3.6.22 \(\int \frac {1}{x^5 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [522]

Optimal. Leaf size=140 \[ -\frac {1}{4 a^6 x^4}+\frac {3 b}{a^7 x^2}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {21 b^2 \log (x)}{a^8}-\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8} \]

[Out]

-1/4/a^6/x^4+3*b/a^7/x^2+1/10*b^2/a^3/(b*x^2+a)^5+3/8*b^2/a^4/(b*x^2+a)^4+b^2/a^5/(b*x^2+a)^3+5/2*b^2/a^6/(b*x
^2+a)^2+15/2*b^2/a^7/(b*x^2+a)+21*b^2*ln(x)/a^8-21/2*b^2*ln(b*x^2+a)/a^8

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Rubi [A]
time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 46} \begin {gather*} -\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac {21 b^2 \log (x)}{a^8}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {3 b}{a^7 x^2}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}-\frac {1}{4 a^6 x^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-1/4*1/(a^6*x^4) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(a + b*x^2)^4) + b^2/(a^5*(a
+ b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^2)/(2*a^7*(a + b*x^2)) + (21*b^2*Log[x])/a^8 - (21*b^2*Log
[a + b*x^2])/(2*a^8)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{x^5 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \frac {1}{x^3 \left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \left (\frac {1}{a^6 b^6 x^3}-\frac {6}{a^7 b^5 x^2}+\frac {21}{a^8 b^4 x}-\frac {1}{a^3 b^3 (a+b x)^6}-\frac {3}{a^4 b^3 (a+b x)^5}-\frac {6}{a^5 b^3 (a+b x)^4}-\frac {10}{a^6 b^3 (a+b x)^3}-\frac {15}{a^7 b^3 (a+b x)^2}-\frac {21}{a^8 b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^6 x^4}+\frac {3 b}{a^7 x^2}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {21 b^2 \log (x)}{a^8}-\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 107, normalized size = 0.76 \begin {gather*} \frac {\frac {a \left (-10 a^6+70 a^5 b x^2+959 a^4 b^2 x^4+2695 a^3 b^3 x^6+3290 a^2 b^4 x^8+1890 a b^5 x^{10}+420 b^6 x^{12}\right )}{x^4 \left (a+b x^2\right )^5}+840 b^2 \log (x)-420 b^2 \log \left (a+b x^2\right )}{40 a^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

((a*(-10*a^6 + 70*a^5*b*x^2 + 959*a^4*b^2*x^4 + 2695*a^3*b^3*x^6 + 3290*a^2*b^4*x^8 + 1890*a*b^5*x^10 + 420*b^
6*x^12))/(x^4*(a + b*x^2)^5) + 840*b^2*Log[x] - 420*b^2*Log[a + b*x^2])/(40*a^8)

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Maple [A]
time = 0.07, size = 134, normalized size = 0.96

method	result	size
norman	\(\frac {-\frac {1}{4 a}+\frac {7 b \,x^{2}}{4 a^{2}}-\frac {105 b^{3} x^{6}}{2 a^{4}}-\frac {315 b^{4} x^{8}}{2 a^{5}}-\frac {385 b^{5} x^{10}}{2 a^{6}}-\frac {875 b^{6} x^{12}}{8 a^{7}}-\frac {959 b^{7} x^{14}}{40 a^{8}}}{x^{4} \left (b \,x^{2}+a \right )^{5}}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}-\frac {21 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{8}}\)	\(111\)
risch	\(\frac {\frac {21 b^{6} x^{12}}{2 a^{7}}+\frac {189 b^{5} x^{10}}{4 a^{6}}+\frac {329 b^{4} x^{8}}{4 a^{5}}+\frac {539 b^{3} x^{6}}{8 a^{4}}+\frac {959 b^{2} x^{4}}{40 a^{3}}+\frac {7 b \,x^{2}}{4 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}-\frac {21 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{8}}\)	\(131\)
default	\(-\frac {b^{3} \left (-\frac {5 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {3 a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}-\frac {15 a}{b \left (b \,x^{2}+a \right )}-\frac {2 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}+\frac {21 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{8}}-\frac {1}{4 a^{6} x^{4}}+\frac {3 b}{a^{7} x^{2}}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^3/a^8*(-5*a^2/b/(b*x^2+a)^2-3/4*a^4/b/(b*x^2+a)^4-15*a/b/(b*x^2+a)-2*a^3/b/(b*x^2+a)^3-1/5*a^5/b/(b*x^2
+a)^5+21/b*ln(b*x^2+a))-1/4/a^6/x^4+3*b/a^7/x^2+21*b^2*ln(x)/a^8

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Maxima [A]
time = 0.29, size = 158, normalized size = 1.13 \begin {gather*} \frac {420 \, b^{6} x^{12} + 1890 \, a b^{5} x^{10} + 3290 \, a^{2} b^{4} x^{8} + 2695 \, a^{3} b^{3} x^{6} + 959 \, a^{4} b^{2} x^{4} + 70 \, a^{5} b x^{2} - 10 \, a^{6}}{40 \, {\left (a^{7} b^{5} x^{14} + 5 \, a^{8} b^{4} x^{12} + 10 \, a^{9} b^{3} x^{10} + 10 \, a^{10} b^{2} x^{8} + 5 \, a^{11} b x^{6} + a^{12} x^{4}\right )}} - \frac {21 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{8}} + \frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/40*(420*b^6*x^12 + 1890*a*b^5*x^10 + 3290*a^2*b^4*x^8 + 2695*a^3*b^3*x^6 + 959*a^4*b^2*x^4 + 70*a^5*b*x^2 -
10*a^6)/(a^7*b^5*x^14 + 5*a^8*b^4*x^12 + 10*a^9*b^3*x^10 + 10*a^10*b^2*x^8 + 5*a^11*b*x^6 + a^12*x^4) - 21/2*b
^2*log(b*x^2 + a)/a^8 + 21/2*b^2*log(x^2)/a^8

