\(\int x^{-4-2 p} (d+e x^2)^p (a+b \arctan (c x)) \, dx\) [1242]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=-\frac {a x^{-3-2 p} \left (d+e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2} (-1-2 p),-\frac {e x^2}{d}\right )}{d (3+2 p)}+b \text {Int}\left (x^{-4-2 p} \left (d+e x^2\right )^p \arctan (c x),x\right ) \]

[Out]

-a*x^(-3-2*p)*(e*x^2+d)^(p+1)*hypergeom([-1/2, 1],[-1/2-p],-e*x^2/d)/d/(3+2*p)+b*Unintegrable(x^(-4-2*p)*(e*x^
2+d)^p*arctan(c*x),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 25, 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 = {} \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]

[In]

Int[x^(-4 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]

[Out]

-((a*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-3 - 2*p)/2, -p, (-1 - 2*p)/2, -((e*x^2)/d)])/((3 + 2*p)*(1
+ (e*x^2)/d)^p)) + b*Defer[Int][x^(-4 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]

Rubi steps \begin{align*} \text {integral}& = a \int x^{-4-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-4-2 p} \left (d+e x^2\right )^p \arctan (c x) \, dx \\ & = b \int x^{-4-2 p} \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^{-4-2 p} \left (1+\frac {e x^2}{d}\right )^p \, dx \\ & = -\frac {a x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-p,\frac {1}{2} (-1-2 p),-\frac {e x^2}{d}\right )}{3+2 p}+b \int x^{-4-2 p} \left (d+e x^2\right )^p \arctan (c x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]

[In]

Integrate[x^(-4 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]

[Out]

Integrate[x^(-4 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[\int x^{-4-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )d x\]

[In]

int(x^(-4-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x)

[Out]

int(x^(-4-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 4} \,d x } \]

[In]

integrate(x^(-4-2*p)*(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^(-2*p - 4), x)

Sympy [F(-1)]

Timed out. \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**(-4-2*p)*(e*x**2+d)**p*(a+b*atan(c*x)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 4} \,d x } \]

[In]

integrate(x^(-4-2*p)*(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^(-2*p - 4), x)

Giac [N/A]

Not integrable

Time = 3.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 4} \,d x } \]

[In]

integrate(x^(-4-2*p)*(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^(-2*p - 4), x)

Mupad [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+4}} \,d x \]

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^p)/x^(2*p + 4),x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^p)/x^(2*p + 4), x)