\(\int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx\) [36]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))^3}{e (1+m)}-\frac {3 b c \text {Int}\left (\frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}},x\right )}{e (1+m)} \]

[Out]

(e*x+d)^(1+m)*(a+b*arccosh(c*x))^3/e/(1+m)-3*b*c*Unintegrable((e*x+d)^(1+m)*(a+b*arccosh(c*x))^2/(c*x-1)^(1/2)
/(c*x+1)^(1/2),x)/e/(1+m)

Rubi [N/A]

Not integrable

Time = 0.26 (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 = {} \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx \]

[In]

Int[(d + e*x)^m*(a + b*ArcCosh[c*x])^3,x]

[Out]

((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^3)/(e*(1 + m)) - (3*b*c*Defer[Int][((d + e*x)^(1 + m)*(a + b*ArcCosh[c
*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))^3}{e (1+m)}-\frac {(3 b c) \int \frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e (1+m)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx \]

[In]

Integrate[(d + e*x)^m*(a + b*ArcCosh[c*x])^3,x]

[Out]

Integrate[(d + e*x)^m*(a + b*ArcCosh[c*x])^3, x]

Maple [N/A] (verified)

Not integrable

Time = 2.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \left (e x +d \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{3}d x\]

[In]

int((e*x+d)^m*(a+b*arccosh(c*x))^3,x)

[Out]

int((e*x+d)^m*(a+b*arccosh(c*x))^3,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="fricas")

[Out]

integral((b^3*arccosh(c*x)^3 + 3*a*b^2*arccosh(c*x)^2 + 3*a^2*b*arccosh(c*x) + a^3)*(e*x + d)^m, x)

Sympy [N/A]

Not integrable

Time = 36.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{m}\, dx \]

[In]

integrate((e*x+d)**m*(a+b*acosh(c*x))**3,x)

[Out]

Integral((a + b*acosh(c*x))**3*(d + e*x)**m, x)

Maxima [N/A]

Not integrable

Time = 2.10 (sec) , antiderivative size = 405, normalized size of antiderivative = 22.50 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^3*(e*x + d)^m, x)

Mupad [N/A]

Not integrable

Time = 3.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m (a+b \text {arccosh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^m \,d x \]

[In]

int((a + b*acosh(c*x))^3*(d + e*x)^m,x)

[Out]

int((a + b*acosh(c*x))^3*(d + e*x)^m, x)