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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x \left (a^2 c x^2+c\right )^3}{\sqrt {\tan ^{-1}(a x)}},x\right ) \]

[Out]

Unintegrable(x*(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.30, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], 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(x*(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

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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*(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 2.81, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a^{2} c \,x^{2}+c \right )^{3}}{\sqrt {\arctan \left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

[Out]

int(x*(a^2*c*x^2+c)^3/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*(a^2*c*x^2+c)^3/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 \[ \int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^3}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + a^2*c*x^2)^3)/atan(a*x)^(1/2),x)

[Out]

int((x*(c + a^2*c*x^2)^3)/atan(a*x)^(1/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {x}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {3 a^{2} x^{3}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {3 a^{4} x^{5}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {a^{6} x^{7}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x/sqrt(atan(a*x)), x) + Integral(3*a**2*x**3/sqrt(atan(a*x)), x) + Integral(3*a**4*x**5/sqrt(at
an(a*x)), x) + Integral(a**6*x**7/sqrt(atan(a*x)), x))

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