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

Optimal. Leaf size=100 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.101341, antiderivative size = 100, 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} \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0945337, size = 98, normalized size = 0.98 \[ \frac{1}{2} \left (-\frac{b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}-\frac{d}{c \left (c+d x^2\right ) (b c-a d)}+\frac{2 \log (x)}{a c^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 139, normalized size = 1.4 \begin{align*} -{\frac{{d}^{2}\ln \left ( d{x}^{2}+c \right ) a}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{d\ln \left ( d{x}^{2}+c \right ) b}{c \left ( ad-bc \right ) ^{2}}}+{\frac{a{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{bd}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{\ln \left ( x \right ) }{a{c}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19167, size = 186, normalized size = 1.86 \begin{align*} -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac{d}{2 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 6.3035, size = 444, normalized size = 4.44 \begin{align*} -\frac{a b c^{2} d - a^{2} c d^{2} +{\left (b^{2} c^{2} d x^{2} + b^{2} c^{3}\right )} \log \left (b x^{2} + a\right ) -{\left (2 \, a b c^{2} d - a^{2} c d^{2} +{\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} +{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x^{2}\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)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.21068, size = 250, normalized size = 2.5 \begin{align*} -\frac{b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}} + \frac{{\left (2 \, b c d^{2} - a d^{3}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} - \frac{2 \, b c d^{2} x^{2} - a d^{3} x^{2} + 3 \, b c^{2} d - 2 \, a c d^{2}}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}{\left (d x^{2} + c\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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