3.7.51 \(\int \frac {(c+a^2 c x^2)^{5/2}}{\text {ArcTan}(a x)^3} \, dx\) [651]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{5/2}}{\text {ArcTan}(a x)^3},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

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Rubi [A]
time = 0.03, 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 {\left (c+a^2 c x^2\right )^{5/2}}{\text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3,x]

[Out]

Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x]

Rubi steps

\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^3} \, dx &=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3,x]

[Out]

Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x]

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Maple [A]
time = 2.80, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arctan \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

[Out]

int((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,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((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/arctan(a*x)^3, 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((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)/atan(a*x)**3, x)

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Giac [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((a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)^(5/2)/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)^(5/2)/atan(a*x)^3, x)

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