\(\int (f x)^m (d+c^2 d x^2)^q (a+b \arctan (c x))^p \, dx\) [1113]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\text {Int}\left ((f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p,x\right ) \]

[Out]

Unintegrable((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 28, 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 = {} \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx \]

[In]

Int[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p,x]

[Out]

Defer[Int][(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x]

Rubi steps \begin{align*} \text {integral}& = \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx \]

[In]

Integrate[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p,x]

[Out]

Integrate[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x]

Maple [N/A] (verified)

Not integrable

Time = 1.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \left (f x \right )^{m} \left (c^{2} d \,x^{2}+d \right )^{q} \left (a +b \arctan \left (c x \right )\right )^{p}d x\]

[In]

int((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x)

[Out]

int((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{q} \left (f x\right )^{m} {\left (b \arctan \left (c x\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x, algorithm="fricas")

[Out]

integral((c^2*d*x^2 + d)^q*(f*x)^m*(b*arctan(c*x) + a)^p, x)

Sympy [F(-1)]

Timed out. \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\text {Timed out} \]

[In]

integrate((f*x)**m*(c**2*d*x**2+d)**q*(a+b*atan(c*x))**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{q} \left (f x\right )^{m} {\left (b \arctan \left (c x\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^q*(f*x)^m*(b*arctan(c*x) + a)^p, x)

Giac [N/A]

Not integrable

Time = 80.87 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.11 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{q} \left (f x\right )^{m} {\left (b \arctan \left (c x\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((f*x)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q (a+b \arctan (c x))^p \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^p\,{\left (d\,c^2\,x^2+d\right )}^q\,{\left (f\,x\right )}^m \,d x \]

[In]

int((a + b*atan(c*x))^p*(d + c^2*d*x^2)^q*(f*x)^m,x)

[Out]

int((a + b*atan(c*x))^p*(d + c^2*d*x^2)^q*(f*x)^m, x)