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

Optimal. Leaf size=57 \[ \frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{4 a^2}-\frac {3 \text {Int}\left (\left (a^2 c x^2+c\right ) \sqrt {\tan ^{-1}(a x)},x\right )}{8 a} \]

[Out]

1/4*c*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a^2-3/8*Unintegrable((a^2*c*x^2+c)*arctan(a*x)^(1/2),x)/a

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

Verification is Not applicable to the result.

[In]

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

[Out]

(c*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(4*a^2) - (3*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/(8*a)

Rubi steps

\begin {align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx &=\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{4 a^2}-\frac {3 \int \left (c+a^2 c x^2\right ) \sqrt {\tan ^{-1}(a x)} \, dx}{8 a}\\ \end {align*}

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Mathematica [A]  time = 1.36, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 2.19, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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