3.234 \(\int \frac{1}{x (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac{d \log \left (c+d x^2\right )}{2 c (b c-a d)}+\frac{\log (x)}{a c} \]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^2])/(2*a*(b*c - a*d)) + (d*Log[c + d*x^2])/(2*c*(b*c - a*d))

________________________________________________________________________________________

Rubi [A]  time = 0.0611619, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 72} \[ -\frac{b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac{d \log \left (c+d x^2\right )}{2 c (b c-a d)}+\frac{\log (x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2)*(c + d*x^2)),x]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^2])/(2*a*(b*c - a*d)) + (d*Log[c + d*x^2])/(2*c*(b*c - a*d))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1}{x \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a c x}+\frac{b^2}{a (-b c+a d) (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac{\log (x)}{a c}-\frac{b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac{d \log \left (c+d x^2\right )}{2 c (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.0273714, size = 54, normalized size = 0.87 \[ \frac{-b c \log \left (a+b x^2\right )+a d \log \left (c+d x^2\right )-2 a d \log (x)+2 b c \log (x)}{2 a b c^2-2 a^2 c d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^2)*(c + d*x^2)),x]

[Out]

(2*b*c*Log[x] - 2*a*d*Log[x] - b*c*Log[a + b*x^2] + a*d*Log[c + d*x^2])/(2*a*b*c^2 - 2*a^2*c*d)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 59, normalized size = 1. \begin{align*} -{\frac{d\ln \left ( d{x}^{2}+c \right ) }{2\,c \left ( ad-bc \right ) }}+{\frac{\ln \left ( x \right ) }{ac}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d/c/(a*d-b*c)*ln(d*x^2+c)+ln(x)/a/c+1/2*b/a/(a*d-b*c)*ln(b*x^2+a)

________________________________________________________________________________________

Maxima [A]  time = 1.07348, size = 82, normalized size = 1.32 \begin{align*} -\frac{b \log \left (b x^{2} + a\right )}{2 \,{\left (a b c - a^{2} d\right )}} + \frac{d \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{2} - a c d\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*log(b*x^2 + a)/(a*b*c - a^2*d) + 1/2*d*log(d*x^2 + c)/(b*c^2 - a*c*d) + 1/2*log(x^2)/(a*c)

________________________________________________________________________________________

Fricas [A]  time = 1.86139, size = 123, normalized size = 1.98 \begin{align*} -\frac{b c \log \left (b x^{2} + a\right ) - a d \log \left (d x^{2} + c\right ) - 2 \,{\left (b c - a d\right )} \log \left (x\right )}{2 \,{\left (a b c^{2} - a^{2} c d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*c*log(b*x^2 + a) - a*d*log(d*x^2 + c) - 2*(b*c - a*d)*log(x))/(a*b*c^2 - a^2*c*d)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError