\(\int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx\) [1235]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\frac {a x^{1+m} \left (d+e x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (1,\frac {4+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (x^m \sqrt {d+e x^2} \arctan (c x),x\right ) \]

[Out]

a*x^(1+m)*(e*x^2+d)^(3/2)*hypergeom([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)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 23, 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 = {} \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

int(x^m*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^m*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 42.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

integrate(x**m*(e*x**2+d)**(1/2)*(a+b*atan(c*x)),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)*x^m, x)

Giac [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)*x^m, x)

Mupad [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]

[In]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^(1/2),x)

[Out]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^(1/2), x)