\(\int x^m (d+e x^2)^{3/2} (a+b \arctan (c x)) \, dx\) [1234]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\frac {a x^{1+m} \left (d+e x^2\right )^{5/2} \operatorname {Hypergeometric2F1}\left (1,\frac {6+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (x^m \left (d+e x^2\right )^{3/2} \arctan (c x),x\right ) \]

[Out]

a*x^(1+m)*(e*x^2+d)^(5/2)*hypergeom([1, 3+1/2*m],[3/2+1/2*m],-e*x^2/d)/d/(1+m)+b*Unintegrable(x^m*(e*x^2+d)^(3
/2)*arctan(c*x),x)

Rubi [N/A]

Not integrable

Time = 0.11 (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 = {} \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx \]

[In]

Integrate[x^m*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]

[Out]

Integrate[x^m*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int x^{m} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )d x\]

[In]

int(x^m*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^m*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**m*(e*x**2+d)**(3/2)*(a+b*atan(c*x)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arctan(c*x) + a)*x^m, x)

Giac [N/A]

Not integrable

Time = 2.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arctan(c*x) + a)*x^m, x)

Mupad [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \]

[In]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^(3/2),x)

[Out]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^(3/2), x)