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

Optimal. Leaf size=59 \[ \frac{\text{Unintegrable}\left (x \tan ^{-1}(a x)^{5/2},x\right )}{a^2 c}+\frac{2 \text{Unintegrable}\left (\tan ^{-1}(a x)^{7/2},x\right )}{7 a^3 c}-\frac{2 x \tan ^{-1}(a x)^{7/2}}{7 a^3 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.119754, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx \]

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*}

Mathematica [A]  time = 4.25563, size = 0, normalized size = 0. \[ \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.609, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{a}^{2}c{x}^{2}+c} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{\frac{5}{2}}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

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]

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