3.208 \(\int \frac{1}{x^7 (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=226 \[ -\frac{165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac{30 b^3}{a^{11} \left (a+b x^2\right )^2}-\frac{14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac{7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac{7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac{5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac{5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac{b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac{b^3}{18 a^4 \left (a+b x^2\right )^9}-\frac{55 b^2}{2 a^{12} x^2}+\frac{110 b^3 \log \left (a+b x^2\right )}{a^{13}}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{5 b}{2 a^{11} x^4}-\frac{1}{6 a^{10} x^6} \]

[Out]

-1/(6*a^10*x^6) + (5*b)/(2*a^11*x^4) - (55*b^2)/(2*a^12*x^2) - b^3/(18*a^4*(a + b*x^2)^9) - b^3/(4*a^5*(a + b*
x^2)^8) - (5*b^3)/(7*a^6*(a + b*x^2)^7) - (5*b^3)/(3*a^7*(a + b*x^2)^6) - (7*b^3)/(2*a^8*(a + b*x^2)^5) - (7*b
^3)/(a^9*(a + b*x^2)^4) - (14*b^3)/(a^10*(a + b*x^2)^3) - (30*b^3)/(a^11*(a + b*x^2)^2) - (165*b^3)/(2*a^12*(a
+ b*x^2)) - (220*b^3*Log[x])/a^13 + (110*b^3*Log[a + b*x^2])/a^13

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Rubi [A]  time = 0.233352, antiderivative size = 226, 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} \[ -\frac{165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac{30 b^3}{a^{11} \left (a+b x^2\right )^2}-\frac{14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac{7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac{7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac{5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac{5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac{b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac{b^3}{18 a^4 \left (a+b x^2\right )^9}-\frac{55 b^2}{2 a^{12} x^2}+\frac{110 b^3 \log \left (a+b x^2\right )}{a^{13}}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{5 b}{2 a^{11} x^4}-\frac{1}{6 a^{10} x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)^10),x]

[Out]

-1/(6*a^10*x^6) + (5*b)/(2*a^11*x^4) - (55*b^2)/(2*a^12*x^2) - b^3/(18*a^4*(a + b*x^2)^9) - b^3/(4*a^5*(a + b*
x^2)^8) - (5*b^3)/(7*a^6*(a + b*x^2)^7) - (5*b^3)/(3*a^7*(a + b*x^2)^6) - (7*b^3)/(2*a^8*(a + b*x^2)^5) - (7*b
^3)/(a^9*(a + b*x^2)^4) - (14*b^3)/(a^10*(a + b*x^2)^3) - (30*b^3)/(a^11*(a + b*x^2)^2) - (165*b^3)/(2*a^12*(a
+ b*x^2)) - (220*b^3*Log[x])/a^13 + (110*b^3*Log[a + b*x^2])/a^13

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^7 \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^{10} x^4}-\frac{10 b}{a^{11} x^3}+\frac{55 b^2}{a^{12} x^2}-\frac{220 b^3}{a^{13} x}+\frac{b^4}{a^4 (a+b x)^{10}}+\frac{4 b^4}{a^5 (a+b x)^9}+\frac{10 b^4}{a^6 (a+b x)^8}+\frac{20 b^4}{a^7 (a+b x)^7}+\frac{35 b^4}{a^8 (a+b x)^6}+\frac{56 b^4}{a^9 (a+b x)^5}+\frac{84 b^4}{a^{10} (a+b x)^4}+\frac{120 b^4}{a^{11} (a+b x)^3}+\frac{165 b^4}{a^{12} (a+b x)^2}+\frac{220 b^4}{a^{13} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 a^{10} x^6}+\frac{5 b}{2 a^{11} x^4}-\frac{55 b^2}{2 a^{12} x^2}-\frac{b^3}{18 a^4 \left (a+b x^2\right )^9}-\frac{b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac{5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac{5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac{7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac{7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac{14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac{30 b^3}{a^{11} \left (a+b x^2\right )^2}-\frac{165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{110 b^3 \log \left (a+b x^2\right )}{a^{13}}\\ \end{align*}

