\(\int (d+e x^2)^{5/2} (a+b \arctan (c x)) \, dx\) [1196]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\text {Int}\left (\left (d+e x^2\right )^{5/2} (a+b \arctan (c x)),x\right ) \]

[Out]

Unintegrable((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx \]

[In]

Int[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]

[Out]

Defer[Int][(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx \]

[In]

Integrate[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]

[Out]

Integrate[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.62 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )d x\]

[In]

int((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x)

[Out]

int((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arctan(c*x))*sqrt(e*x^2 + d), x)

Sympy [N/A]

Not integrable

Time = 77.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((e*x**2+d)**(5/2)*(a+b*atan(c*x)),x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 94.14 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^{5/2} (a+b \arctan (c x)) \, dx=\int \left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2} \,d x \]

[In]

int((a + b*atan(c*x))*(d + e*x^2)^(5/2),x)

[Out]

int((a + b*atan(c*x))*(d + e*x^2)^(5/2), x)