3.2.87 \(\int \frac {1}{(a+b x^2)^3} \, dx\) [187]

Optimal. Leaf size=62 \[ \frac {x}{4 a \left (a+b x^2\right )^2}+\frac {3 x}{8 a^2 \left (a+b x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {3 x}{8 a^2 \left (a+b x^2\right )}+\frac {x}{4 a \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-3),x]

[Out]

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

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

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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.89 \begin {gather*} \frac {5 a x+3 b x^3}{8 a^2 \left (a+b x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-3),x]

[Out]

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

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Maple [A]
time = 0.05, size = 57, normalized size = 0.92

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 58, normalized size = 0.94 \begin {gather*} \frac {3 \, b x^{3} + 5 \, a x}{8 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.79, size = 188, normalized size = 3.03 \begin {gather*} \left [\frac {6 \, a b^{2} x^{3} + 10 \, a^{2} b x - 3 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{16 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, \frac {3 \, a b^{2} x^{3} + 5 \, a^{2} b x + 3 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{8 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.13, size = 105, normalized size = 1.69 \begin {gather*} - \frac {3 \sqrt {- \frac {1}{a^{5} b}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} b}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} b}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} b}} + x \right )}}{16} + \frac {5 a x + 3 b x^{3}}{8 a^{4} + 16 a^{3} b x^{2} + 8 a^{2} b^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.35, size = 45, normalized size = 0.73 \begin {gather*} \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} + \frac {3 \, b x^{3} + 5 \, a x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.66, size = 55, normalized size = 0.89 \begin {gather*} \frac {\frac {5\,x}{8\,a}+\frac {3\,b\,x^3}{8\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^2)^3,x)

[Out]

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

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