3.1059 \(\int \frac {x^3}{(c+a^2 c x^2)^2 \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac {8}{3} a^2 \text {Int}\left (\frac {x^5}{\left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {16}{3} \text {Int}\left (\frac {x^3}{\left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4 c^2}-\frac {4 x^2}{a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {4 x^4}{3 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2 x^3}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}} \]

[Out]

-2/3*x^3/a/c^2/(a^2*x^2+1)/arctan(a*x)^(3/2)+4*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^4/c^2-4*x^2/a
^2/c^2/(a^2*x^2+1)/arctan(a*x)^(1/2)-4/3*x^4/c^2/(a^2*x^2+1)/arctan(a*x)^(1/2)+16/3*Unintegrable(x^3/(a^2*c*x^
2+c)^2/arctan(a*x)^(1/2),x)+8/3*a^2*Unintegrable(x^5/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.42, 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 = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*x^3)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (4*x^
4)/(3*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) +
(16*Defer[Int][x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + (8*a^2*Defer[Int][x^5/((c + a^2*c*x^2)^2*Sq
rt[ArcTan[a*x]]), x])/3

Rubi steps

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

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Mathematica [A]  time = 4.81, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

Timed out

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maple [A]  time = 1.62, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3/(a^2*c*x^2+c)^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(x^3/(a^2*c*x^2+c)^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 = -0.01 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

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)^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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