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

Optimal. Leaf size=141 \[ -\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{2 a \left (a+b x^2\right ) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^3}+\frac{d^2}{2 c \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.16858, antiderivative size = 141, 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, 88} \[ -\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{2 a \left (a+b x^2\right ) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^3}+\frac{d^2}{2 c \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.23333, size = 133, normalized size = 0.94 \[ \frac{1}{2} \left (\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{a^2 (a d-b c)^3}+\frac{2 \log (x)}{a^2 c^2}+\frac{b^2}{a \left (a+b x^2\right ) (b c-a d)^2}+\frac{d^2 (a d-3 b c) \log \left (c+d x^2\right )}{c^2 (b c-a d)^3}+\frac{d^2}{c \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 225, normalized size = 1.6 \begin{align*} -{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) a}{2\,{c}^{2} \left ( ad-bc \right ) ^{3}}}+{\frac{3\,{d}^{2}\ln \left ( d{x}^{2}+c \right ) b}{2\,c \left ( ad-bc \right ) ^{3}}}+{\frac{{d}^{3}a}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{2}b}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}{c}^{2}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,a \left ( ad-bc \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) c}{2\,{a}^{2} \left ( ad-bc \right ) ^{3}}}+{\frac{{b}^{2}d}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}c}{2\,a \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(1/x/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

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Maxima [B]  time = 1.05212, size = 398, normalized size = 2.82 \begin{align*} -\frac{{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac{{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} + \frac{b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x^{2}}{2 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*c - 3*a*b^2*d)*log(b*x^2 + a)/(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 1/2*(3*b*c
*d^2 - a*d^3)*log(d*x^2 + c)/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) + 1/2*(b^2*c^2 + a^2*d^
2 + (b^2*c*d + a*b*d^2)*x^2)/(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a
^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2) + 1/2*log(x^2)/(a^2*c^2)

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Fricas [B]  time = 25.7502, size = 1042, normalized size = 7.39 \begin{align*} \frac{a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x^{2} -{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) -{\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\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 )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} +{\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{4} +{\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*b^3*c^4 - a^2*b^2*c^3*d + a^3*b*c^2*d^2 - a^4*c*d^3 + (a*b^3*c^3*d - a^3*b*c*d^3)*x^2 - (a*b^3*c^4 - 3*
a^2*b^2*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2*d^2)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2)*x^2)*log(b*x
^2 + a) - (3*a^3*b*c^2*d^2 - a^4*c*d^3 + (3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (3*a^2*b^2*c^2*d^2 + 2*a^3*b*c*d^
3 - a^4*d^4)*x^2)*log(d*x^2 + c) + 2*(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)
*log(x))/(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3 + (a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 +
3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4)*x^4 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^2)

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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)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError