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

Optimal. Leaf size=61 \[ \frac {c^2 \left (a^2 x^2+1\right )^3 \sqrt {\tan ^{-1}(a x)}}{6 a^2}-\frac {\text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2}{\sqrt {\tan ^{-1}(a x)}},x\right )}{12 a} \]

[Out]

1/6*c^2*(a^2*x^2+1)^3*arctan(a*x)^(1/2)/a^2-1/12*Unintegrable((a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)/a

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Rubi [A]  time = 0.07, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c^2*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(6*a^2) - Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]/(12*a)

Rubi steps

\begin {align*} \int x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx &=\frac {c^2 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{6 a^2}-\frac {\int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{12 a}\\ \end {align*}

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Mathematica [A]  time = 1.97, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 2.43, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{2} \sqrt {\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(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*(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 = -0.02 \[ \int x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(x*sqrt(atan(a*x)), x) + Integral(2*a**2*x**3*sqrt(atan(a*x)), x) + Integral(a**4*x**5*sqrt(atan
(a*x)), x))

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