3.257 \(\int \frac{1}{x (a+b x^2) (c+d x^2)^3} \, dx\)

Optimal. Leaf size=149 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3}-\frac{b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}-\frac{d (2 b c-a d)}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c \left (c+d x^2\right )^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

[Out]

-d/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(2*b*c - a*d))/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) + Log[x]/(a*c^3) - (b
^3*Log[a + b*x^2])/(2*a*(b*c - a*d)^3) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(2*c^3*(b*c - a*
d)^3)

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Rubi [A]  time = 0.152368, antiderivative size = 149, 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} \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3}-\frac{b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}-\frac{d (2 b c-a d)}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c \left (c+d x^2\right )^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-d/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(2*b*c - a*d))/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) + Log[x]/(a*c^3) - (b
^3*Log[a + b*x^2])/(2*a*(b*c - a*d)^3) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(2*c^3*(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 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 )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x) (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a c^3 x}+\frac{b^4}{a (-b c+a d)^3 (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)^3}+\frac{d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)^2}+\frac{d^2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{d}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (2 b c-a d)}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\log (x)}{a c^3}-\frac{b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.28008, size = 141, normalized size = 0.95 \[ \frac{\frac{d \left (\frac{c \left (a^2 d^2 \left (3 c+2 d x^2\right )-2 a b c d \left (4 c+3 d x^2\right )+b^2 c^2 \left (5 c+4 d x^2\right )\right )}{\left (c+d x^2\right )^2}-2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )\right )}{c^3}+\frac{2 b^3 \log \left (a+b x^2\right )}{a}}{4 (a d-b c)^3}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

Log[x]/(a*c^3) + ((2*b^3*Log[a + b*x^2])/a + (d*((c*(a^2*d^2*(3*c + 2*d*x^2) - 2*a*b*c*d*(4*c + 3*d*x^2) + b^2
*c^2*(5*c + 4*d*x^2)))/(c + d*x^2)^2 - 2*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x^2]))/c^3)/(4*(-(b*c) +
a*d)^3)

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Maple [B]  time = 0.016, size = 286, normalized size = 1.9 \begin{align*} -{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{3} \left ( ad-bc \right ) ^{3}}}+{\frac{3\,{d}^{2}\ln \left ( d{x}^{2}+c \right ) ab}{2\,{c}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{3\,d\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,c \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{2}{d}^{3}}{4\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{a{d}^{2}b}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}cd}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,a{d}^{2}b}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{\ln \left ( x \right ) }{a{c}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.19304, size = 375, normalized size = 2.52 \begin{align*} -\frac{b^{3} \log \left (b x^{2} + a\right )}{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac{{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}} - \frac{5 \, b c^{2} d - 3 \, a c d^{2} + 2 \,{\left (2 \, b c d^{2} - a d^{3}\right )} x^{2}}{4 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 23.2272, size = 1027, normalized size = 6.89 \begin{align*} -\frac{5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \,{\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} d^{2} x^{4} + 2 \, b^{3} c^{4} d x^{2} + b^{3} c^{5}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} +{\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \,{\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} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} +{\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{4} + 2 \,{\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} 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)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/4*(5*a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 3*a^3*c^2*d^3 + 2*(2*a*b^2*c^3*d^2 - 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x^2
+ 2*(b^3*c^3*d^2*x^4 + 2*b^3*c^4*d*x^2 + b^3*c^5)*log(b*x^2 + a) - 2*(3*a*b^2*c^4*d - 3*a^2*b*c^3*d^2 + a^3*c^
2*d^3 + (3*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4 + a^3*d^5)*x^4 + 2*(3*a*b^2*c^3*d^2 - 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x^
2)*log(d*x^2 + c) - 4*(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 + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^
3 + 3*a^2*b*c*d^4 - a^3*d^5)*x^4 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x^2)*log(x))/
(a*b^3*c^8 - 3*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*c^5*d^3 + (a*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^3*b*c^
4*d^4 - a^4*c^3*d^5)*x^4 + 2*(a*b^3*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^5*d^3 - a^4*c^4*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)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.13324, size = 425, normalized size = 2.85 \begin{align*} -\frac{b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )}} + \frac{{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )}} - \frac{9 \, b^{2} c^{2} d^{3} x^{4} - 9 \, a b c d^{4} x^{4} + 3 \, a^{2} d^{5} x^{4} + 22 \, b^{2} c^{3} d^{2} x^{2} - 24 \, a b c^{2} d^{3} x^{2} + 8 \, a^{2} c d^{4} x^{2} + 14 \, b^{2} c^{4} d - 17 \, a b c^{3} d^{2} + 6 \, a^{2} c^{2} d^{3}}{4 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-1/2*b^4*log(abs(b*x^2 + a))/(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3) + 1/2*(3*b^2*c^2*d^2
- 3*a*b*c*d^3 + a^2*d^4)*log(abs(d*x^2 + c))/(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4) - 1
/4*(9*b^2*c^2*d^3*x^4 - 9*a*b*c*d^4*x^4 + 3*a^2*d^5*x^4 + 22*b^2*c^3*d^2*x^2 - 24*a*b*c^2*d^3*x^2 + 8*a^2*c*d^
4*x^2 + 14*b^2*c^4*d - 17*a*b*c^3*d^2 + 6*a^2*c^2*d^3)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d
^3)*(d*x^2 + c)^2) + 1/2*log(x^2)/(a*c^3)