Mathematica [A]  time = 0.114229, size = 162, normalized size = 0.72 \[ -\frac{\frac{a \left (882420 a^2 b^9 x^{18}+1905750 a^3 b^8 x^{16}+2604294 a^4 b^7 x^{14}+2318316 a^5 b^6 x^{12}+1326204 a^6 b^5 x^{10}+456291 a^7 b^4 x^8+78419 a^8 b^3 x^6+2772 a^9 b^2 x^4-252 a^{10} b x^2+42 a^{11}+235620 a b^{10} x^{20}+27720 b^{11} x^{22}\right )}{x^6 \left (a+b x^2\right )^9}-27720 b^3 \log \left (a+b x^2\right )+55440 b^3 \log (x)}{252 a^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^10),x]

[Out]

-((a*(42*a^11 - 252*a^10*b*x^2 + 2772*a^9*b^2*x^4 + 78419*a^8*b^3*x^6 + 456291*a^7*b^4*x^8 + 1326204*a^6*b^5*x
^10 + 2318316*a^5*b^6*x^12 + 2604294*a^4*b^7*x^14 + 1905750*a^3*b^8*x^16 + 882420*a^2*b^9*x^18 + 235620*a*b^10
*x^20 + 27720*b^11*x^22))/(x^6*(a + b*x^2)^9) + 55440*b^3*Log[x] - 27720*b^3*Log[a + b*x^2])/(252*a^13)

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Maple [A]  time = 0.019, size = 209, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{a}^{10}{x}^{6}}}+{\frac{5\,b}{2\,{a}^{11}{x}^{4}}}-{\frac{55\,{b}^{2}}{2\,{a}^{12}{x}^{2}}}-{\frac{{b}^{3}}{18\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{{b}^{3}}{4\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{5\,{b}^{3}}{7\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{5\,{b}^{3}}{3\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{7\,{b}^{3}}{2\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-7\,{\frac{{b}^{3}}{{a}^{9} \left ( b{x}^{2}+a \right ) ^{4}}}-14\,{\frac{{b}^{3}}{{a}^{10} \left ( b{x}^{2}+a \right ) ^{3}}}-30\,{\frac{{b}^{3}}{{a}^{11} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{165\,{b}^{3}}{2\,{a}^{12} \left ( b{x}^{2}+a \right ) }}-220\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{13}}}+110\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{{a}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^2+a)^10,x)

[Out]

-1/6/a^10/x^6+5/2*b/a^11/x^4-55/2*b^2/a^12/x^2-1/18*b^3/a^4/(b*x^2+a)^9-1/4*b^3/a^5/(b*x^2+a)^8-5/7*b^3/a^6/(b
*x^2+a)^7-5/3*b^3/a^7/(b*x^2+a)^6-7/2*b^3/a^8/(b*x^2+a)^5-7*b^3/a^9/(b*x^2+a)^4-14*b^3/a^10/(b*x^2+a)^3-30*b^3
/a^11/(b*x^2+a)^2-165/2*b^3/a^12/(b*x^2+a)-220*b^3*ln(x)/a^13+110*b^3*ln(b*x^2+a)/a^13

