3.42 \(\int (e+f x)^m (a+b \tan ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left ((e+f x)^m \left (a+b \tan ^{-1}(c+d x)\right )^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int (e+f x)^m \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[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*} \int (e+f x)^m \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^m \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 5.00, size = 0, normalized size = 0.00 \[ \int (e+f x)^m \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arctan \left (d x + c\right )^{2} + 2 \, a b \arctan \left (d x + c\right ) + a^{2}\right )} {\left (f x + e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 1.48, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \arctan \left (d x +c \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (f x + e\right )}^{m + 1} a^{2}}{f {\left (m + 1\right )}} + \frac {7 \, {\left (b^{2} f x + b^{2} e\right )} {\left (f x + e\right )}^{m} \arctan \left (d x + c\right )^{2} - \frac {3}{4} \, {\left (b^{2} f x + b^{2} e\right )} {\left (f x + e\right )}^{m} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + {\left (f m + f\right )} \int \frac {36 \, {\left ({\left (b^{2} c^{2} + b^{2}\right )} f m + {\left (b^{2} d^{2} f m + b^{2} d^{2} f\right )} x^{2} + {\left (b^{2} c^{2} + b^{2}\right )} f + 2 \, {\left (b^{2} c d f m + b^{2} c d f\right )} x\right )} {\left (f x + e\right )}^{m} \arctan \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{2} c^{2} + b^{2}\right )} f m + {\left (b^{2} d^{2} f m + b^{2} d^{2} f\right )} x^{2} + {\left (b^{2} c^{2} + b^{2}\right )} f + 2 \, {\left (b^{2} c d f m + b^{2} c d f\right )} x\right )} {\left (f x + e\right )}^{m} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} - 8 \, {\left (7 \, b^{2} d e - 16 \, {\left (a b c^{2} + a b\right )} f m - 16 \, {\left (a b d^{2} f m + a b d^{2} f\right )} x^{2} - 16 \, {\left (a b c^{2} + a b\right )} f - {\left (32 \, a b c d f m + {\left (32 \, a b c - 7 \, b^{2}\right )} d f\right )} x\right )} {\left (f x + e\right )}^{m} \arctan \left (d x + c\right ) + 12 \, {\left (b^{2} d^{2} f x^{2} + b^{2} c d e + {\left (b^{2} d^{2} e + b^{2} c d f\right )} x\right )} {\left (f x + e\right )}^{m} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{4 \, {\left ({\left (c^{2} + 1\right )} f m + {\left (d^{2} f m + d^{2} f\right )} x^{2} + {\left (c^{2} + 1\right )} f + 2 \, {\left (c d f m + c d f\right )} x\right )}}\,{d x}}{16 \, {\left (f m + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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