sqrt(d)*integrate(sqrt(x)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^
2*d^3*x^5 - d^3*x^3), x))*d^(5/2)*x^(3/2))/(d^(5/2)*x^(3/2))

Giac [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^3/(d*x)^(5/2), x)

Mupad [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^3}{(d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{{\left (d\,x\right )}^{5/2}} \,d x \]

[In]

int((a + b*asin(c*x))^3/(d*x)^(5/2),x)

[Out]

int((a + b*asin(c*x))^3/(d*x)^(5/2), x)