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

Optimal. Leaf size=156 \[ \frac {b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {1}{2 a^2 c^2 x^2}+\frac {d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac {d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2} \]

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Rubi [A]  time = 0.21, antiderivative size = 156, 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, 88} \begin {gather*} -\frac {b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {1}{2 a^2 c^2 x^2}-\frac {d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}+\frac {d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

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

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Mathematica [A]  time = 0.22, size = 157, normalized size = 1.01 \begin {gather*} \frac {1}{2} \left (\frac {2 b^3 (2 a d-b c) \log \left (a+b x^2\right )}{a^3 (a d-b c)^3}-\frac {4 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {1}{a^2 c^2 x^2}+\frac {2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac {d^3}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 25.06, size = 667, normalized size = 4.28 \begin {gather*} -\frac {a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \, {\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x^{2} - 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left ({\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{6} + {\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{4} + {\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.39, size = 333, normalized size = 2.13 \begin {gather*} \frac {{\left (b^{5} c - 2 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}} + \frac {{\left (2 \, b c d^{4} - a d^{5}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{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}} - \frac {2 \, b^{3} c^{2} d x^{4} - 2 \, a b^{2} c d^{2} x^{4} + 2 \, a^{2} b d^{3} x^{4} + 2 \, b^{3} c^{3} x^{2} - a b^{2} c^{2} d x^{2} - a^{2} b c d^{2} x^{2} + 2 \, a^{3} d^{3} x^{2} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{6} + b c x^{4} + a d x^{4} + a c x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 254, normalized size = 1.63 \begin {gather*} -\frac {a \,d^{4}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right ) c^{2}}+\frac {a \,d^{4} \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{3} c^{3}}-\frac {b^{3} d}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right ) a}+\frac {b^{4} c}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right ) a^{2}}+\frac {2 b^{3} d \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{3} a^{2}}-\frac {b^{4} c \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{3} a^{3}}+\frac {b \,d^{3}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right ) c}-\frac {2 b \,d^{3} \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{3} c^{2}}-\frac {2 d \ln \relax (x )}{a^{2} c^{3}}-\frac {2 b \ln \relax (x )}{a^{3} c^{2}}-\frac {1}{2 a^{2} c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.21, size = 381, normalized size = 2.44 \begin {gather*} \frac {{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {{\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x^{2} + c\right )}{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}} - \frac {a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{4} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.64, size = 313, normalized size = 2.01 \begin {gather*} -\frac {\frac {1}{2\,a\,c}+\frac {x^4\,\left (a^2\,b\,d^3-a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (a\,d+b\,c\right )\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^4+a\,c\,x^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (b^4\,c-2\,a\,b^3\,d\right )}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\ln \left (d\,x^2+c\right )\,\left (a\,d^4-2\,b\,c\,d^3\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}-\frac {\ln \relax (x)\,\left (2\,a\,d+2\,b\,c\right )}{a^3\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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