3.206 \(\int \frac {1}{x^3 (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=184 \[ \frac {5 b \log \left (a+b x^2\right )}{a^{11}}-\frac {10 b \log (x)}{a^{11}}-\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {1}{2 a^{10} x^2}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {b}{18 a^2 \left (a+b x^2\right )^9} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 184, 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} \[ -\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {b}{18 a^2 \left (a+b x^2\right )^9}+\frac {5 b \log \left (a+b x^2\right )}{a^{11}}-\frac {10 b \log (x)}{a^{11}}-\frac {1}{2 a^{10} x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2)^10),x]

[Out]

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

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

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Mathematica [A]  time = 0.13, size = 136, normalized size = 0.74 \[ -\frac {\frac {a \left (252 a^9+7129 a^8 b x^2+41481 a^7 b^2 x^4+120564 a^6 b^3 x^6+210756 a^5 b^4 x^8+236754 a^4 b^5 x^{10}+173250 a^3 b^6 x^{12}+80220 a^2 b^7 x^{14}+21420 a b^8 x^{16}+2520 b^9 x^{18}\right )}{x^2 \left (a+b x^2\right )^9}-2520 b \log \left (a+b x^2\right )+5040 b \log (x)}{504 a^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^2)^10),x]

[Out]

-1/504*((a*(252*a^9 + 7129*a^8*b*x^2 + 41481*a^7*b^2*x^4 + 120564*a^6*b^3*x^6 + 210756*a^5*b^4*x^8 + 236754*a^
4*b^5*x^10 + 173250*a^3*b^6*x^12 + 80220*a^2*b^7*x^14 + 21420*a*b^8*x^16 + 2520*b^9*x^18))/(x^2*(a + b*x^2)^9)
+ 5040*b*Log[x] - 2520*b*Log[a + b*x^2])/a^11

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fricas [B]  time = 0.95, size = 427, normalized size = 2.32 \[ -\frac {2520 \, a b^{9} x^{18} + 21420 \, a^{2} b^{8} x^{16} + 80220 \, a^{3} b^{7} x^{14} + 173250 \, a^{4} b^{6} x^{12} + 236754 \, a^{5} b^{5} x^{10} + 210756 \, a^{6} b^{4} x^{8} + 120564 \, a^{7} b^{3} x^{6} + 41481 \, a^{8} b^{2} x^{4} + 7129 \, a^{9} b x^{2} + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 5040 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \relax (x)}{504 \, {\left (a^{11} b^{9} x^{20} + 9 \, a^{12} b^{8} x^{18} + 36 \, a^{13} b^{7} x^{16} + 84 \, a^{14} b^{6} x^{14} + 126 \, a^{15} b^{5} x^{12} + 126 \, a^{16} b^{4} x^{10} + 84 \, a^{17} b^{3} x^{8} + 36 \, a^{18} b^{2} x^{6} + 9 \, a^{19} b x^{4} + a^{20} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/504*(2520*a*b^9*x^18 + 21420*a^2*b^8*x^16 + 80220*a^3*b^7*x^14 + 173250*a^4*b^6*x^12 + 236754*a^5*b^5*x^10
+ 210756*a^6*b^4*x^8 + 120564*a^7*b^3*x^6 + 41481*a^8*b^2*x^4 + 7129*a^9*b*x^2 + 252*a^10 - 2520*(b^10*x^20 +
9*a*b^9*x^18 + 36*a^2*b^8*x^16 + 84*a^3*b^7*x^14 + 126*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 84*a^6*b^4*x^8 + 36*a
^7*b^3*x^6 + 9*a^8*b^2*x^4 + a^9*b*x^2)*log(b*x^2 + a) + 5040*(b^10*x^20 + 9*a*b^9*x^18 + 36*a^2*b^8*x^16 + 84
*a^3*b^7*x^14 + 126*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 84*a^6*b^4*x^8 + 36*a^7*b^3*x^6 + 9*a^8*b^2*x^4 + a^9*b*
x^2)*log(x))/(a^11*b^9*x^20 + 9*a^12*b^8*x^18 + 36*a^13*b^7*x^16 + 84*a^14*b^6*x^14 + 126*a^15*b^5*x^12 + 126*
a^16*b^4*x^10 + 84*a^17*b^3*x^8 + 36*a^18*b^2*x^6 + 9*a^19*b*x^4 + a^20*x^2)

