\(\int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\frac {(d+e x)^{1+m} (a+b \text {arcsinh}(c x))^2}{e (1+m)}-\frac {2 b c \text {Int}\left (\frac {(d+e x)^{1+m} (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}},x\right )}{e (1+m)} \]

[Out]

(e*x+d)^(1+m)*(a+b*arcsinh(c*x))^2/e/(1+m)-2*b*c*Unintegrable((e*x+d)^(1+m)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/
2),x)/e/(1+m)

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx \]

[In]

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

[Out]

((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x])^2)/(e*(1 + m)) - (2*b*c*Defer[Int][((d + e*x)^(1 + m)*(a + b*ArcSinh[c
*x]))/Sqrt[1 + c^2*x^2], x])/(e*(1 + m))

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} (a+b \text {arcsinh}(c x))^2}{e (1+m)}-\frac {(2 b c) \int \frac {(d+e x)^{1+m} (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{e (1+m)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx \]

[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 = 1.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[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 = 32, normalized size of antiderivative = 1.78 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} {\left (e x + d\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{m}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 271, normalized size of antiderivative = 15.06 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} {\left (e x + d\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} {\left (e x + d\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^m \,d x \]

[In]

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

[Out]

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