3.5.61 \(\int \frac {1}{x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=133 \[ \frac {b \left (a+b x^2\right )}{a^2 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {-a-b x^2}{3 a x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.04, antiderivative size = 130, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1112, 325, 205} \begin {gather*} \frac {b \left (a+b x^2\right )}{a^2 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{3 a x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

Rule 205

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 70, normalized size = 0.53 \begin {gather*} -\frac {\left (a+b x^2\right ) \left (\sqrt {a} \left (a-3 b x^2\right )-3 b^{3/2} x^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{3 a^{5/2} x^3 \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

-1/3*((a + b*x^2)*(Sqrt[a]*(a - 3*b*x^2) - 3*b^(3/2)*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(5/2)*x^3*Sqrt[(a +
b*x^2)^2])

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IntegrateAlgebraic [A]  time = 17.68, size = 66, normalized size = 0.50 \begin {gather*} \frac {\left (a+b x^2\right ) \left (\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {3 b x^2-a}{3 a^2 x^3}\right )}{\sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

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fricas [A]  time = 1.20, size = 106, normalized size = 0.80 \begin {gather*} \left [\frac {3 \, b x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b x^{2} - 2 \, a}{6 \, a^{2} x^{3}}, \frac {3 \, b x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, b x^{2} - a}{3 \, a^{2} x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/((b*x^2+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 50, normalized size = 0.38 \begin {gather*} \frac {1}{3} \, {\left (\frac {3 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{a^{2} x^{3}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/((b*x^2+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*(3*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + (3*b*x^2 - a)/(a^2*x^3))*sgn(b*x^2 + a)

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maple [A]  time = 0.01, size = 69, normalized size = 0.52 \begin {gather*} \frac {\left (b \,x^{2}+a \right ) \left (3 b^{2} x^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3 \sqrt {a b}\, b \,x^{2}-\sqrt {a b}\, a \right )}{3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}\, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/((b*x^2+a)^2)^(1/2),x)

[Out]

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

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maxima [A]  time = 3.01, size = 40, normalized size = 0.30 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/((b*x^2+a)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((a + b*x^2)^2)^(1/2)),x)

[Out]

int(1/(x^4*((a + b*x^2)^2)^(1/2)), x)

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sympy [A]  time = 0.28, size = 87, normalized size = 0.65 \begin {gather*} - \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac {- a + 3 b x^{2}}{3 a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/((b*x**2+a)**2)**(1/2),x)

[Out]

-sqrt(-b**3/a**5)*log(-a**3*sqrt(-b**3/a**5)/b**2 + x)/2 + sqrt(-b**3/a**5)*log(a**3*sqrt(-b**3/a**5)/b**2 + x
)/2 + (-a + 3*b*x**2)/(3*a**2*x**3)

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