\(\int \frac {1}{(a+c x^2)^3} \, dx\) [55]

Optimal result

Integrand size = 9, antiderivative size = 62 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {x}{4 a \left (a+c x^2\right )^2}+\frac {3 x}{8 a^2 \left (a+c x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \]

[Out]

1/4*x/a/(c*x^2+a)^2+3/8*x/a^2/(c*x^2+a)+3/8*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/c^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 211} \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}+\frac {3 x}{8 a^2 \left (a+c x^2\right )}+\frac {x}{4 a \left (a+c x^2\right )^2} \]

[In]

Int[(a + c*x^2)^(-3),x]

[Out]

x/(4*a*(a + c*x^2)^2) + (3*x)/(8*a^2*(a + c*x^2)) + (3*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c])

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{4 a \left (a+c x^2\right )^2}+\frac {3 \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{4 a} \\ & = \frac {x}{4 a \left (a+c x^2\right )^2}+\frac {3 x}{8 a^2 \left (a+c x^2\right )}+\frac {3 \int \frac {1}{a+c x^2} \, dx}{8 a^2} \\ & = \frac {x}{4 a \left (a+c x^2\right )^2}+\frac {3 x}{8 a^2 \left (a+c x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {5 a x+3 c x^3}{8 a^2 \left (a+c x^2\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \]

[In]

Integrate[(a + c*x^2)^(-3),x]

[Out]

(5*a*x + 3*c*x^3)/(8*a^2*(a + c*x^2)^2) + (3*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c])

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {x}{4 a \left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 x}{8 a \left (c \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 a \sqrt {a c}}}{a}\)	\(57\)
risch	\(\frac {\frac {3 c \,x^{3}}{8 a^{2}}+\frac {5 x}{8 a}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}\)	\(73\)

[In]

int(1/(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*x/a/(c*x^2+a)^2+3/4/a*(1/2*x/a/(c*x^2+a)+1/2/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 188, normalized size of antiderivative = 3.03 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\left [\frac {6 \, a c^{2} x^{3} + 10 \, a^{2} c x - 3 \, {\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{16 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac {3 \, a c^{2} x^{3} + 5 \, a^{2} c x + 3 \, {\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{8 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \]

[In]

integrate(1/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/16*(6*a*c^2*x^3 + 10*a^2*c*x - 3*(c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c
*x^2 + a)))/(a^3*c^3*x^4 + 2*a^4*c^2*x^2 + a^5*c), 1/8*(3*a*c^2*x^3 + 5*a^2*c*x + 3*(c^2*x^4 + 2*a*c*x^2 + a^2
)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a^3*c^3*x^4 + 2*a^4*c^2*x^2 + a^5*c)]

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=- \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16} + \frac {5 a x + 3 c x^{3}}{8 a^{4} + 16 a^{3} c x^{2} + 8 a^{2} c^{2} x^{4}} \]

[In]

integrate(1/(c*x**2+a)**3,x)

[Out]

-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x)/16 + 3*sqrt(-1/(a**5*c))*log(a**3*sqrt(-1/(a**5*c)) + x)
/16 + (5*a*x + 3*c*x**3)/(8*a**4 + 16*a**3*c*x**2 + 8*a**2*c**2*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, c x^{3} + 5 \, a x}{8 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} \]

[In]

integrate(1/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/8*(3*c*x^3 + 5*a*x)/(a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4) + 3/8*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{3} + 5 \, a x}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2}} \]

[In]

integrate(1/(c*x^2+a)^3,x, algorithm="giac")

[Out]

3/8*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/8*(3*c*x^3 + 5*a*x)/((c*x^2 + a)^2*a^2)

Mupad [B] (verification not implemented)

Time = 9.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {5\,x}{8\,a}+\frac {3\,c\,x^3}{8\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}} \]

[In]

int(1/(a + c*x^2)^3,x)

[Out]

((5*x)/(8*a) + (3*c*x^3)/(8*a^2))/(a^2 + c^2*x^4 + 2*a*c*x^2) + (3*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(5/2)*c^(1/
2))