\(\int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx\) [197]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 22.46 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}{c + d x}\, dx + \int \frac {b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{d e x + c e} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{d e x + c e} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.54 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}}{c\,e+d\,e\,x} \,d x \]

[In]

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

[Out]

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