\(\int x^m (d+e x^2)^p (a+b \arctan (c x)) \, dx\) [1239]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\frac {a x^{1+m} \left (d+e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (3+m+2 p),\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (x^m \left (d+e x^2\right )^p \arctan (c x),x\right ) \]

[Out]

a*x^(1+m)*(e*x^2+d)^(p+1)*hypergeom([1, 3/2+1/2*m+p],[3/2+1/2*m],-e*x^2/d)/d/(1+m)+b*Unintegrable(x^m*(e*x^2+d
)^p*arctan(c*x),x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 21, 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 = {} \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

(a*x^(1 + m)*(d + e*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*(1 + (e*x^2)/d)
^p) + b*Defer[Int][x^m*(d + e*x^2)^p*ArcTan[c*x], x]

Rubi steps \begin{align*} \text {integral}& = a \int x^m \left (d+e x^2\right )^p \, dx+b \int x^m \left (d+e x^2\right )^p \arctan (c x) \, dx \\ & = b \int x^m \left (d+e x^2\right )^p \arctan (c x) \, dx+\left (a \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p}\right ) \int x^m \left (1+\frac {e x^2}{d}\right )^p \, dx \\ & = \frac {a x^{1+m} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {e x^2}{d}\right )}{1+m}+b \int x^m \left (d+e x^2\right )^p \arctan (c x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.74 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int x^m \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p \,d x \]

[In]

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

[Out]

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