\(\int \frac {x^2}{(c+a^2 c x^2)^{3/2} \arctan (a x)^{5/2}} \, dx\) [1095]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {8 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c \sqrt {c+a^2 c x^2}}+4 \text {Int}\left (\frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right )+\frac {8}{3} a^2 \text {Int}\left (\frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-2/3*x^2/a/c/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2)+8/3*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi
^(1/2)*(a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)-8/3*x/a^2/c/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)-4/3*x^3/c
/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)+4*Unintegrable(x^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)+8/3*a^2*Uni
ntegrable(x^4/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

(-2*x^2)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
- (4*x^3)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*S
qrt[ArcTan[a*x]]])/(3*a^3*c*Sqrt[c + a^2*c*x^2]) + 4*Defer[Int][x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),
x] + (8*a^2*Defer[Int][x^4/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{3 a^2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^3 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a^3 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 x^3}{3 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {8 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^3 c \sqrt {c+a^2 c x^2}}+4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{3} \left (8 a^2\right ) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 159.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 296.32 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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