\(\int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx\) [155]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.78 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{m}}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

integral((d*e*x + c*e)^m/(b*arcsinh(d*x + c) + a), x)

Sympy [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{m}}{a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{m}}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{m}}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.79 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c e+d e x)^m}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^m}{a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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