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

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

[Out]

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

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.87, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 3.70, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sqrt {\arctan \left (a x \right )}}{a^{2} c \,x^{2}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c),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^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c),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 \[ \int \frac {x^3\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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