\(\int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx\) [188]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\text {Int}\left (\frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d} \,d x } \]

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)*x^m/(c^2*d*x^2 + d), x)

Sympy [N/A]

Not integrable

Time = 2.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {\int \frac {a x^{m}}{c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \]

[In]

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

[Out]

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