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

Optimal. Leaf size=70 \[ -\frac{1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac{d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

[Out]

-1/(2*(b*c - a*d)*(a + b*x^2)) - (d*Log[a + b*x^2])/(2*(b*c - a*d)^2) + (d*Log[c + d*x^2])/(2*(b*c - a*d)^2)

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Rubi [A]  time = 0.0516852, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 44} \[ -\frac{1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac{d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/(2*(b*c - a*d)*(a + b*x^2)) - (d*Log[a + b*x^2])/(2*(b*c - a*d)^2) + (d*Log[c + d*x^2])/(2*(b*c - a*d)^2)

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0280295, size = 66, normalized size = 0.94 \[ \frac{d \left (a+b x^2\right ) \log \left (c+d x^2\right )-d \left (a+b x^2\right ) \log \left (a+b x^2\right )+a d-b c}{2 \left (a+b x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 90, normalized size = 1.3 \begin{align*}{\frac{d\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{ad}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{bc}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971497, size = 134, normalized size = 1.91 \begin{align*} -\frac{d \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{d \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{1}{2 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60039, size = 219, normalized size = 3.13 \begin{align*} -\frac{b c - a d +{\left (b d x^{2} + a d\right )} \log \left (b x^{2} + a\right ) -{\left (b d x^{2} + a d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*c - a*d + (b*d*x^2 + a*d)*log(b*x^2 + a) - (b*d*x^2 + a*d)*log(d*x^2 + c))/(a*b^2*c^2 - 2*a^2*b*c*d +
a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^2)

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Sympy [B]  time = 2.18057, size = 248, normalized size = 3.54 \begin{align*} \frac{d \log{\left (x^{2} + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{d \log{\left (x^{2} + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{1}{2 a^{2} d - 2 a b c + x^{2} \left (2 a b d - 2 b^{2} c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(x**2 + (-a**3*d**4/(a*d - b*c)**2 + 3*a**2*b*c*d**3/(a*d - b*c)**2 - 3*a*b**2*c**2*d**2/(a*d - b*c)**2 +
a*d**2 + b**3*c**3*d/(a*d - b*c)**2 + b*c*d)/(2*b*d**2))/(2*(a*d - b*c)**2) - d*log(x**2 + (a**3*d**4/(a*d -
b*c)**2 - 3*a**2*b*c*d**3/(a*d - b*c)**2 + 3*a*b**2*c**2*d**2/(a*d - b*c)**2 + a*d**2 - b**3*c**3*d/(a*d - b*c
)**2 + b*c*d)/(2*b*d**2))/(2*(a*d - b*c)**2) + 1/(2*a**2*d - 2*a*b*c + x**2*(2*a*b*d - 2*b**2*c))

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Giac [A]  time = 1.14854, size = 115, normalized size = 1.64 \begin{align*} \frac{b d \log \left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac{b}{2 \,{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*d*log(abs(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) + d))/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) - 1/2*b/((b^2*c -
a*b*d)*(b*x^2 + a))