3.11.75 \(\int \frac {1}{x^4 (c+a^2 c x^2)^3 \text {ArcTan}(a x)^{5/2}} \, dx\) [1075]

Optimal. Leaf size=191 \[ -\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^{3/2}}+\frac {16}{3 a^2 c^3 x^5 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}+\frac {32}{3 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}+\frac {80 \text {Int}\left (\frac {1}{x^6 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right )}{3 a^2}+80 \text {Int}\left (\frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right )+\frac {224}{3} a^2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

-2/3/a/c^3/x^4/(a^2*x^2+1)^2/arctan(a*x)^(3/2)+16/3/a^2/c^3/x^5/(a^2*x^2+1)^2/arctan(a*x)^(1/2)+32/3/c^3/x^3/(
a^2*x^2+1)^2/arctan(a*x)^(1/2)+80/3*Unintegrable(1/x^6/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a^2+80*Unintegrabl
e(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)+224/3*a^2*Unintegrable(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

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Rubi [A]
time = 0.32, 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 = {} \begin {gather*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2/(3*a*c^3*x^4*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c^3*x^5*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 32
/(3*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x
])/(3*a^2) + 80*Defer[Int][1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x] + (224*a^2*Defer[Int][1/(x^2*(c + a
^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (16 a) \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16}{3 a^2 c^3 x^5 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {32}{3 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+32 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+48 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {80 \int \frac {1}{x^6 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (224 a^2\right ) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]
time = 9.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]

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Maple [A]
time = 1.60, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(5/2),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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{6} x^{10} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{8} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a**2*c*x**2+c)**3/atan(a*x)**(5/2),x)

[Out]

Integral(1/(a**6*x**10*atan(a*x)**(5/2) + 3*a**4*x**8*atan(a*x)**(5/2) + 3*a**2*x**6*atan(a*x)**(5/2) + x**4*a
tan(a*x)**(5/2)), x)/c**3

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*atan(a*x)^(5/2)*(c + a^2*c*x^2)^3),x)

[Out]

int(1/(x^4*atan(a*x)^(5/2)*(c + a^2*c*x^2)^3), x)

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