\(\int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx\) [246]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {2 (e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))^3}{5 d e}-\frac {6 b \text {Int}\left (\frac {(e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))^2}{\sqrt {1+(c+d x)^2}},x\right )}{5 e} \]

[Out]

2/5*(e*(d*x+c))^(5/2)*(a+b*arcsinh(d*x+c))^3/d/e-6/5*b*Unintegrable((e*(d*x+c))^(5/2)*(a+b*arcsinh(d*x+c))^2/(
1+(d*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.15 (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 = {} \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 66.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.00 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 20.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arcsinh(d*x+c))^3,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 = 1.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

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

[Out]

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