3.300 \(\int \frac{1}{x^7 (a+b x^2)^2 (c+d x^2)} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^2 d^2+2 a b c d+3 b^2 c^2}{2 a^4 c^3 x^2}-\frac{\log (x) \left (2 a^2 b c d^2+a^3 d^3+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}-\frac{b^4}{2 a^4 \left (a+b x^2\right ) (b c-a d)}+\frac{b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac{a d+2 b c}{4 a^3 c^2 x^4}-\frac{1}{6 a^2 c x^6}+\frac{d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.248792, antiderivative size = 210, 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} \[ -\frac{a^2 d^2+2 a b c d+3 b^2 c^2}{2 a^4 c^3 x^2}-\frac{\log (x) \left (2 a^2 b c d^2+a^3 d^3+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}-\frac{b^4}{2 a^4 \left (a+b x^2\right ) (b c-a d)}+\frac{b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac{a d+2 b c}{4 a^3 c^2 x^4}-\frac{1}{6 a^2 c x^6}+\frac{d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/(6*a^2*c*x^6) + (2*b*c + a*d)/(4*a^3*c^2*x^4) - (3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)/(2*a^4*c^3*x^2) - b^4/(2*
a^4*(b*c - a*d)*(a + b*x^2)) - ((4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(a^5*c^4) + (b^4
*(4*b*c - 5*a*d)*Log[a + b*x^2])/(2*a^5*(b*c - a*d)^2) + (d^5*Log[c + d*x^2])/(2*c^4*(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 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^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 c x^4}+\frac{-2 b c-a d}{a^3 c^2 x^3}+\frac{3 b^2 c^2+2 a b c d+a^2 d^2}{a^4 c^3 x^2}+\frac{-4 b^3 c^3-3 a b^2 c^2 d-2 a^2 b c d^2-a^3 d^3}{a^5 c^4 x}-\frac{b^5}{a^4 (-b c+a d) (a+b x)^2}-\frac{b^5 (-4 b c+5 a d)}{a^5 (-b c+a d)^2 (a+b x)}+\frac{d^6}{c^4 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 a^2 c x^6}+\frac{2 b c+a d}{4 a^3 c^2 x^4}-\frac{3 b^2 c^2+2 a b c d+a^2 d^2}{2 a^4 c^3 x^2}-\frac{b^4}{2 a^4 (b c-a d) \left (a+b x^2\right )}-\frac{\left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac{b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac{d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.265752, size = 202, normalized size = 0.96 \[ \frac{1}{12} \left (-\frac{6 \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3 x^2}-\frac{12 \log (x) \left (2 a^2 b c d^2+a^3 d^3+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}+\frac{6 b^4}{a^4 \left (a+b x^2\right ) (a d-b c)}+\frac{6 b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{a^5 (b c-a d)^2}+\frac{3 a d+6 b c}{a^3 c^2 x^4}-\frac{2}{a^2 c x^6}+\frac{6 d^5 \log \left (c+d x^2\right )}{c^4 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

(-2/(a^2*c*x^6) + (6*b*c + 3*a*d)/(a^3*c^2*x^4) - (6*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2))/(a^4*c^3*x^2) + (6*b^4
)/(a^4*(-(b*c) + a*d)*(a + b*x^2)) - (12*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(a^5*c^
4) + (6*b^4*(4*b*c - 5*a*d)*Log[a + b*x^2])/(a^5*(b*c - a*d)^2) + (6*d^5*Log[c + d*x^2])/(c^4*(b*c - a*d)^2))/
12

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Maple [A]  time = 0.021, size = 268, normalized size = 1.3 \begin{align*}{\frac{{d}^{5}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{4} \left ( ad-bc \right ) ^{2}}}-{\frac{1}{6\,{a}^{2}c{x}^{6}}}+{\frac{d}{4\,{a}^{2}{c}^{2}{x}^{4}}}+{\frac{b}{2\,{a}^{3}c{x}^{4}}}-{\frac{{d}^{2}}{2\,{a}^{2}{c}^{3}{x}^{2}}}-{\frac{bd}{{a}^{3}{c}^{2}{x}^{2}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}c{x}^{2}}}-{\frac{\ln \left ( x \right ){d}^{3}}{{a}^{2}{c}^{4}}}-2\,{\frac{\ln \left ( x \right ){d}^{2}b}{{a}^{3}{c}^{3}}}-3\,{\frac{\ln \left ( x \right ) d{b}^{2}}{{a}^{4}{c}^{2}}}-4\,{\frac{\ln \left ( x \right ){b}^{3}}{{a}^{5}c}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{4} \left ( ad-bc \right ) ^{2}}}+2\,{\frac{{b}^{5}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{5} \left ( ad-bc \right ) ^{2}}}+{\frac{{b}^{4}d}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}c}{2\,{a}^{4} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^2+a)^2/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.04319, size = 458, normalized size = 2.18 \begin{align*} \frac{d^{5} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )}} + \frac{{\left (4 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )}} - \frac{2 \, a^{3} b c^{3} - 2 \, a^{4} c^{2} d + 6 \,{\left (4 \, b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{6} + 3 \,{\left (4 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{4} -{\left (4 \, a^{2} b^{2} c^{3} - a^{3} b c^{2} d - 3 \, a^{4} c d^{2}\right )} x^{2}}{12 \,{\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{8} +{\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{6}\right )}} - \frac{{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

