3.207 \(\int \frac {1}{x^5 (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=217 \[ -\frac {55 b^2 \log \left (a+b x^2\right )}{2 a^{12}}+\frac {55 b^2 \log (x)}{a^{12}}+\frac {45 b^2}{2 a^{11} \left (a+b x^2\right )}+\frac {5 b}{a^{11} x^2}+\frac {9 b^2}{a^{10} \left (a+b x^2\right )^2}-\frac {1}{4 a^{10} x^4}+\frac {14 b^2}{3 a^9 \left (a+b x^2\right )^3}+\frac {21 b^2}{8 a^8 \left (a+b x^2\right )^4}+\frac {3 b^2}{2 a^7 \left (a+b x^2\right )^5}+\frac {5 b^2}{6 a^6 \left (a+b x^2\right )^6}+\frac {3 b^2}{7 a^5 \left (a+b x^2\right )^7}+\frac {3 b^2}{16 a^4 \left (a+b x^2\right )^8}+\frac {b^2}{18 a^3 \left (a+b x^2\right )^9} \]

[Out]

-1/4/a^10/x^4+5*b/a^11/x^2+1/18*b^2/a^3/(b*x^2+a)^9+3/16*b^2/a^4/(b*x^2+a)^8+3/7*b^2/a^5/(b*x^2+a)^7+5/6*b^2/a
^6/(b*x^2+a)^6+3/2*b^2/a^7/(b*x^2+a)^5+21/8*b^2/a^8/(b*x^2+a)^4+14/3*b^2/a^9/(b*x^2+a)^3+9*b^2/a^10/(b*x^2+a)^
2+45/2*b^2/a^11/(b*x^2+a)+55*b^2*ln(x)/a^12-55/2*b^2*ln(b*x^2+a)/a^12

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Rubi [A]  time = 0.22, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac {45 b^2}{2 a^{11} \left (a+b x^2\right )}+\frac {9 b^2}{a^{10} \left (a+b x^2\right )^2}+\frac {14 b^2}{3 a^9 \left (a+b x^2\right )^3}+\frac {21 b^2}{8 a^8 \left (a+b x^2\right )^4}+\frac {3 b^2}{2 a^7 \left (a+b x^2\right )^5}+\frac {5 b^2}{6 a^6 \left (a+b x^2\right )^6}+\frac {3 b^2}{7 a^5 \left (a+b x^2\right )^7}+\frac {3 b^2}{16 a^4 \left (a+b x^2\right )^8}+\frac {b^2}{18 a^3 \left (a+b x^2\right )^9}-\frac {55 b^2 \log \left (a+b x^2\right )}{2 a^{12}}+\frac {55 b^2 \log (x)}{a^{12}}+\frac {5 b}{a^{11} x^2}-\frac {1}{4 a^{10} x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^2)^10),x]

[Out]

-1/(4*a^10*x^4) + (5*b)/(a^11*x^2) + b^2/(18*a^3*(a + b*x^2)^9) + (3*b^2)/(16*a^4*(a + b*x^2)^8) + (3*b^2)/(7*
a^5*(a + b*x^2)^7) + (5*b^2)/(6*a^6*(a + b*x^2)^6) + (3*b^2)/(2*a^7*(a + b*x^2)^5) + (21*b^2)/(8*a^8*(a + b*x^
2)^4) + (14*b^2)/(3*a^9*(a + b*x^2)^3) + (9*b^2)/(a^10*(a + b*x^2)^2) + (45*b^2)/(2*a^11*(a + b*x^2)) + (55*b^
2*Log[x])/a^12 - (55*b^2*Log[a + b*x^2])/(2*a^12)

