3.13.46 \(\int x^{-8-2 p} (d+e x^2)^p (a+b \text {ArcTan}(c x)) \, dx\) [1246]

Optimal. Leaf size=81 \[ -\frac {a x^{-7-2 p} \left (d+e x^2\right )^{1+p} \, _2F_1\left (-\frac {5}{2},1;\frac {1}{2} (-5-2 p);-\frac {e x^2}{d}\right )}{d (7+2 p)}+b \text {Int}\left (x^{-8-2 p} \left (d+e x^2\right )^p \text {ArcTan}(c x),x\right ) \]

[Out]

-a*x^(-7-2*p)*(e*x^2+d)^(1+p)*hypergeom([-5/2, 1],[-5/2-p],-e*x^2/d)/d/(7+2*p)+b*Unintegrable(x^(-8-2*p)*(e*x^
2+d)^p*arctan(c*x),x)

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Rubi [A]
time = 0.10, 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 = {} \begin {gather*} \int x^{-8-2 p} \left (d+e x^2\right )^p (a+b \text {ArcTan}(c x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^(-8 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]

[Out]

-((a*x^(-7 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-7 - 2*p)/2, -p, (-5 - 2*p)/2, -((e*x^2)/d)])/((7 + 2*p)*(1
 + (e*x^2)/d)^p)) + b*Defer[Int][x^(-8 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]

Rubi steps

\begin {align*} \int x^{-8-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^{-8-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-8-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ &=b \int x^{-8-2 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^{-8-2 p} \left (1+\frac {e x^2}{d}\right )^p \, dx\\ &=-\frac {a x^{-7-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \, _2F_1\left (\frac {1}{2} (-7-2 p),-p;\frac {1}{2} (-5-2 p);-\frac {e x^2}{d}\right )}{7+2 p}+b \int x^{-8-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ \end {align*}

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Mathematica [A]
time = 2.16, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-8-2 p} \left (d+e x^2\right )^p (a+b \text {ArcTan}(c x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^(-8 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]

[Out]

Integrate[x^(-8 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]

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Maple [A]
time = 1.25, size = 0, normalized size = 0.00 \[\int x^{-8-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-8-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x)

[Out]

int(x^(-8-2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

integrate((b*arctan(c*x) + a)*(x^2*e + d)^p*x^(-2*p - 8), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)*(x^2*e + d)^p*x^(-2*p - 8), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-8-2*p)*(e*x**2+d)**p*(a+b*atan(c*x)),x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-8-2*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 - 8), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+8}} \,d x \end {gather*}

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 + 8),x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^p)/x^(2*p + 8), x)

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