\(\int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx\) [215]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\frac {2 (d x)^{5/2} (a+b \arcsin (c x))^3}{5 d}-\frac {6 b c \text {Int}\left (\frac {(d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{5 d} \]

[Out]

2/5*(d*x)^(5/2)*(a+b*arcsin(c*x))^3/d-6/5*b*c*Unintegrable((d*x)^(5/2)*(a+b*arcsin(c*x))^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 = {} \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx \]

[In]

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

[Out]

(2*(d*x)^(5/2)*(a + b*ArcSin[c*x])^3)/(5*d) - (6*b*c*Defer[Int][((d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c
^2*x^2], x])/(5*d)

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{5/2} (a+b \arcsin (c x))^3}{5 d}-\frac {(6 b c) \int \frac {(d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{5 d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 58.95 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.94 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 78.70 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.55 (sec) , antiderivative size = 421, normalized size of antiderivative = 23.39 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^3 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

2/5*b^3*d^(3/2)*x^(5/2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3 + 1/10*a^3*c^2*d^(3/2)*(4*(c^2*x^(5/2) +
5*sqrt(x))/c^4 - 10*arctan(sqrt(c)*sqrt(x))/c^(9/2) + 5*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(9/
2)) + 15*a*b^2*c^2*d^(3/2)*integrate(1/5*x^(7/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^2/(c^2*x^2 - 1), x
) + 15*a^2*b*c^2*d^(3/2)*integrate(1/5*x^(7/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*x^2 - 1), x) +
6*b^3*c*d^(3/2)*integrate(1/5*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^
2/(c^2*x^2 - 1), x) - 1/2*a^3*d^(3/2)*(4*sqrt(x)/c^2 - 2*arctan(sqrt(c)*sqrt(x))/c^(5/2) + log((c*sqrt(x) - sq
rt(c))/(c*sqrt(x) + sqrt(c)))/c^(5/2)) - 15*a*b^2*d^(3/2)*integrate(1/5*x^(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt
(-c*x + 1)))^2/(c^2*x^2 - 1), x) - 15*a^2*b*d^(3/2)*integrate(1/5*x^(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x
+ 1)))/(c^2*x^2 - 1), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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