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 266

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+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^{10} x^3}-\frac {10 b}{a^{11} x^2}+\frac {55 b^2}{a^{12} x}-\frac {b^3}{a^3 (a+b x)^{10}}-\frac {3 b^3}{a^4 (a+b x)^9}-\frac {6 b^3}{a^5 (a+b x)^8}-\frac {10 b^3}{a^6 (a+b x)^7}-\frac {15 b^3}{a^7 (a+b x)^6}-\frac {21 b^3}{a^8 (a+b x)^5}-\frac {28 b^3}{a^9 (a+b x)^4}-\frac {36 b^3}{a^{10} (a+b x)^3}-\frac {45 b^3}{a^{11} (a+b x)^2}-\frac {55 b^3}{a^{12} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^{10} x^4}+\frac {5 b}{a^{11} x^2}+\frac {b^2}{18 a^3 \left (a+b x^2\right )^9}+\frac {3 b^2}{16 a^4 \left (a+b x^2\right )^8}+\frac {3 b^2}{7 a^5 \left (a+b x^2\right )^7}+\frac {5 b^2}{6 a^6 \left (a+b x^2\right )^6}+\frac {3 b^2}{2 a^7 \left (a+b x^2\right )^5}+\frac {21 b^2}{8 a^8 \left (a+b x^2\right )^4}+\frac {14 b^2}{3 a^9 \left (a+b x^2\right )^3}+\frac {9 b^2}{a^{10} \left (a+b x^2\right )^2}+\frac {45 b^2}{2 a^{11} \left (a+b x^2\right )}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log \left (a+b x^2\right )}{2 a^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 151, normalized size = 0.70 \[ \frac {\frac {a \left (-252 a^{10}+2772 a^9 b x^2+78419 a^8 b^2 x^4+456291 a^7 b^3 x^6+1326204 a^6 b^4 x^8+2318316 a^5 b^5 x^{10}+2604294 a^4 b^6 x^{12}+1905750 a^3 b^7 x^{14}+882420 a^2 b^8 x^{16}+235620 a b^9 x^{18}+27720 b^{10} x^{20}\right )}{x^4 \left (a+b x^2\right )^9}-27720 b^2 \log \left (a+b x^2\right )+55440 b^2 \log (x)}{1008 a^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2)^10),x]

[Out]

((a*(-252*a^10 + 2772*a^9*b*x^2 + 78419*a^8*b^2*x^4 + 456291*a^7*b^3*x^6 + 1326204*a^6*b^4*x^8 + 2318316*a^5*b
^5*x^10 + 2604294*a^4*b^6*x^12 + 1905750*a^3*b^7*x^14 + 882420*a^2*b^8*x^16 + 235620*a*b^9*x^18 + 27720*b^10*x
^20))/(x^4*(a + b*x^2)^9) + 55440*b^2*Log[x] - 27720*b^2*Log[a + b*x^2])/(1008*a^12)

