\(\int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx\) [42]

Optimal result

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

[Out]

Unintegrable((f*x+e)^m*(a+b*arctan(d*x+c))^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 8.77 (sec) , antiderivative size = 504, normalized size of antiderivative = 25.20 \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

(f*x + e)^(m + 1)*a^2/(f*(m + 1)) + 1/16*(4*(b^2*f*x + b^2*e)*(f*x + e)^m*arctan(d*x + c)^2 - (b^2*f*x + b^2*e
)*(f*x + e)^m*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 16*(f*m + f)*integrate(1/16*(12*((b^2*c^2 + b^2)*f*m + (b^2
*d^2*f*m + b^2*d^2*f)*x^2 + (b^2*c^2 + b^2)*f + 2*(b^2*c*d*f*m + b^2*c*d*f)*x)*(f*x + e)^m*arctan(d*x + c)^2 +
((b^2*c^2 + b^2)*f*m + (b^2*d^2*f*m + b^2*d^2*f)*x^2 + (b^2*c^2 + b^2)*f + 2*(b^2*c*d*f*m + b^2*c*d*f)*x)*(f*
x + e)^m*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - 8*(b^2*d*e - 4*(a*b*c^2 + a*b)*f*m - 4*(a*b*d^2*f*m + a*b*d^2*f)
*x^2 - 4*(a*b*c^2 + a*b)*f - (8*a*b*c*d*f*m + (8*a*b*c - b^2)*d*f)*x)*(f*x + e)^m*arctan(d*x + c) + 4*(b^2*d^2
*f*x^2 + b^2*c*d*e + (b^2*d^2*e + b^2*c*d*f)*x)*(f*x + e)^m*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/((c^2 + 1)*f*m +
(d^2*f*m + d^2*f)*x^2 + (c^2 + 1)*f + 2*(c*d*f*m + c*d*f)*x), x))/(f*m + f)

Giac [N/A]

Not integrable

Time = 111.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m (a+b \arctan (c+d x))^2 \, dx=\int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

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

[Out]

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