3.1101 \(\int \frac {1}{x^4 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=206 \[ \frac {80}{3} a^2 \text {Int}\left (\frac {1}{x^2 \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {80 \text {Int}\left (\frac {1}{x^6 \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )}{3 a^2}+52 \text {Int}\left (\frac {1}{x^4 \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {16}{3 a^2 c x^5 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c x^3 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]

[Out]

-2/3/a/c/x^4/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2)+16/3/a^2/c/x^5/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)+20/3/c
/x^3/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)+80/3*Unintegrable(1/x^6/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)/a^
2+52*Unintegrable(1/x^4/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)+80/3*a^2*Unintegrable(1/x^2/(a^2*c*x^2+c)^(3/
2)/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.85, 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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(3*a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
+ 20/(3*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan
[a*x]]), x])/(3*a^2) + 52*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + (80*a^2*Defer[Int][
1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac {16}{3 a^2 c x^5 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+\frac {20}{3 c x^3 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+20 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+32 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {80 \int \frac {1}{x^6 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]  time = 28.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 7.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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)^(3/2)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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