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (128) = 256\).
time = 0.38, size = 266, normalized size = 1.90 \begin {gather*} \frac {420 \, a b^{6} x^{12} + 1890 \, a^{2} b^{5} x^{10} + 3290 \, a^{3} b^{4} x^{8} + 2695 \, a^{4} b^{3} x^{6} + 959 \, a^{5} b^{2} x^{4} + 70 \, a^{6} b x^{2} - 10 \, a^{7} - 420 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 840 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (x\right )}{40 \, {\left (a^{8} b^{5} x^{14} + 5 \, a^{9} b^{4} x^{12} + 10 \, a^{10} b^{3} x^{10} + 10 \, a^{11} b^{2} x^{8} + 5 \, a^{12} b x^{6} + a^{13} x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/40*(420*a*b^6*x^12 + 1890*a^2*b^5*x^10 + 3290*a^3*b^4*x^8 + 2695*a^4*b^3*x^6 + 959*a^5*b^2*x^4 + 70*a^6*b*x^
2 - 10*a^7 - 420*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*lo
g(b*x^2 + a) + 840*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*
log(x))/(a^8*b^5*x^14 + 5*a^9*b^4*x^12 + 10*a^10*b^3*x^10 + 10*a^11*b^2*x^8 + 5*a^12*b*x^6 + a^13*x^4)

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Sympy [A]
time = 0.47, size = 165, normalized size = 1.18 \begin {gather*} \frac {- 10 a^{6} + 70 a^{5} b x^{2} + 959 a^{4} b^{2} x^{4} + 2695 a^{3} b^{3} x^{6} + 3290 a^{2} b^{4} x^{8} + 1890 a b^{5} x^{10} + 420 b^{6} x^{12}}{40 a^{12} x^{4} + 200 a^{11} b x^{6} + 400 a^{10} b^{2} x^{8} + 400 a^{9} b^{3} x^{10} + 200 a^{8} b^{4} x^{12} + 40 a^{7} b^{5} x^{14}} + \frac {21 b^{2} \log {\left (x \right )}}{a^{8}} - \frac {21 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

(-10*a**6 + 70*a**5*b*x**2 + 959*a**4*b**2*x**4 + 2695*a**3*b**3*x**6 + 3290*a**2*b**4*x**8 + 1890*a*b**5*x**1
0 + 420*b**6*x**12)/(40*a**12*x**4 + 200*a**11*b*x**6 + 400*a**10*b**2*x**8 + 400*a**9*b**3*x**10 + 200*a**8*b
**4*x**12 + 40*a**7*b**5*x**14) + 21*b**2*log(x)/a**8 - 21*b**2*log(a/b + x**2)/(2*a**8)

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Giac [A]
time = 3.31, size = 130, normalized size = 0.93 \begin {gather*} \frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} - \frac {21 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{8}} - \frac {63 \, b^{2} x^{4} - 12 \, a b x^{2} + a^{2}}{4 \, a^{8} x^{4}} + \frac {959 \, b^{7} x^{10} + 5095 \, a b^{6} x^{8} + 10890 \, a^{2} b^{5} x^{6} + 11730 \, a^{3} b^{4} x^{4} + 6390 \, a^{4} b^{3} x^{2} + 1418 \, a^{5} b^{2}}{40 \, {\left (b x^{2} + a\right )}^{5} a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

21/2*b^2*log(x^2)/a^8 - 21/2*b^2*log(abs(b*x^2 + a))/a^8 - 1/4*(63*b^2*x^4 - 12*a*b*x^2 + a^2)/(a^8*x^4) + 1/4
0*(959*b^7*x^10 + 5095*a*b^6*x^8 + 10890*a^2*b^5*x^6 + 11730*a^3*b^4*x^4 + 6390*a^4*b^3*x^2 + 1418*a^5*b^2)/((
b*x^2 + a)^5*a^8)

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Mupad [B]
time = 4.91, size = 155, normalized size = 1.11 \begin {gather*} \frac {\frac {7\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {959\,b^2\,x^4}{40\,a^3}+\frac {539\,b^3\,x^6}{8\,a^4}+\frac {329\,b^4\,x^8}{4\,a^5}+\frac {189\,b^5\,x^{10}}{4\,a^6}+\frac {21\,b^6\,x^{12}}{2\,a^7}}{a^5\,x^4+5\,a^4\,b\,x^6+10\,a^3\,b^2\,x^8+10\,a^2\,b^3\,x^{10}+5\,a\,b^4\,x^{12}+b^5\,x^{14}}-\frac {21\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^8}+\frac {21\,b^2\,\ln \left (x\right )}{a^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)

[Out]

((7*b*x^2)/(4*a^2) - 1/(4*a) + (959*b^2*x^4)/(40*a^3) + (539*b^3*x^6)/(8*a^4) + (329*b^4*x^8)/(4*a^5) + (189*b
^5*x^10)/(4*a^6) + (21*b^6*x^12)/(2*a^7))/(a^5*x^4 + b^5*x^14 + 5*a^4*b*x^6 + 5*a*b^4*x^12 + 10*a^3*b^2*x^8 +
10*a^2*b^3*x^10) - (21*b^2*log(a + b*x^2))/(2*a^8) + (21*b^2*log(x))/a^8

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