\(\int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx\) [43]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\text {Int}\left ((e+f x)^m (a+b \arctan (c+d x))^3,x\right ) \]

[Out]

Unintegrable((f*x+e)^m*(a+b*arctan(d*x+c))^3,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 = {} \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx \]

[In]

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

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTan[x])^3, 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))^3 \, dx,x,c+d x\right )}{d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

int((f*x+e)^m*(a+b*arctan(d*x+c))^3,x)

[Out]

int((f*x+e)^m*(a+b*arctan(d*x+c))^3,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arctan(d*x + c) + a^3)*(f*x + e)^m, x)

Sympy [F(-1)]

Timed out. \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**m*(a+b*atan(d*x+c))**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 11.47 (sec) , antiderivative size = 659, normalized size of antiderivative = 32.95 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

integrate((f*x+e)^m*(a+b*arctan(d*x+c))^3,x, algorithm="maxima")

[Out]

(f*x + e)^(m + 1)*a^3/(f*(m + 1)) + 1/32*(4*(b^3*f*x + b^3*e)*(f*x + e)^m*arctan(d*x + c)^3 - 3*(b^3*f*x + b^3
*e)*(f*x + e)^m*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 32*(f*m + f)*integrate(1/32*(28*((b^3*c^2
+ b^3)*f*m + (b^3*d^2*f*m + b^3*d^2*f)*x^2 + (b^3*c^2 + b^3)*f + 2*(b^3*c*d*f*m + b^3*c*d*f)*x)*(f*x + e)^m*a
rctan(d*x + c)^3 - 12*(b^3*d*e - 8*(a*b^2*c^2 + a*b^2)*f*m - 8*(a*b^2*d^2*f*m + a*b^2*d^2*f)*x^2 - 8*(a*b^2*c^
2 + a*b^2)*f - (16*a*b^2*c*d*f*m + (16*a*b^2*c - b^3)*d*f)*x)*(f*x + e)^m*arctan(d*x + c)^2 + 12*(b^3*d^2*f*x^
2 + b^3*c*d*e + (b^3*d^2*e + b^3*c*d*f)*x)*(f*x + e)^m*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1) + 96*(
(a^2*b*c^2 + a^2*b)*f*m + (a^2*b*d^2*f*m + a^2*b*d^2*f)*x^2 + (a^2*b*c^2 + a^2*b)*f + 2*(a^2*b*c*d*f*m + a^2*b
*c*d*f)*x)*(f*x + e)^m*arctan(d*x + c) + 3*(((b^3*c^2 + b^3)*f*m + (b^3*d^2*f*m + b^3*d^2*f)*x^2 + (b^3*c^2 +
b^3)*f + 2*(b^3*c*d*f*m + b^3*c*d*f)*x)*(f*x + e)^m*arctan(d*x + c) + (b^3*d*f*x + b^3*d*e)*(f*x + e)^m)*log(d
^2*x^2 + 2*c*d*x + c^2 + 1)^2)/((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 = 112.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m (a+b \arctan (c+d x))^3 \, dx=\int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((e + f*x)^m*(a + b*atan(c + d*x))^3,x)

[Out]

int((e + f*x)^m*(a + b*atan(c + d*x))^3, x)