3.9.56 \(\int \frac {x^3 \text {ArcTan}(a x)^{5/2}}{c+a^2 c x^2} \, dx\) [856]

Optimal. Leaf size=61 \[ -\frac {2 x \text {ArcTan}(a x)^{7/2}}{7 a^3 c}+\frac {\text {Int}\left (x \text {ArcTan}(a x)^{5/2},x\right )}{a^2 c}+\frac {2 \text {Int}\left (\text {ArcTan}(a x)^{7/2},x\right )}{7 a^3 c} \]

[Out]

-2/7*x*arctan(a*x)^(7/2)/a^3/c+Unintegrable(x*arctan(a*x)^(5/2),x)/a^2/c+2/7*Unintegrable(arctan(a*x)^(7/2),x)
/a^3/c

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Rubi [A]
time = 0.08, 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 {x^3 \text {ArcTan}(a x)^{5/2}}{c+a^2 c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*x*ArcTan[a*x]^(7/2))/(7*a^3*c) + Defer[Int][x*ArcTan[a*x]^(5/2), x]/(a^2*c) + (2*Defer[Int][ArcTan[a*x]^(7
/2), x])/(7*a^3*c)

Rubi steps

\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \tan ^{-1}(a x)^{5/2} \, dx}{a^2 c}\\ &=-\frac {2 x \tan ^{-1}(a x)^{7/2}}{7 a^3 c}+\frac {2 \int \tan ^{-1}(a x)^{7/2} \, dx}{7 a^3 c}+\frac {\int x \tan ^{-1}(a x)^{5/2} \, dx}{a^2 c}\\ \end {align*}

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Mathematica [A]
time = 3.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \text {ArcTan}(a x)^{5/2}}{c+a^2 c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.52, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \arctan \left (a x \right )^{\frac {5}{2}}}{a^{2} c \,x^{2}+c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c),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(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c),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(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c),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 {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{c\,a^2\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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