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fricas [B]  time = 0.80, size = 442, normalized size = 2.04 \[ \frac {27720 \, a b^{10} x^{20} + 235620 \, a^{2} b^{9} x^{18} + 882420 \, a^{3} b^{8} x^{16} + 1905750 \, a^{4} b^{7} x^{14} + 2604294 \, a^{5} b^{6} x^{12} + 2318316 \, a^{6} b^{5} x^{10} + 1326204 \, a^{7} b^{4} x^{8} + 456291 \, a^{8} b^{3} x^{6} + 78419 \, a^{9} b^{2} x^{4} + 2772 \, a^{10} b x^{2} - 252 \, a^{11} - 27720 \, {\left (b^{11} x^{22} + 9 \, a b^{10} x^{20} + 36 \, a^{2} b^{9} x^{18} + 84 \, a^{3} b^{8} x^{16} + 126 \, a^{4} b^{7} x^{14} + 126 \, a^{5} b^{6} x^{12} + 84 \, a^{6} b^{5} x^{10} + 36 \, a^{7} b^{4} x^{8} + 9 \, a^{8} b^{3} x^{6} + a^{9} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 55440 \, {\left (b^{11} x^{22} + 9 \, a b^{10} x^{20} + 36 \, a^{2} b^{9} x^{18} + 84 \, a^{3} b^{8} x^{16} + 126 \, a^{4} b^{7} x^{14} + 126 \, a^{5} b^{6} x^{12} + 84 \, a^{6} b^{5} x^{10} + 36 \, a^{7} b^{4} x^{8} + 9 \, a^{8} b^{3} x^{6} + a^{9} b^{2} x^{4}\right )} \log \relax (x)}{1008 \, {\left (a^{12} b^{9} x^{22} + 9 \, a^{13} b^{8} x^{20} + 36 \, a^{14} b^{7} x^{18} + 84 \, a^{15} b^{6} x^{16} + 126 \, a^{16} b^{5} x^{14} + 126 \, a^{17} b^{4} x^{12} + 84 \, a^{18} b^{3} x^{10} + 36 \, a^{19} b^{2} x^{8} + 9 \, a^{20} b x^{6} + a^{21} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/1008*(27720*a*b^10*x^20 + 235620*a^2*b^9*x^18 + 882420*a^3*b^8*x^16 + 1905750*a^4*b^7*x^14 + 2604294*a^5*b^6
*x^12 + 2318316*a^6*b^5*x^10 + 1326204*a^7*b^4*x^8 + 456291*a^8*b^3*x^6 + 78419*a^9*b^2*x^4 + 2772*a^10*b*x^2
- 252*a^11 - 27720*(b^11*x^22 + 9*a*b^10*x^20 + 36*a^2*b^9*x^18 + 84*a^3*b^8*x^16 + 126*a^4*b^7*x^14 + 126*a^5
*b^6*x^12 + 84*a^6*b^5*x^10 + 36*a^7*b^4*x^8 + 9*a^8*b^3*x^6 + a^9*b^2*x^4)*log(b*x^2 + a) + 55440*(b^11*x^22
+ 9*a*b^10*x^20 + 36*a^2*b^9*x^18 + 84*a^3*b^8*x^16 + 126*a^4*b^7*x^14 + 126*a^5*b^6*x^12 + 84*a^6*b^5*x^10 +
36*a^7*b^4*x^8 + 9*a^8*b^3*x^6 + a^9*b^2*x^4)*log(x))/(a^12*b^9*x^22 + 9*a^13*b^8*x^20 + 36*a^14*b^7*x^18 + 84
*a^15*b^6*x^16 + 126*a^16*b^5*x^14 + 126*a^17*b^4*x^12 + 84*a^18*b^3*x^10 + 36*a^19*b^2*x^8 + 9*a^20*b*x^6 + a
^21*x^4)

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giac [A]  time = 0.61, size = 174, normalized size = 0.80 \[ \frac {55 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{12}} - \frac {55 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{12}} - \frac {165 \, b^{2} x^{4} - 20 \, a b x^{2} + a^{2}}{4 \, a^{12} x^{4}} + \frac {78419 \, b^{11} x^{18} + 728451 \, a b^{10} x^{16} + 3013596 \, a^{2} b^{9} x^{14} + 7290444 \, a^{3} b^{8} x^{12} + 11372256 \, a^{4} b^{7} x^{10} + 11871216 \, a^{5} b^{6} x^{8} + 8302224 \, a^{6} b^{5} x^{6} + 3757680 \, a^{7} b^{4} x^{4} + 1001790 \, a^{8} b^{3} x^{2} + 120550 \, a^{9} b^{2}}{1008 \, {\left (b x^{2} + a\right )}^{9} a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^10,x, algorithm="giac")

[Out]

55/2*b^2*log(x^2)/a^12 - 55/2*b^2*log(abs(b*x^2 + a))/a^12 - 1/4*(165*b^2*x^4 - 20*a*b*x^2 + a^2)/(a^12*x^4) +
1/1008*(78419*b^11*x^18 + 728451*a*b^10*x^16 + 3013596*a^2*b^9*x^14 + 7290444*a^3*b^8*x^12 + 11372256*a^4*b^7
*x^10 + 11871216*a^5*b^6*x^8 + 8302224*a^6*b^5*x^6 + 3757680*a^7*b^4*x^4 + 1001790*a^8*b^3*x^2 + 120550*a^9*b^
2)/((b*x^2 + a)^9*a^12)

