3.9.100 \(\int \frac {\text {ArcTan}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx\) [900]

Optimal. Leaf size=138 \[ -\frac {5 a \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{4 c x}-\frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}}{2 c x^2}+\frac {15}{8} a^2 \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x \sqrt {c+a^2 c x^2}},x\right )-\frac {1}{2} a^2 \text {Int}\left (\frac {\text {ArcTan}(a x)^{5/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-5/4*a*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x-1/2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x^2-1/2*a^2*Unint
egrable(arctan(a*x)^(5/2)/x/(a^2*c*x^2+c)^(1/2),x)+15/8*a^2*Unintegrable(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^(1/
2),x)

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Rubi [A]
time = 0.27, 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 {\text {ArcTan}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*c*x^2) + (15
*a^2*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(5/2)/(x*Sqrt[c
+ a^2*c*x^2]), x])/2

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}+\frac {1}{4} (5 a) \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} a^2 \int \frac {\tan ^{-1}(a x)^{5/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 c x}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac {1}{2} a^2 \int \frac {\tan ^{-1}(a x)^{5/2}}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (15 a^2\right ) \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 = 3.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 2.26, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{3} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(5/2)/x^3/(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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^3/(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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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(arctan(a*x)^(5/2)/x^3/(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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Sympy [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(atan(a*x)**(5/2)/x**3/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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