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

Optimal. Leaf size=81 \[ b \text {Int}\left (x^{-2 p-2} \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right )-\frac {a x^{-2 p-1} \left (d+e x^2\right )^{p+1} \, _2F_1\left (\frac {1}{2},1;\frac {1}{2} (1-2 p);-\frac {e x^2}{d}\right )}{d (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 3.25, size = 0, normalized size = 0.00 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

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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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}\,{d x} \]

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)

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maple [A]  time = 1.61, size = 0, normalized size = 0.00 \[ \int x^{-2-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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