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maple [A]  time = 0.02, size = 198, normalized size = 0.91 \[ \frac {b^{2}}{18 \left (b \,x^{2}+a \right )^{9} a^{3}}+\frac {3 b^{2}}{16 \left (b \,x^{2}+a \right )^{8} a^{4}}+\frac {3 b^{2}}{7 \left (b \,x^{2}+a \right )^{7} a^{5}}+\frac {5 b^{2}}{6 \left (b \,x^{2}+a \right )^{6} a^{6}}+\frac {3 b^{2}}{2 \left (b \,x^{2}+a \right )^{5} a^{7}}+\frac {21 b^{2}}{8 \left (b \,x^{2}+a \right )^{4} a^{8}}+\frac {14 b^{2}}{3 \left (b \,x^{2}+a \right )^{3} a^{9}}+\frac {9 b^{2}}{\left (b \,x^{2}+a \right )^{2} a^{10}}+\frac {45 b^{2}}{2 \left (b \,x^{2}+a \right ) a^{11}}+\frac {55 b^{2} \ln \relax (x )}{a^{12}}-\frac {55 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{12}}+\frac {5 b}{a^{11} x^{2}}-\frac {1}{4 a^{10} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^2+a)^10,x)

[Out]

-1/4/a^10/x^4+5*b/a^11/x^2+1/18*b^2/a^3/(b*x^2+a)^9+3/16*b^2/a^4/(b*x^2+a)^8+3/7*b^2/a^5/(b*x^2+a)^7+5/6*b^2/a
^6/(b*x^2+a)^6+3/2*b^2/a^7/(b*x^2+a)^5+21/8*b^2/a^8/(b*x^2+a)^4+14/3*b^2/a^9/(b*x^2+a)^3+9*b^2/a^10/(b*x^2+a)^
2+45/2*b^2/a^11/(b*x^2+a)+55*b^2*ln(x)/a^12-55/2*b^2*ln(b*x^2+a)/a^12

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maxima [A]  time = 1.71, size = 246, normalized size = 1.13 \[ \frac {27720 \, b^{10} x^{20} + 235620 \, a b^{9} x^{18} + 882420 \, a^{2} b^{8} x^{16} + 1905750 \, a^{3} b^{7} x^{14} + 2604294 \, a^{4} b^{6} x^{12} + 2318316 \, a^{5} b^{5} x^{10} + 1326204 \, a^{6} b^{4} x^{8} + 456291 \, a^{7} b^{3} x^{6} + 78419 \, a^{8} b^{2} x^{4} + 2772 \, a^{9} b x^{2} - 252 \, a^{10}}{1008 \, {\left (a^{11} b^{9} x^{22} + 9 \, a^{12} b^{8} x^{20} + 36 \, a^{13} b^{7} x^{18} + 84 \, a^{14} b^{6} x^{16} + 126 \, a^{15} b^{5} x^{14} + 126 \, a^{16} b^{4} x^{12} + 84 \, a^{17} b^{3} x^{10} + 36 \, a^{18} b^{2} x^{8} + 9 \, a^{19} b x^{6} + a^{20} x^{4}\right )}} - \frac {55 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{12}} + \frac {55 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/1008*(27720*b^10*x^20 + 235620*a*b^9*x^18 + 882420*a^2*b^8*x^16 + 1905750*a^3*b^7*x^14 + 2604294*a^4*b^6*x^1
2 + 2318316*a^5*b^5*x^10 + 1326204*a^6*b^4*x^8 + 456291*a^7*b^3*x^6 + 78419*a^8*b^2*x^4 + 2772*a^9*b*x^2 - 252
*a^10)/(a^11*b^9*x^22 + 9*a^12*b^8*x^20 + 36*a^13*b^7*x^18 + 84*a^14*b^6*x^16 + 126*a^15*b^5*x^14 + 126*a^16*b
^4*x^12 + 84*a^17*b^3*x^10 + 36*a^18*b^2*x^8 + 9*a^19*b*x^6 + a^20*x^4) - 55/2*b^2*log(b*x^2 + a)/a^12 + 55/2*
b^2*log(x^2)/a^12

