3.820 \(\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=173 \[ \frac {1}{8} a^2 \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {5}{4} a^3 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {a^2 c x^2+c}},x\right )+\frac {2 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 c x}-\frac {a \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{4 c x^2}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 c x^3} \]

[Out]

-1/3*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x^3+2/3*a^2*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x-1/4*a*(a^2*
c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x^2+1/8*a^2*Unintegrable(1/x^2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)-5/4
*a^3*Unintegrable(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^(1/2),x)

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Rubi [A]  time = 0.66, 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 = {} \[ \int \frac {\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x^3) + (2*
a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x) + (a^2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*
x]]), x])/8 - (5*a^3*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/4

Rubi steps

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

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Mathematica [A]  time = 19.24, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(3/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

Integrate[ArcTan[a*x]^(3/2)/(x^4*Sqrt[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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c)^(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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 8.20, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c)^(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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c)^(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 = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(3/2)/(x^4*(c + a^2*c*x^2)^(1/2)),x)

[Out]

int(atan(a*x)^(3/2)/(x^4*(c + a^2*c*x^2)^(1/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(3/2)/x**4/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(atan(a*x)**(3/2)/(x**4*sqrt(c*(a**2*x**2 + 1))), x)

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