3.597 \(\int \frac{\sqrt{f x}}{(d+c^2 d x^2)^2 (a+b \tan ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{\sqrt{f x}}{\left (c^2 d x^2+d\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]

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Rubi [A]  time = 0.100492, 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 = {} \[ \int \frac{\sqrt{f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2),x]

[Out]

Defer[Int][Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]

Rubi steps

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

Mathematica [A]  time = 28.9607, size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2),x]

[Out]

Integrate[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]

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Maple [A]  time = 1.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({c}^{2}d{x}^{2}+d \right ) ^{2} \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}}\sqrt{fx}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(1/2)/(c^2*d*x^2+d)^2/(a+b*arctan(c*x))^2,x)

[Out]

int((f*x)^(1/2)/(c^2*d*x^2+d)^2/(a+b*arctan(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{1}{2} \,{\left (a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )\right )} \sqrt{f} \int \frac{{\left (a c^{2} x^{2} + 4 \, b c x +{\left (b c^{2} x^{2} + b\right )} \arctan \left (c x\right ) + a\right )} \sqrt{x}}{a^{3} c^{4} d^{2} x^{4} + 2 \, a^{3} c^{2} d^{2} x^{2} + a^{3} d^{2} +{\left (b^{3} c^{4} d^{2} x^{4} + 2 \, b^{3} c^{2} d^{2} x^{2} + b^{3} d^{2}\right )} \arctan \left (c x\right )^{3} + 3 \,{\left (a b^{2} c^{4} d^{2} x^{4} + 2 \, a b^{2} c^{2} d^{2} x^{2} + a b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 3 \,{\left (a^{2} b c^{4} d^{2} x^{4} + 2 \, a^{2} b c^{2} d^{2} x^{2} + a^{2} b d^{2}\right )} \arctan \left (c x\right )}\,{d x} + \sqrt{f} x^{\frac{3}{2}}}{2 \,{\left (a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(1/2)/(c^2*d*x^2+d)^2/(a+b*arctan(c*x))^2,x, algorithm="maxima")

[Out]

1/2*(2*(a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^2*d^2*x^2 + b^2*d^2)*arctan(c*x)^2 + 2*(a*b*c^2*d^2*x^2 + a*b*d^2)*
arctan(c*x))*sqrt(f)*integrate(1/4*(a*c^2*x^2 + 4*b*c*x + (b*c^2*x^2 + b)*arctan(c*x) + a)*sqrt(x)/(a^3*c^4*d^
2*x^4 + 2*a^3*c^2*d^2*x^2 + a^3*d^2 + (b^3*c^4*d^2*x^4 + 2*b^3*c^2*d^2*x^2 + b^3*d^2)*arctan(c*x)^3 + 3*(a*b^2
*c^4*d^2*x^4 + 2*a*b^2*c^2*d^2*x^2 + a*b^2*d^2)*arctan(c*x)^2 + 3*(a^2*b*c^4*d^2*x^4 + 2*a^2*b*c^2*d^2*x^2 + a
^2*b*d^2)*arctan(c*x)), x) + sqrt(f)*x^(3/2))/(a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^2*d^2*x^2 + b^2*d^2)*arctan(
c*x)^2 + 2*(a*b*c^2*d^2*x^2 + a*b*d^2)*arctan(c*x))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{f x}}{a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(1/2)/(c^2*d*x^2+d)^2/(a+b*arctan(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(f*x)/(a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2
*d^2)*arctan(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arctan(c*x)), x)

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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)**(1/2)/(c**2*d*x**2+d)**2/(a+b*atan(c*x))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f x}}{{\left (c^{2} d x^{2} + d\right )}^{2}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(1/2)/(c^2*d*x^2+d)^2/(a+b*arctan(c*x))^2,x, algorithm="giac")

[Out]

integrate(sqrt(f*x)/((c^2*d*x^2 + d)^2*(b*arctan(c*x) + a)^2), x)