3.918 \(\int \frac {x^m (c+a^2 c x^2)^2}{\sqrt {\tan ^{-1}(a x)}} \, dx\)

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

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.39, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} x^{m}}{\sqrt {\arctan \left (a x\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 3.78, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{2}}{\sqrt {\arctan \left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

[Out]

int(x^m*(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),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(x^m*(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),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 \[ \int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^2}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(c + a^2*c*x^2)^2)/atan(a*x)^(1/2),x)

[Out]

int((x^m*(c + a^2*c*x^2)^2)/atan(a*x)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a**2*c*x**2+c)**2/atan(a*x)**(1/2),x)

[Out]

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

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