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

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\frac {a x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (\frac {x^m \arctan (c x)}{d+e x^2},x\right ) \]

[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)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx \]

[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*} \text {integral}& = a \int \frac {x^m}{d+e x^2} \, dx+b \int \frac {x^m \arctan (c x)}{d+e x^2} \, dx \\ & = \frac {a x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \int \frac {x^m \arctan (c x)}{d+e x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.57 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{e \,x^{2}+d}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 160.77 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int \frac {x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 154.14 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x^m (a+b \arctan (c x))}{d+e x^2} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \]

[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)