\(\int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx\) [1174]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\frac {a d x \sqrt {d+e x^2}}{8 e}+\frac {1}{4} a x^3 \sqrt {d+e x^2}-\frac {a d^2 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 e^{3/2}}+b \text {Int}\left (x^2 \sqrt {d+e x^2} \arctan (c x),x\right ) \]

[Out]

-1/8*a*d^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(3/2)+1/8*a*d*x*(e*x^2+d)^(1/2)/e+1/4*a*x^3*(e*x^2+d)^(1/2)+b*
Unintegrable(x^2*arctan(c*x)*(e*x^2+d)^(1/2),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 = {} \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

(a*d*x*Sqrt[d + e*x^2])/(8*e) + (a*x^3*Sqrt[d + e*x^2])/4 - (a*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*e^
(3/2)) + b*Defer[Int][x^2*Sqrt[d + e*x^2]*ArcTan[c*x], x]

Rubi steps \begin{align*} \text {integral}& = a \int x^2 \sqrt {d+e x^2} \, dx+b \int x^2 \sqrt {d+e x^2} \arctan (c x) \, dx \\ & = \frac {1}{4} a x^3 \sqrt {d+e x^2}+b \int x^2 \sqrt {d+e x^2} \arctan (c x) \, dx+\frac {1}{4} (a d) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx \\ & = \frac {a d x \sqrt {d+e x^2}}{8 e}+\frac {1}{4} a x^3 \sqrt {d+e x^2}+b \int x^2 \sqrt {d+e x^2} \arctan (c x) \, dx-\frac {\left (a d^2\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{8 e} \\ & = \frac {a d x \sqrt {d+e x^2}}{8 e}+\frac {1}{4} a x^3 \sqrt {d+e x^2}+b \int x^2 \sqrt {d+e x^2} \arctan (c x) \, dx-\frac {\left (a d^2\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{8 e} \\ & = \frac {a d x \sqrt {d+e x^2}}{8 e}+\frac {1}{4} a x^3 \sqrt {d+e x^2}-\frac {a d^2 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 e^{3/2}}+b \int x^2 \sqrt {d+e x^2} \arctan (c x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 23.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^2*(e*x^2+d)^(1/2)*(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 = 95.51 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {d+e x^2} (a+b \arctan (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]

[In]

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

[Out]

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