3.6.97 \(\int \frac {\sqrt {f x}}{(d+c^2 d x^2)^2 (a+b \text {ArcTan}(c x))^2} \, dx\) [597]

Optimal. Leaf size=33 \[ \text {Int}\left (\frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, 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 \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 19.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2} \, dx \end {gather*}

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 = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {f x}}{\left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arctan \left (c x \right )\right )^{2}}\, dx\]

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 = 0.00, 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)^(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 = 0.00, 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)^(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)] 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)**(1/2)/(c**2*d*x**2+d)**2/(a+b*atan(c*x))**2,x)

[Out]

Timed out

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

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {f\,x}}{{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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