\(\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx\) [221]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx \]

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2)),x]

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcSinh[x])^(5/2)), 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))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx \]

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2)),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

[In]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(5/2),x)

[Out]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 8.74 (sec) , antiderivative size = 155, normalized size of antiderivative = 6.20 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\frac {\int \frac {1}{a^{2} c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + a^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 2 a b d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx}{e} \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*asinh(d*x+c))**(5/2),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(5/2)), x)

Giac [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(5/2)), x)

Mupad [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(1/((c*e + d*e*x)*(a + b*asinh(c + d*x))^(5/2)),x)

[Out]

int(1/((c*e + d*e*x)*(a + b*asinh(c + d*x))^(5/2)), x)