3.1.43 \(\int (e+f x)^m (a+b \text {ArcTan}(c+d x))^3 \, dx\) [43]

Optimal. Leaf size=23 \[ \text {Int}\left ((e+f x)^m (a+b \text {ArcTan}(c+d x))^3,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (e+f x)^m (a+b \text {ArcTan}(c+d x))^3 \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int (e+f x)^m \left (a+b \tan ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^m \left (a+b \tan ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int (e+f x)^m (a+b \text {ArcTan}(c+d x))^3 \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (f x +e \right )^{m} \left (a +b \arctan \left (d x +c \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*c*d*f + b^3*d^2*e)*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)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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