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giac [A]  time = 0.59, size = 159, normalized size = 0.86 \[ -\frac {5 \, b \log \left (x^{2}\right )}{a^{11}} + \frac {5 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{11}} + \frac {10 \, b x^{2} - a}{2 \, a^{11} x^{2}} - \frac {7129 \, b^{10} x^{18} + 66429 \, a b^{9} x^{16} + 275796 \, a^{2} b^{8} x^{14} + 669984 \, a^{3} b^{7} x^{12} + 1050336 \, a^{4} b^{6} x^{10} + 1103256 \, a^{5} b^{5} x^{8} + 777840 \, a^{6} b^{4} x^{6} + 356040 \, a^{7} b^{3} x^{4} + 96570 \, a^{8} b^{2} x^{2} + 11990 \, a^{9} b}{504 \, {\left (b x^{2} + a\right )}^{9} a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-5*b*log(x^2)/a^11 + 5*b*log(abs(b*x^2 + a))/a^11 + 1/2*(10*b*x^2 - a)/(a^11*x^2) - 1/504*(7129*b^10*x^18 + 66
429*a*b^9*x^16 + 275796*a^2*b^8*x^14 + 669984*a^3*b^7*x^12 + 1050336*a^4*b^6*x^10 + 1103256*a^5*b^5*x^8 + 7778
40*a^6*b^4*x^6 + 356040*a^7*b^3*x^4 + 96570*a^8*b^2*x^2 + 11990*a^9*b)/((b*x^2 + a)^9*a^11)

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maple [A]  time = 0.02, size = 167, normalized size = 0.91 \[ -\frac {b}{18 \left (b \,x^{2}+a \right )^{9} a^{2}}-\frac {b}{8 \left (b \,x^{2}+a \right )^{8} a^{3}}-\frac {3 b}{14 \left (b \,x^{2}+a \right )^{7} a^{4}}-\frac {b}{3 \left (b \,x^{2}+a \right )^{6} a^{5}}-\frac {b}{2 \left (b \,x^{2}+a \right )^{5} a^{6}}-\frac {3 b}{4 \left (b \,x^{2}+a \right )^{4} a^{7}}-\frac {7 b}{6 \left (b \,x^{2}+a \right )^{3} a^{8}}-\frac {2 b}{\left (b \,x^{2}+a \right )^{2} a^{9}}-\frac {9 b}{2 \left (b \,x^{2}+a \right ) a^{10}}-\frac {10 b \ln \relax (x )}{a^{11}}+\frac {5 b \ln \left (b \,x^{2}+a \right )}{a^{11}}-\frac {1}{2 a^{10} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^2+a)^10,x)

[Out]

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

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maxima [A]  time = 1.67, size = 231, normalized size = 1.26 \[ -\frac {2520 \, b^{9} x^{18} + 21420 \, a b^{8} x^{16} + 80220 \, a^{2} b^{7} x^{14} + 173250 \, a^{3} b^{6} x^{12} + 236754 \, a^{4} b^{5} x^{10} + 210756 \, a^{5} b^{4} x^{8} + 120564 \, a^{6} b^{3} x^{6} + 41481 \, a^{7} b^{2} x^{4} + 7129 \, a^{8} b x^{2} + 252 \, a^{9}}{504 \, {\left (a^{10} b^{9} x^{20} + 9 \, a^{11} b^{8} x^{18} + 36 \, a^{12} b^{7} x^{16} + 84 \, a^{13} b^{6} x^{14} + 126 \, a^{14} b^{5} x^{12} + 126 \, a^{15} b^{4} x^{10} + 84 \, a^{16} b^{3} x^{8} + 36 \, a^{17} b^{2} x^{6} + 9 \, a^{18} b x^{4} + a^{19} x^{2}\right )}} + \frac {5 \, b \log \left (b x^{2} + a\right )}{a^{11}} - \frac {5 \, b \log \left (x^{2}\right )}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/504*(2520*b^9*x^18 + 21420*a*b^8*x^16 + 80220*a^2*b^7*x^14 + 173250*a^3*b^6*x^12 + 236754*a^4*b^5*x^10 + 21
0756*a^5*b^4*x^8 + 120564*a^6*b^3*x^6 + 41481*a^7*b^2*x^4 + 7129*a^8*b*x^2 + 252*a^9)/(a^10*b^9*x^20 + 9*a^11*
b^8*x^18 + 36*a^12*b^7*x^16 + 84*a^13*b^6*x^14 + 126*a^14*b^5*x^12 + 126*a^15*b^4*x^10 + 84*a^16*b^3*x^8 + 36*
a^17*b^2*x^6 + 9*a^18*b*x^4 + a^19*x^2) + 5*b*log(b*x^2 + a)/a^11 - 5*b*log(x^2)/a^11

