3.1231 \(\int \frac {x^m (a+b \tan ^{-1}(c x))}{d+e x^2} \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx \]

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*}

________________________________________________________________________________________

Mathematica [A]  time = 2.39, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d}, x\right ) \]

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/(e*x^2 + d), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 2.01, 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)

________________________________________________________________________________________

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

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/(e*x^2 + d), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________