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mupad [B]  time = 5.78, size = 243, normalized size = 1.12 \[ \frac {\frac {11\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {78419\,b^2\,x^4}{1008\,a^3}+\frac {50699\,b^3\,x^6}{112\,a^4}+\frac {36839\,b^4\,x^8}{28\,a^5}+\frac {27599\,b^5\,x^{10}}{12\,a^6}+\frac {20669\,b^6\,x^{12}}{8\,a^7}+\frac {15125\,b^7\,x^{14}}{8\,a^8}+\frac {10505\,b^8\,x^{16}}{12\,a^9}+\frac {935\,b^9\,x^{18}}{4\,a^{10}}+\frac {55\,b^{10}\,x^{20}}{2\,a^{11}}}{a^9\,x^4+9\,a^8\,b\,x^6+36\,a^7\,b^2\,x^8+84\,a^6\,b^3\,x^{10}+126\,a^5\,b^4\,x^{12}+126\,a^4\,b^5\,x^{14}+84\,a^3\,b^6\,x^{16}+36\,a^2\,b^7\,x^{18}+9\,a\,b^8\,x^{20}+b^9\,x^{22}}-\frac {55\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^{12}}+\frac {55\,b^2\,\ln \relax (x)}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^2)^10),x)

[Out]

((11*b*x^2)/(4*a^2) - 1/(4*a) + (78419*b^2*x^4)/(1008*a^3) + (50699*b^3*x^6)/(112*a^4) + (36839*b^4*x^8)/(28*a
^5) + (27599*b^5*x^10)/(12*a^6) + (20669*b^6*x^12)/(8*a^7) + (15125*b^7*x^14)/(8*a^8) + (10505*b^8*x^16)/(12*a
^9) + (935*b^9*x^18)/(4*a^10) + (55*b^10*x^20)/(2*a^11))/(a^9*x^4 + b^9*x^22 + 9*a^8*b*x^6 + 9*a*b^8*x^20 + 36
*a^7*b^2*x^8 + 84*a^6*b^3*x^10 + 126*a^5*b^4*x^12 + 126*a^4*b^5*x^14 + 84*a^3*b^6*x^16 + 36*a^2*b^7*x^18) - (5
5*b^2*log(a + b*x^2))/(2*a^12) + (55*b^2*log(x))/a^12

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sympy [A]  time = 1.55, size = 260, normalized size = 1.20 \[ \frac {- 252 a^{10} + 2772 a^{9} b x^{2} + 78419 a^{8} b^{2} x^{4} + 456291 a^{7} b^{3} x^{6} + 1326204 a^{6} b^{4} x^{8} + 2318316 a^{5} b^{5} x^{10} + 2604294 a^{4} b^{6} x^{12} + 1905750 a^{3} b^{7} x^{14} + 882420 a^{2} b^{8} x^{16} + 235620 a b^{9} x^{18} + 27720 b^{10} x^{20}}{1008 a^{20} x^{4} + 9072 a^{19} b x^{6} + 36288 a^{18} b^{2} x^{8} + 84672 a^{17} b^{3} x^{10} + 127008 a^{16} b^{4} x^{12} + 127008 a^{15} b^{5} x^{14} + 84672 a^{14} b^{6} x^{16} + 36288 a^{13} b^{7} x^{18} + 9072 a^{12} b^{8} x^{20} + 1008 a^{11} b^{9} x^{22}} + \frac {55 b^{2} \log {\relax (x )}}{a^{12}} - \frac {55 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**2+a)**10,x)

[Out]

(-252*a**10 + 2772*a**9*b*x**2 + 78419*a**8*b**2*x**4 + 456291*a**7*b**3*x**6 + 1326204*a**6*b**4*x**8 + 23183
16*a**5*b**5*x**10 + 2604294*a**4*b**6*x**12 + 1905750*a**3*b**7*x**14 + 882420*a**2*b**8*x**16 + 235620*a*b**
9*x**18 + 27720*b**10*x**20)/(1008*a**20*x**4 + 9072*a**19*b*x**6 + 36288*a**18*b**2*x**8 + 84672*a**17*b**3*x
**10 + 127008*a**16*b**4*x**12 + 127008*a**15*b**5*x**14 + 84672*a**14*b**6*x**16 + 36288*a**13*b**7*x**18 + 9
072*a**12*b**8*x**20 + 1008*a**11*b**9*x**22) + 55*b**2*log(x)/a**12 - 55*b**2*log(a/b + x**2)/(2*a**12)

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