\(\int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx\) [32]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\text {Int}\left (\frac {(d+e x)^m}{a+b \text {arcsinh}(c x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.95 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (d + e x\right )^{m}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]

[In]

int((d + e*x)^m/(a + b*asinh(c*x)),x)

[Out]

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