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

Optimal. Leaf size=155 \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}+\frac{b^4 \log \left (a+b x^2\right )}{2 a^4 (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)}-\frac{1}{6 a c x^6} \]

[Out]

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

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Rubi [A]  time = 0.169009, antiderivative size = 155, 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} \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}+\frac{b^4 \log \left (a+b x^2\right )}{2 a^4 (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)}-\frac{1}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0643291, size = 147, normalized size = 0.95 \[ \frac{12 x^6 \log (x) \left (b^4 c^4-a^4 d^4\right )+a \left (2 a^2 b c^4+a^3 c d \left (-2 c^2+3 c d x^2-6 d^2 x^4\right )+6 a^3 d^4 x^6 \log \left (c+d x^2\right )-3 a b^2 c^4 x^2+6 b^3 c^4 x^4\right )-6 b^4 c^4 x^6 \log \left (a+b x^2\right )}{12 a^4 c^4 x^6 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 184, normalized size = 1.2 \begin{align*}{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{4} \left ( ad-bc \right ) }}-{\frac{1}{6\,ac{x}^{6}}}+{\frac{d}{4\,a{c}^{2}{x}^{4}}}+{\frac{b}{4\,{a}^{2}c{x}^{4}}}-{\frac{{d}^{2}}{2\,a{c}^{3}{x}^{2}}}-{\frac{bd}{2\,{a}^{2}{c}^{2}{x}^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3}c{x}^{2}}}-{\frac{\ln \left ( x \right ){d}^{3}}{a{c}^{4}}}-{\frac{\ln \left ( x \right ) b{d}^{2}}{{a}^{2}{c}^{3}}}-{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}{c}^{2}}}-{\frac{\ln \left ( x \right ){b}^{3}}{{a}^{4}c}}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03975, size = 223, normalized size = 1.44 \begin{align*} \frac{b^{4} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{4} b c - a^{5} d\right )}} - \frac{d^{4} \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{5} - a c^{4} d\right )}} - \frac{{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{4}} - \frac{6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{12 \, a^{3} c^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 10.5954, size = 311, normalized size = 2.01 \begin{align*} \frac{6 \, b^{4} c^{4} x^{6} \log \left (b x^{2} + a\right ) - 6 \, a^{4} d^{4} x^{6} \log \left (d x^{2} + c\right ) - 2 \, a^{3} b c^{4} + 2 \, a^{4} c^{3} d - 12 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x^{6} \log \left (x\right ) - 6 \,{\left (a b^{3} c^{4} - a^{4} c d^{3}\right )} x^{4} + 3 \,{\left (a^{2} b^{2} c^{4} - a^{4} c^{2} d^{2}\right )} x^{2}}{12 \,{\left (a^{4} b c^{5} - a^{5} c^{4} d\right )} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError