\(\int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx\) [307]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c+d x))^4}{3 d e (e (c+d x))^{3/2}}+\frac {8 b \text {Int}\left (\frac {(a+b \arcsin (c+d x))^3}{(e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}},x\right )}{3 e} \]

[Out]

-2/3*(a+b*arcsin(d*x+c))^4/d/e/(e*(d*x+c))^(3/2)+8/3*b*Unintegrable((a+b*arcsin(d*x+c))^3/(e*(d*x+c))^(3/2)/(1
-(d*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx \]

[In]

Int[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2),x]

[Out]

(-2*(a + b*ArcSin[c + d*x])^4)/(3*d*e*(e*(c + d*x))^(3/2)) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcSin[x])^3/
((e*x)^(3/2)*Sqrt[1 - x^2]), x], x, c + d*x])/(3*d*e)

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^4}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \arcsin (c+d x))^4}{3 d e (e (c+d x))^{3/2}}+\frac {(8 b) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 10.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx \]

[In]

Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.80 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {5}{2}}}d x\]

[In]

int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x)

[Out]

int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

[Out]

integral((b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x
+ c) + a^4)*sqrt(d*e*x + c*e)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

Sympy [N/A]

Not integrable

Time = 40.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((a+b*asin(d*x+c))**4/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*asin(c + d*x))**4/(e*(c + d*x))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^4/(d*e*x + c*e)^(5/2), x)

Mupad [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]

[In]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(5/2), x)