3.1240 \(\int x^{-2-2 p} (d+e x^2)^p (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=80 \[ b \text{Unintegrable}\left (x^{-2 p-2} \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right )-\frac{a x^{-2 p-1} \left (d+e x^2\right )^{p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},1,\frac{1}{2} (1-2 p),-\frac{e x^2}{d}\right )}{d (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.14238, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^{-2-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int x^{-2-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^{-2-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-2-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ &=b \int x^{-2-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx+\left (a \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p}\right ) \int x^{-2-2 p} \left (1+\frac{e x^2}{d}\right )^p \, dx\\ &=-\frac{a x^{-1-2 p} \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p} \, _2F_1\left (\frac{1}{2} (-1-2 p),-p;\frac{1}{2} (1-2 p);-\frac{e x^2}{d}\right )}{1+2 p}+b \int x^{-2-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ \end{align*}

Mathematica [A]  time = 3.01047, size = 0, normalized size = 0. \[ \int x^{-2-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.871, size = 0, normalized size = 0. \begin{align*} \int{x}^{-2-2\,p} \left ( e{x}^{2}+d \right ) ^{p} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-2*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 - 2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-2*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 - 2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-2*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 - 2), x)