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

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

[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)

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Rubi [A]  time = 0.220003, antiderivative size = 217, normalized size of antiderivative = 1., 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 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]]

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])

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*}

Mathematica [A]  time = 0.0930556, size = 151, normalized size = 0.7 \[ \frac{\frac{a \left (882420 a^2 b^8 x^{16}+1905750 a^3 b^7 x^{14}+2604294 a^4 b^6 x^{12}+2318316 a^5 b^5 x^{10}+1326204 a^6 b^4 x^8+456291 a^7 b^3 x^6+78419 a^8 b^2 x^4+2772 a^9 b x^2-252 a^{10}+235620 a b^9 x^{18}+27720 b^{10} x^{20}\right )}{x^4 \left (a+b x^2\right )^9}-27720 b^2 \log \left (a+b x^2\right )+55440 b^2 \log (x)}{1008 a^{12}} \]

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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Maple [A]  time = 0.021, size = 198, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{10}{x}^{4}}}+5\,{\frac{b}{{a}^{11}{x}^{2}}}+{\frac{{b}^{2}}{18\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{3\,{b}^{2}}{16\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{3\,{b}^{2}}{7\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{5\,{b}^{2}}{6\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{3\,{b}^{2}}{2\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{21\,{b}^{2}}{8\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{14\,{b}^{2}}{3\,{a}^{9} \left ( b{x}^{2}+a \right ) ^{3}}}+9\,{\frac{{b}^{2}}{{a}^{10} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{45\,{b}^{2}}{2\,{a}^{11} \left ( b{x}^{2}+a \right ) }}+55\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{12}}}-{\frac{55\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{12}}} \end{align*}

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 = 3.03586, size = 332, normalized size = 1.53 \begin{align*} \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}} \end{align*}

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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Fricas [B]  time = 1.32716, size = 1053, normalized size = 4.85 \begin{align*} \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 \left (x\right )}{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 )}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 2.69591, size = 235, normalized size = 1.08 \begin{align*} \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}} \end{align*}

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)