1/2*d^5*log(d*x^2 + c)/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2) + 1/2*(4*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^5*b
^2*c^2 - 2*a^6*b*c*d + a^7*d^2) - 1/12*(2*a^3*b*c^3 - 2*a^4*c^2*d + 6*(4*b^4*c^3 - a*b^3*c^2*d - a^2*b^2*c*d^2
- a^3*b*d^3)*x^6 + 3*(4*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 - 2*a^4*d^3)*x^4 - (4*a^2*b^2*c^3 - a^3*b*c^2
*d - 3*a^4*c*d^2)*x^2)/((a^4*b^2*c^4 - a^5*b*c^3*d)*x^8 + (a^5*b*c^4 - a^6*c^3*d)*x^6) - 1/2*(4*b^3*c^3 + 3*a*
b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(x^2)/(a^5*c^4)

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Fricas [B]  time = 54.1615, size = 817, normalized size = 3.89 \begin{align*} -\frac{2 \, a^{4} b^{2} c^{5} - 4 \, a^{5} b c^{4} d + 2 \, a^{6} c^{3} d^{2} + 6 \,{\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{5} b c d^{4}\right )} x^{6} + 3 \,{\left (4 \, a^{2} b^{4} c^{5} - 5 \, a^{3} b^{3} c^{4} d - a^{5} b c^{2} d^{3} + 2 \, a^{6} c d^{4}\right )} x^{4} -{\left (4 \, a^{3} b^{3} c^{5} - 5 \, a^{4} b^{2} c^{4} d - 2 \, a^{5} b c^{3} d^{2} + 3 \, a^{6} c^{2} d^{3}\right )} x^{2} - 6 \,{\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d\right )} x^{8} +{\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (a^{5} b d^{5} x^{8} + a^{6} d^{5} x^{6}\right )} \log \left (d x^{2} + c\right ) + 12 \,{\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d + a^{5} b d^{5}\right )} x^{8} +{\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{6} d^{5}\right )} x^{6}\right )} \log \left (x\right )}{12 \,{\left ({\left (a^{5} b^{3} c^{6} - 2 \, a^{6} b^{2} c^{5} d + a^{7} b c^{4} d^{2}\right )} x^{8} +{\left (a^{6} b^{2} c^{6} - 2 \, a^{7} b c^{5} d + a^{8} c^{4} d^{2}\right )} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

-1/12*(2*a^4*b^2*c^5 - 4*a^5*b*c^4*d + 2*a^6*c^3*d^2 + 6*(4*a*b^5*c^5 - 5*a^2*b^4*c^4*d + a^5*b*c*d^4)*x^6 + 3
*(4*a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - a^5*b*c^2*d^3 + 2*a^6*c*d^4)*x^4 - (4*a^3*b^3*c^5 - 5*a^4*b^2*c^4*d - 2*a^
5*b*c^3*d^2 + 3*a^6*c^2*d^3)*x^2 - 6*((4*b^6*c^5 - 5*a*b^5*c^4*d)*x^8 + (4*a*b^5*c^5 - 5*a^2*b^4*c^4*d)*x^6)*l
og(b*x^2 + a) - 6*(a^5*b*d^5*x^8 + a^6*d^5*x^6)*log(d*x^2 + c) + 12*((4*b^6*c^5 - 5*a*b^5*c^4*d + a^5*b*d^5)*x
^8 + (4*a*b^5*c^5 - 5*a^2*b^4*c^4*d + a^6*d^5)*x^6)*log(x))/((a^5*b^3*c^6 - 2*a^6*b^2*c^5*d + a^7*b*c^4*d^2)*x
^8 + (a^6*b^2*c^6 - 2*a^7*b*c^5*d + a^8*c^4*d^2)*x^6)

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

[Out]

Timed out

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Giac [A]  time = 1.12494, size = 478, normalized size = 2.28 \begin{align*} \frac{d^{6} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )}} + \frac{{\left (4 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{5} b^{3} c^{2} - 2 \, a^{6} b^{2} c d + a^{7} b d^{2}\right )}} - \frac{4 \, b^{6} c x^{2} - 5 \, a b^{5} d x^{2} + 5 \, a b^{5} c - 6 \, a^{2} b^{4} d}{2 \,{\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )}{\left (b x^{2} + a\right )}} - \frac{{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} + \frac{44 \, b^{3} c^{3} x^{6} + 33 \, a b^{2} c^{2} d x^{6} + 22 \, a^{2} b c d^{2} x^{6} + 11 \, a^{3} d^{3} x^{6} - 18 \, a b^{2} c^{3} x^{4} - 12 \, a^{2} b c^{2} d x^{4} - 6 \, a^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{3} x^{2} + 3 \, a^{3} c^{2} d x^{2} - 2 \, a^{3} c^{3}}{12 \, a^{5} c^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

1/2*d^6*log(abs(d*x^2 + c))/(b^2*c^6*d - 2*a*b*c^5*d^2 + a^2*c^4*d^3) + 1/2*(4*b^6*c - 5*a*b^5*d)*log(abs(b*x^
2 + a))/(a^5*b^3*c^2 - 2*a^6*b^2*c*d + a^7*b*d^2) - 1/2*(4*b^6*c*x^2 - 5*a*b^5*d*x^2 + 5*a*b^5*c - 6*a^2*b^4*d
)/((a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2)*(b*x^2 + a)) - 1/2*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d
^3)*log(x^2)/(a^5*c^4) + 1/12*(44*b^3*c^3*x^6 + 33*a*b^2*c^2*d*x^6 + 22*a^2*b*c*d^2*x^6 + 11*a^3*d^3*x^6 - 18*
a*b^2*c^3*x^4 - 12*a^2*b*c^2*d*x^4 - 6*a^3*c*d^2*x^4 + 6*a^2*b*c^3*x^2 + 3*a^3*c^2*d*x^2 - 2*a^3*c^3)/(a^5*c^4
*x^6)