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

Optimal. Leaf size=92 \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

[Out]

-b/(2*(b*c - a*d)^2*(a + b*x^2)) - d/(2*(b*c - a*d)^2*(c + d*x^2)) - (b*d*Log[a + b*x^2])/(b*c - a*d)^3 + (b*d
*Log[c + d*x^2])/(b*c - a*d)^3

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Rubi [A]  time = 0.0792166, antiderivative size = 92, 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} \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b/(2*(b*c - a*d)^2*(a + b*x^2)) - d/(2*(b*c - a*d)^2*(c + d*x^2)) - (b*d*Log[a + b*x^2])/(b*c - a*d)^3 + (b*d
*Log[c + d*x^2])/(b*c - a*d)^3

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 )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^2 (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{(b c-a d)^2 (a+b x)^2}-\frac{2 b^2 d}{(b c-a d)^3 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)^2}+\frac{2 b d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b}{2 (b c-a d)^2 \left (a+b x^2\right )}-\frac{d}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.0679679, size = 77, normalized size = 0.84 \[ \frac{\frac{b (a d-b c)}{a+b x^2}+\frac{d (a d-b c)}{c+d x^2}-2 b d \log \left (a+b x^2\right )+2 b d \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*(-(b*c) + a*d))/(a + b*x^2) + (d*(-(b*c) + a*d))/(c + d*x^2) - 2*b*d*Log[a + b*x^2] + 2*b*d*Log[c + d*x^2]
)/(2*(b*c - a*d)^3)

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Maple [A]  time = 0.016, size = 143, normalized size = 1.6 \begin{align*} -{\frac{bd\ln \left ( d{x}^{2}+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{bdc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{b\ln \left ( b{x}^{2}+a \right ) d}{ \left ( ad-bc \right ) ^{3}}}-{\frac{abd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{3} \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)^2,x)

[Out]

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

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Maxima [B]  time = 0.988063, size = 290, normalized size = 3.15 \begin{align*} -\frac{b d \log \left (b x^{2} + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{b d \log \left (d x^{2} + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x^{2} + b c + a d}{2 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.6321, size = 508, normalized size = 5.52 \begin{align*} -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.84365, size = 408, normalized size = 4.43 \begin{align*} - \frac{b d \log{\left (x^{2} + \frac{- \frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} - \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{b d \log{\left (x^{2} + \frac{\frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} + \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x^{2}}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\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)**2,x)

[Out]

-b*d*log(x**2 + (-a**4*b*d**5/(a*d - b*c)**3 + 4*a**3*b**2*c*d**4/(a*d - b*c)**3 - 6*a**2*b**3*c**2*d**3/(a*d
- b*c)**3 + 4*a*b**4*c**3*d**2/(a*d - b*c)**3 + a*b*d**2 - b**5*c**4*d/(a*d - b*c)**3 + b**2*c*d)/(2*b**2*d**2
))/(a*d - b*c)**3 + b*d*log(x**2 + (a**4*b*d**5/(a*d - b*c)**3 - 4*a**3*b**2*c*d**4/(a*d - b*c)**3 + 6*a**2*b*
*3*c**2*d**3/(a*d - b*c)**3 - 4*a*b**4*c**3*d**2/(a*d - b*c)**3 + a*b*d**2 + b**5*c**4*d/(a*d - b*c)**3 + b**2
*c*d)/(2*b**2*d**2))/(a*d - b*c)**3 - (a*d + b*c + 2*b*d*x**2)/(2*a**3*c*d**2 - 4*a**2*b*c**2*d + 2*a*b**2*c**
3 + x**4*(2*a**2*b*d**3 - 4*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**2*(2*a**3*d**3 - 2*a**2*b*c*d**2 - 2*a*b**2*c*
*2*d + 2*b**3*c**3))

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Giac [A]  time = 1.15689, size = 220, normalized size = 2.39 \begin{align*} \frac{b^{2} d \log \left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{3}}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x^{2} + a\right )}} + \frac{b d^{2}}{2 \,{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\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)^2,x, algorithm="giac")

[Out]

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