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

Optimal. Leaf size=208 \[ \frac {5}{16} a^3 \text {Int}\left (\frac {1}{x \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {25}{12} a^3 \text {Int}\left (\frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {a^2 c x^2+c}},x\right )+\frac {2 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x}-\frac {5 a^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{8 c x}-\frac {5 a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x^3} \]

[Out]

-5/12*a*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x^2-1/3*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x^3+2/3*a^2*ar
ctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x-5/8*a^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x-25/12*a^3*Unintegrab
le(arctan(a*x)^(3/2)/x/(a^2*c*x^2+c)^(1/2),x)+5/16*a^3*Unintegrable(1/x/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),
x)

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Rubi [A]  time = 0.74, 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 = {} \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x)
+ (5*a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Defer[Int][ArcTan[a*x]^(3/2)
/(x*Sqrt[c + a^2*c*x^2]), x])/12

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {1}{6} (5 a) \int \frac {\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{8 c x}-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 17.02, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 7.52, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(5/2)/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)^(5/2)/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 = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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