\(\int \frac {x^2 (a+b \arctan (c x))}{(d+e x^2)^{3/2}} \, dx\) [1210]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 23.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.51 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 35.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^2*(a+b*arctan(c*x))/(e*x^2+d)^(3/2),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 = 136.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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