\(\int \frac {(a+b \arctan (c x^2))^2}{(d+e x)^2} \, dx\) [27]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 108.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{(d+e x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 2.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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