3.3.40 \(\int \frac {1}{x^7 (a+b x^2) (c+d x^2)} \, dx\) [240]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 155, 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 = {457, 84} \begin {gather*} \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 {\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}-\frac {a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)}-\frac {1}{6 a c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*1/(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 84

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 457

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^7 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 (a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.05, size = 147, normalized size = 0.95 \begin {gather*} \frac {12 \left (b^4 c^4-a^4 d^4\right ) x^6 \log (x)-6 b^4 c^4 x^6 \log \left (a+b x^2\right )+a \left (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 \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 )\right )}{12 a^4 c^4 (-b c+a d) x^6} \end {gather*}

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 = 0.13, size = 162, normalized size = 1.05

method	result	size
norman	\(\frac {-\frac {1}{6 a c}+\frac {\left (a d +b c \right ) x^{2}}{4 a^{2} c^{2}}-\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{4}}{2 a^{3} c^{3}}}{x^{6}}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{4} \left (a d -b c \right )}+\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a d -b c \right )}-\frac {\left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4} c^{4}}\)	\(159\)
default	\(-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{4} \left (a d -b c \right )}+\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a d -b c \right )}-\frac {1}{6 a c \,x^{6}}-\frac {-a d -b c}{4 a^{2} c^{2} x^{4}}-\frac {a^{2} d^{2}+a b c d +b^{2} c^{2}}{2 a^{3} c^{3} x^{2}}+\frac {\left (-a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4} c^{4}}\)	\(162\)
risch	\(\frac {-\frac {1}{6 a c}+\frac {\left (a d +b c \right ) x^{2}}{4 a^{2} c^{2}}-\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{4}}{2 a^{3} c^{3}}}{x^{6}}-\frac {\ln \left (x \right ) d^{3}}{a \,c^{4}}-\frac {\ln \left (x \right ) b \,d^{2}}{a^{2} c^{3}}-\frac {\ln \left (x \right ) b^{2} d}{a^{3} c^{2}}-\frac {\ln \left (x \right ) b^{3}}{a^{4} c}+\frac {d^{4} \ln \left (-d \,x^{2}-c \right )}{2 c^{4} \left (a d -b c \right )}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{4} \left (a d -b c \right )}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 165, normalized size = 1.06 \begin {gather*} \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 {gather*}

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 = 2.24, size = 155, normalized size = 1.00 \begin {gather*} \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 {gather*}

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

[Out]

Timed out

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Giac [A]
time = 1.24, size = 239, normalized size = 1.54 \begin {gather*} \frac {b^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{4} b^{2} c - a^{5} b d\right )}} - \frac {d^{5} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c^{5} d - a c^{4} d^{2}\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 {11 \, b^{3} c^{3} x^{6} + 11 \, a b^{2} c^{2} d x^{6} + 11 \, a^{2} b c d^{2} x^{6} + 11 \, a^{3} d^{3} x^{6} - 6 \, a b^{2} c^{3} x^{4} - 6 \, a^{2} b c^{2} d x^{4} - 6 \, a^{3} c d^{2} x^{4} + 3 \, a^{2} b c^{3} x^{2} + 3 \, a^{3} c^{2} d x^{2} - 2 \, a^{3} c^{3}}{12 \, a^{4} c^{4} x^{6}} \end {gather*}

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]

1/2*b^5*log(abs(b*x^2 + a))/(a^4*b^2*c - a^5*b*d) - 1/2*d^5*log(abs(d*x^2 + c))/(b*c^5*d - a*c^4*d^2) - 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*(11*b^3*c^3*x^6 + 11*a*b^2*c^2*d*x^6 +
11*a^2*b*c*d^2*x^6 + 11*a^3*d^3*x^6 - 6*a*b^2*c^3*x^4 - 6*a^2*b*c^2*d*x^4 - 6*a^3*c*d^2*x^4 + 3*a^2*b*c^3*x^2
+ 3*a^3*c^2*d*x^2 - 2*a^3*c^3)/(a^4*c^4*x^6)

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Mupad [B]
time = 0.27, size = 165, normalized size = 1.06 \begin {gather*} -\frac {\frac {1}{6\,a\,c}-\frac {x^2\,\left (a\,d+b\,c\right )}{4\,a^2\,c^2}+\frac {x^4\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{2\,a^3\,c^3}}{x^6}-\frac {b^4\,\ln \left (b\,x^2+a\right )}{2\,\left (a^5\,d-a^4\,b\,c\right )}-\frac {d^4\,\ln \left (d\,x^2+c\right )}{2\,\left (b\,c^5-a\,c^4\,d\right )}-\frac {\ln \left (x\right )\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{a^4\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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