3.767 \(\int \frac {(c+a^2 c x^2)^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2 \tan ^{-1}(a x)^{3/2}}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 2.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {3}{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int 2 a^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(2*a**2*atan(a*x)**(3/2), x) + Integral(atan(a*x)**(3/2)/x**2, x) + Integral(a**4*x**2*atan(a*x)
**(3/2), x))

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