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mupad [B]  time = 0.52, size = 229, normalized size = 1.24 \[ \frac {5\,b\,\ln \left (b\,x^2+a\right )}{a^{11}}-\frac {\frac {1}{2\,a}+\frac {7129\,b\,x^2}{504\,a^2}+\frac {4609\,b^2\,x^4}{56\,a^3}+\frac {3349\,b^3\,x^6}{14\,a^4}+\frac {2509\,b^4\,x^8}{6\,a^5}+\frac {1879\,b^5\,x^{10}}{4\,a^6}+\frac {1375\,b^6\,x^{12}}{4\,a^7}+\frac {955\,b^7\,x^{14}}{6\,a^8}+\frac {85\,b^8\,x^{16}}{2\,a^9}+\frac {5\,b^9\,x^{18}}{a^{10}}}{a^9\,x^2+9\,a^8\,b\,x^4+36\,a^7\,b^2\,x^6+84\,a^6\,b^3\,x^8+126\,a^5\,b^4\,x^{10}+126\,a^4\,b^5\,x^{12}+84\,a^3\,b^6\,x^{14}+36\,a^2\,b^7\,x^{16}+9\,a\,b^8\,x^{18}+b^9\,x^{20}}-\frac {10\,b\,\ln \relax (x)}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^2)^10),x)

[Out]

(5*b*log(a + b*x^2))/a^11 - (1/(2*a) + (7129*b*x^2)/(504*a^2) + (4609*b^2*x^4)/(56*a^3) + (3349*b^3*x^6)/(14*a
^4) + (2509*b^4*x^8)/(6*a^5) + (1879*b^5*x^10)/(4*a^6) + (1375*b^6*x^12)/(4*a^7) + (955*b^7*x^14)/(6*a^8) + (8
5*b^8*x^16)/(2*a^9) + (5*b^9*x^18)/a^10)/(a^9*x^2 + b^9*x^20 + 9*a^8*b*x^4 + 9*a*b^8*x^18 + 36*a^7*b^2*x^6 + 8
4*a^6*b^3*x^8 + 126*a^5*b^4*x^10 + 126*a^4*b^5*x^12 + 84*a^3*b^6*x^14 + 36*a^2*b^7*x^16) - (10*b*log(x))/a^11

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sympy [A]  time = 1.46, size = 245, normalized size = 1.33 \[ \frac {- 252 a^{9} - 7129 a^{8} b x^{2} - 41481 a^{7} b^{2} x^{4} - 120564 a^{6} b^{3} x^{6} - 210756 a^{5} b^{4} x^{8} - 236754 a^{4} b^{5} x^{10} - 173250 a^{3} b^{6} x^{12} - 80220 a^{2} b^{7} x^{14} - 21420 a b^{8} x^{16} - 2520 b^{9} x^{18}}{504 a^{19} x^{2} + 4536 a^{18} b x^{4} + 18144 a^{17} b^{2} x^{6} + 42336 a^{16} b^{3} x^{8} + 63504 a^{15} b^{4} x^{10} + 63504 a^{14} b^{5} x^{12} + 42336 a^{13} b^{6} x^{14} + 18144 a^{12} b^{7} x^{16} + 4536 a^{11} b^{8} x^{18} + 504 a^{10} b^{9} x^{20}} - \frac {10 b \log {\relax (x )}}{a^{11}} + \frac {5 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**2+a)**10,x)

[Out]

(-252*a**9 - 7129*a**8*b*x**2 - 41481*a**7*b**2*x**4 - 120564*a**6*b**3*x**6 - 210756*a**5*b**4*x**8 - 236754*
a**4*b**5*x**10 - 173250*a**3*b**6*x**12 - 80220*a**2*b**7*x**14 - 21420*a*b**8*x**16 - 2520*b**9*x**18)/(504*
a**19*x**2 + 4536*a**18*b*x**4 + 18144*a**17*b**2*x**6 + 42336*a**16*b**3*x**8 + 63504*a**15*b**4*x**10 + 6350
4*a**14*b**5*x**12 + 42336*a**13*b**6*x**14 + 18144*a**12*b**7*x**16 + 4536*a**11*b**8*x**18 + 504*a**10*b**9*
x**20) - 10*b*log(x)/a**11 + 5*b*log(a/b + x**2)/a**11

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