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

Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac{\log \left (c+d x^2\right )}{2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0262701, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {444, 36, 31} \[ \frac{\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac{\log \left (c+d x^2\right )}{2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0162048, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^2\right )-\log \left (c+d x^2\right )}{2 b c-2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 42, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( d{x}^{2}+c \right ) }{2\,ad-2\,bc}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,ad-2\,bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06689, size = 55, normalized size = 1.22 \begin{align*} \frac{\log \left (b x^{2} + a\right )}{2 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{2} + c\right )}{2 \,{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54995, size = 69, normalized size = 1.53 \begin{align*} \frac{\log \left (b x^{2} + a\right ) - \log \left (d x^{2} + c\right )}{2 \,{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.87919, size = 138, normalized size = 3.07 \begin{align*} \frac{\log{\left (x^{2} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{2 \left (a d - b c\right )} - \frac{\log{\left (x^{2} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{2 \left (a d - b c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: NotImplementedError