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

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

[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

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Rubi [A]  time = 0.185709, antiderivative size = 184, 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} \[ -\frac{9 b}{2 a^{10} \left (a+b x^2\right )}-\frac{2 b}{a^9 \left (a+b x^2\right )^2}-\frac{7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac{3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac{b}{2 a^6 \left (a+b x^2\right )^5}-\frac{b}{3 a^5 \left (a+b x^2\right )^6}-\frac{3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac{b}{8 a^3 \left (a+b x^2\right )^8}-\frac{b}{18 a^2 \left (a+b x^2\right )^9}+\frac{5 b \log \left (a+b x^2\right )}{a^{11}}-\frac{10 b \log (x)}{a^{11}}-\frac{1}{2 a^{10} x^2} \]

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 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^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*}

Mathematica [A]  time = 0.120819, size = 136, normalized size = 0.74 \[ -\frac{\frac{a \left (80220 a^2 b^7 x^{14}+173250 a^3 b^6 x^{12}+236754 a^4 b^5 x^{10}+210756 a^5 b^4 x^8+120564 a^6 b^3 x^6+41481 a^7 b^2 x^4+7129 a^8 b x^2+252 a^9+21420 a b^8 x^{16}+2520 b^9 x^{18}\right )}{x^2 \left (a+b x^2\right )^9}-2520 b \log \left (a+b x^2\right )+5040 b \log (x)}{504 a^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^2)^10),x]

[Out]

-((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) + 504
0*b*Log[x] - 2520*b*Log[a + b*x^2])/(504*a^11)

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Maple [A]  time = 0.019, size = 167, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{10}{x}^{2}}}-{\frac{b}{18\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{b}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{3\,b}{14\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{b}{3\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{b}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{3\,b}{4\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{7\,b}{6\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{3}}}-2\,{\frac{b}{{a}^{9} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,b}{2\,{a}^{10} \left ( b{x}^{2}+a \right ) }}-10\,{\frac{b\ln \left ( x \right ) }{{a}^{11}}}+5\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{11}}} \end{align*}

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 = 1.26732, size = 312, normalized size = 1.7 \begin{align*} -\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}} \end{align*}

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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Fricas [B]  time = 1.38937, size = 996, normalized size = 5.41 \begin{align*} -\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 \left (x\right )}{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 )}} \end{align*}

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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Sympy [A]  time = 168.877, size = 243, normalized size = 1.32 \begin{align*} - \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{\left (x \right )}}{a^{11}} + \frac{5 b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{11}} \end{align*}

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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Giac [A]  time = 1.94949, size = 215, normalized size = 1.17 \begin{align*} -\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}} \end{align*}

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)