\(\int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx\) [33]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2},x\right ) \]

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 14.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 607, normalized size of antiderivative = 33.72 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((c^2*x^2 + 1)^(3/2)*(e*x + d)^m + (c^3*x^3 + c*x)*(e*x + d)^m)/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x +
a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((c^3*e*
(m + 1)*x^3 + c^3*d*x^2 + c*e*(m - 1)*x - c*d)*(c^2*x^2 + 1)*(e*x + d)^m + (2*c^4*e*(m + 1)*x^4 + 2*c^4*d*x^3
+ c^2*e*(3*m + 1)*x^2 + c^2*d*x + e*m)*sqrt(c^2*x^2 + 1)*(e*x + d)^m + (c^5*e*(m + 1)*x^5 + c^5*d*x^4 + 2*c^3*
e*(m + 1)*x^3 + 2*c^3*d*x^2 + c*e*(m + 1)*x + c*d)*(e*x + d)^m)/(a*b*c^5*e*x^5 + a*b*c^5*d*x^4 + 2*a*b*c^3*e*x
^3 + 2*a*b*c^3*d*x^2 + a*b*c*e*x + a*b*c*d + (a*b*c^3*e*x^3 + a*b*c^3*d*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e*x^5 +
b^2*c^5*d*x^4 + 2*b^2*c^3*e*x^3 + 2*b^2*c^3*d*x^2 + b^2*c*e*x + b^2*c*d + (b^2*c^3*e*x^3 + b^2*c^3*d*x^2)*(c^2
*x^2 + 1) + 2*(b^2*c^4*e*x^4 + b^2*c^4*d*x^3 + b^2*c^2*e*x^2 + b^2*c^2*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(
c^2*x^2 + 1)) + 2*(a*b*c^4*e*x^4 + a*b*c^4*d*x^3 + a*b*c^2*e*x^2 + a*b*c^2*d*x)*sqrt(c^2*x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^m}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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