3.1113 \(\int (f x)^m (d+c^2 d x^2)^q (a+b \tan ^{-1}(c x))^p \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left ((f x)^m \left (c^2 d x^2+d\right )^q \left (a+b \tan ^{-1}(c x)\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int (f x)^m \left (d+c^2 d x^2\right )^q \left (a+b \tan ^{-1}(c x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int (f x)^m \left (d+c^2 d x^2\right )^q \left (a+b \tan ^{-1}(c x)\right )^p \, dx &=\int (f x)^m \left (d+c^2 d x^2\right )^q \left (a+b \tan ^{-1}(c x)\right )^p \, dx\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+c^2 d x^2\right )^q \left (a+b \tan ^{-1}(c x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} d x^{2} + d\right )}^{q} \left (f x\right )^{m} {\left (b \arctan \left (c x\right ) + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^2*d*x^2 + d)^q*(f*x)^m*(b*arctan(c*x) + a)^p, 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)^m*(c^2*d*x^2+d)^q*(a+b*arctan(c*x))^p,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 7.56, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (c^{2} d \,x^{2}+d \right )^{q} \left (a +b \arctan \left (c x \right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*d*x^2 + d)^q*(f*x)^m*(b*arctan(c*x) + a)^p, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^p\,{\left (d\,c^2\,x^2+d\right )}^q\,{\left (f\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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