\(\int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx\) [255]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\frac {x^{1+m} \sqrt {c+a^2 c x^2} \arctan (a x)}{2+m}-\frac {a x^{2+m} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{(2+m)^2}+\frac {c \text {Int}\left (\frac {x^m \arctan (a x)}{\sqrt {c+a^2 c x^2}},x\right )}{2+m} \]

[Out]

x^(1+m)*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/(2+m)-a*x^(2+m)*hypergeom([1, 3/2+1/2*m],[2+1/2*m],-a^2*x^2)*(a^2*c*x^
2+c)^(1/2)/(2+m)^2+c*Unintegrable(x^m*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x)/(2+m)

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 22, 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 = {} \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \]

[In]

Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]

[Out]

(x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*c*x^(2 + m)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2,
(2 + m)/2, (4 + m)/2, -(a^2*x^2)])/((2 + m)^2*Sqrt[c + a^2*c*x^2]) + (c*Defer[Int][(x^m*ArcTan[a*x])/Sqrt[c +
a^2*c*x^2], x])/(2 + m)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \sqrt {c+a^2 c x^2} \arctan (a x)}{2+m}+\frac {c \int \frac {x^m \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{2+m}-\frac {(a c) \int \frac {x^{1+m}}{\sqrt {c+a^2 c x^2}} \, dx}{2+m} \\ & = \frac {x^{1+m} \sqrt {c+a^2 c x^2} \arctan (a x)}{2+m}+\frac {c \int \frac {x^m \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{2+m}-\frac {\left (a c \sqrt {1+a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {1+a^2 x^2}} \, dx}{(2+m) \sqrt {c+a^2 c x^2}} \\ & = \frac {x^{1+m} \sqrt {c+a^2 c x^2} \arctan (a x)}{2+m}-\frac {a c x^{2+m} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{(2+m)^2 \sqrt {c+a^2 c x^2}}+\frac {c \int \frac {x^m \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{2+m} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \]

[In]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]

[Out]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int x^{m} \sqrt {a^{2} c \,x^{2}+c}\, \arctan \left (a x \right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right ) \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)^(1/2)*arctan(a*x),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m*arctan(a*x), x)

Sympy [N/A]

Not integrable

Time = 18.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^{m} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]

[In]

integrate(x**m*(a**2*c*x**2+c)**(1/2)*atan(a*x),x)

[Out]

Integral(x**m*sqrt(c*(a**2*x**2 + 1))*atan(a*x), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right ) \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)^(1/2)*arctan(a*x),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*c*x^2 + c)*x^m*arctan(a*x), x)

Giac [F(-2)]

Exception generated. \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^m\,\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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