\(\int (f x)^m (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^n \, dx\) [457]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Int}\left ((f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 31, 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 = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx \]

[In]

Int[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx \]

[In]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]

[Out]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x]

Maple [N/A] (verified)

Not integrable

Time = 1.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]

[In]

int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)

[Out]

int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas")

[Out]

integral(-(c^2*d*x^2 - d)*sqrt(-c^2*d*x^2 + d)*(f*x)^m*(b*arccosh(c*x) + a)^n, x)

Sympy [F(-1)]

Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \]

[In]

integrate((f*x)**m*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**n,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(f*x)^m*(b*arccosh(c*x) + a)^n, x)

Giac [F(-1)]

Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 3.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \]

[In]

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2)*(f*x)^m,x)

[Out]

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2)*(f*x)^m, x)