3.13.31 \(\int \frac {x^m (a+b \text {ArcTan}(c x))}{d+e x^2} \, dx\) [1231]

Optimal. Leaf size=63 \[ \frac {a x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (\frac {x^m \text {ArcTan}(c x)}{d+e x^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, 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 \frac {x^m (a+b \text {ArcTan}(c x))}{d+e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx &=a \int \frac {x^m}{d+e x^2} \, dx+b \int \frac {x^m \tan ^{-1}(c x)}{d+e x^2} \, dx\\ &=\frac {a x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right )}{d (1+m)}+b \int \frac {x^m \tan ^{-1}(c x)}{d+e x^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m (a+b \text {ArcTan}(c x))}{d+e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.79, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*(a + b*atan(c*x))/(d + e*x**2), x)

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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