\(\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx\) [249]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=-\frac {2 (a+b \text {arcsinh}(c+d x))^3}{d e \sqrt {e (c+d x)}}+\frac {6 b \text {Int}\left (\frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}},x\right )}{e} \]

[Out]

-2*(a+b*arcsinh(d*x+c))^3/d/e/(e*(d*x+c))^(1/2)+6*b*Unintegrable((a+b*arcsinh(d*x+c))^2/(e*(d*x+c))^(1/2)/(1+(
d*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx \]

[In]

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

[Out]

(-2*(a + b*ArcSinh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)]) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^2/(S
qrt[e*x]*Sqrt[1 + x^2]), x], x, c + d*x])/(d*e)

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

Mathematica [N/A]

Not integrable

Time = 43.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(3/2),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 = 0.75 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((b*arcsinh(d*x + c) + a)^3/(d*e*x + c*e)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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