3.515 \(\int \frac{x^4}{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{x^4}{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]

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Rubi [A]  time = 0.138177, 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 = {} \[ \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]

[Out]

Defer[Int][x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]

Rubi steps

\begin{align*} \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx &=\int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx\\ \end{align*}

Mathematica [A]  time = 20.0736, size = 0, normalized size = 0. \[ \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]

[Out]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]

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Maple [A]  time = 0.915, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}}{\arctan \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x),x)

[Out]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} x^{4}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^4/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}} \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a**2*c*x**2+c)**(5/2)/atan(a*x),x)

[Out]

Integral(x**4/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x),x, algorithm="giac")

[Out]

integrate(x^4/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)), x)