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

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

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 149, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]

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]]

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 = 0.27, size = 141, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 8.89, size = 520, normalized size = 3.49 \begin {gather*} -\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 \relax (x)}{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 {gather*}

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)

________________________________________________________________________________________

giac [B]  time = 0.43, size = 315, normalized size = 2.11 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 286, normalized size = 1.92 \begin {gather*} \frac {a^{2} d^{3}}{4 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )^{2} c}-\frac {a b \,d^{2}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )^{2}}+\frac {b^{2} c d}{4 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )^{2}}+\frac {a^{2} d^{3}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right ) c^{2}}-\frac {a^{2} d^{3} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3} c^{3}}-\frac {3 a b \,d^{2}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right ) c}+\frac {3 a b \,d^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3} c^{2}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{3} a}-\frac {3 b^{2} d \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3} c}+\frac {b^{2} d}{\left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {\ln \relax (x )}{a \,c^{3}} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 1.25, size = 278, normalized size = 1.87 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 1.40, size = 246, normalized size = 1.65 \begin {gather*} \frac {\frac {3\,a\,d^2-5\,b\,c\,d}{4\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,x^2\,\left (a\,d-2\,b\,c\right )}{2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x^2+d^2\,x^4}+\frac {b^3\,\ln \left (b\,x^2+a\right )}{2\,a^4\,d^3-6\,a^3\,b\,c\,d^2+6\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3}+\frac {\ln \relax (x)}{a\,c^3}+\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^3-3\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{-2\,a^3\,c^3\,d^3+6\,a^2\,b\,c^4\,d^2-6\,a\,b^2\,c^5\,d+2\,b^3\,c^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________