\(\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx\) [179]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \text {arcsinh}(c+d x))^4},x\right )}{e} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx \]

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.67 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx \]

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4), x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{4}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 5.26 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

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

[Out]

integral(1/(a^4*d*e*x + a^4*c*e + (b^4*d*e*x + b^4*c*e)*arcsinh(d*x + c)^4 + 4*(a*b^3*d*e*x + a*b^3*c*e)*arcsi
nh(d*x + c)^3 + 6*(a^2*b^2*d*e*x + a^2*b^2*c*e)*arcsinh(d*x + c)^2 + 4*(a^3*b*d*e*x + a^3*b*c*e)*arcsinh(d*x +
c)), x)

Sympy [N/A]

Not integrable

Time = 7.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.57 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {\int \frac {1}{a^{4} c + a^{4} d x + 4 a^{3} b c \operatorname {asinh}{\left (c + d x \right )} + 4 a^{3} b d x \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )} + 6 a^{2} b^{2} d x \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )} + 4 a b^{3} d x \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} c \operatorname {asinh}^{4}{\left (c + d x \right )} + b^{4} d x \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx}{e} \]

[In]

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

[Out]

Integral(1/(a**4*c + a**4*d*x + 4*a**3*b*c*asinh(c + d*x) + 4*a**3*b*d*x*asinh(c + d*x) + 6*a**2*b**2*c*asinh(
c + d*x)**2 + 6*a**2*b**2*d*x*asinh(c + d*x)**2 + 4*a*b**3*c*asinh(c + d*x)**3 + 4*a*b**3*d*x*asinh(c + d*x)**
3 + b**4*c*asinh(c + d*x)**4 + b**4*d*x*asinh(c + d*x)**4), x)/e

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.65 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]

[In]

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

[Out]

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