\(\int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx\) [255]

Optimal result

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

[Out]

2/3*(e*(d*x+c))^(3/2)*(a+b*arcsinh(d*x+c))^4/d/e-8/3*b*Unintegrable((e*(d*x+c))^(3/2)*(a+b*arcsinh(d*x+c))^3/(
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 = {} \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx \]

[In]

Int[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 57.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx \]

[In]

Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4, x]

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{4} \sqrt {d e x +c e}d x\]

[In]

int((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \]

[In]

integrate((a+b*asinh(d*x+c))**4*(d*e*x+c*e)**(1/2),x)

[Out]

Integral(sqrt(e*(c + d*x))*(a + b*asinh(c + d*x))**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),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 = 1.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^4, x)

Mupad [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^4, x)