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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left ((f x)^m \left (c^2 d x^2+d\right )^q \left (a+b \tan ^{-1}(c x)\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0986459, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.630113, size = 0, normalized size = 0. \[ \int (f x)^m \left (d+c^2 d x^2\right )^q \left (a+b \tan ^{-1}(c x)\right )^p \, dx \]

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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Maple [A]  time = 5.934, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ({c}^{2}d{x}^{2}+d \right ) ^{q} \left ( a+b\arctan \left ( cx \right ) \right ) ^{p}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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]

Exception raised: UnboundLocalError

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

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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]

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