________________________________________________________________________________________

Maxima [A]  time = 2.58007, size = 347, normalized size = 1.54 \begin{align*} -\frac{27720 \, b^{11} x^{22} + 235620 \, a b^{10} x^{20} + 882420 \, a^{2} b^{9} x^{18} + 1905750 \, a^{3} b^{8} x^{16} + 2604294 \, a^{4} b^{7} x^{14} + 2318316 \, a^{5} b^{6} x^{12} + 1326204 \, a^{6} b^{5} x^{10} + 456291 \, a^{7} b^{4} x^{8} + 78419 \, a^{8} b^{3} x^{6} + 2772 \, a^{9} b^{2} x^{4} - 252 \, a^{10} b x^{2} + 42 \, a^{11}}{252 \,{\left (a^{12} b^{9} x^{24} + 9 \, a^{13} b^{8} x^{22} + 36 \, a^{14} b^{7} x^{20} + 84 \, a^{15} b^{6} x^{18} + 126 \, a^{16} b^{5} x^{16} + 126 \, a^{17} b^{4} x^{14} + 84 \, a^{18} b^{3} x^{12} + 36 \, a^{19} b^{2} x^{10} + 9 \, a^{20} b x^{8} + a^{21} x^{6}\right )}} + \frac{110 \, b^{3} \log \left (b x^{2} + a\right )}{a^{13}} - \frac{110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/252*(27720*b^11*x^22 + 235620*a*b^10*x^20 + 882420*a^2*b^9*x^18 + 1905750*a^3*b^8*x^16 + 2604294*a^4*b^7*x^
14 + 2318316*a^5*b^6*x^12 + 1326204*a^6*b^5*x^10 + 456291*a^7*b^4*x^8 + 78419*a^8*b^3*x^6 + 2772*a^9*b^2*x^4 -
252*a^10*b*x^2 + 42*a^11)/(a^12*b^9*x^24 + 9*a^13*b^8*x^22 + 36*a^14*b^7*x^20 + 84*a^15*b^6*x^18 + 126*a^16*b
^5*x^16 + 126*a^17*b^4*x^14 + 84*a^18*b^3*x^12 + 36*a^19*b^2*x^10 + 9*a^20*b*x^8 + a^21*x^6) + 110*b^3*log(b*x
^2 + a)/a^13 - 110*b^3*log(x^2)/a^13

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Fricas [B]  time = 1.39205, size = 1087, normalized size = 4.81 \begin{align*} -\frac{27720 \, a b^{11} x^{22} + 235620 \, a^{2} b^{10} x^{20} + 882420 \, a^{3} b^{9} x^{18} + 1905750 \, a^{4} b^{8} x^{16} + 2604294 \, a^{5} b^{7} x^{14} + 2318316 \, a^{6} b^{6} x^{12} + 1326204 \, a^{7} b^{5} x^{10} + 456291 \, a^{8} b^{4} x^{8} + 78419 \, a^{9} b^{3} x^{6} + 2772 \, a^{10} b^{2} x^{4} - 252 \, a^{11} b x^{2} + 42 \, a^{12} - 27720 \,{\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 55440 \,{\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (x\right )}{252 \,{\left (a^{13} b^{9} x^{24} + 9 \, a^{14} b^{8} x^{22} + 36 \, a^{15} b^{7} x^{20} + 84 \, a^{16} b^{6} x^{18} + 126 \, a^{17} b^{5} x^{16} + 126 \, a^{18} b^{4} x^{14} + 84 \, a^{19} b^{3} x^{12} + 36 \, a^{20} b^{2} x^{10} + 9 \, a^{21} b x^{8} + a^{22} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/252*(27720*a*b^11*x^22 + 235620*a^2*b^10*x^20 + 882420*a^3*b^9*x^18 + 1905750*a^4*b^8*x^16 + 2604294*a^5*b^
7*x^14 + 2318316*a^6*b^6*x^12 + 1326204*a^7*b^5*x^10 + 456291*a^8*b^4*x^8 + 78419*a^9*b^3*x^6 + 2772*a^10*b^2*
x^4 - 252*a^11*b*x^2 + 42*a^12 - 27720*(b^12*x^24 + 9*a*b^11*x^22 + 36*a^2*b^10*x^20 + 84*a^3*b^9*x^18 + 126*a
^4*b^8*x^16 + 126*a^5*b^7*x^14 + 84*a^6*b^6*x^12 + 36*a^7*b^5*x^10 + 9*a^8*b^4*x^8 + a^9*b^3*x^6)*log(b*x^2 +
a) + 55440*(b^12*x^24 + 9*a*b^11*x^22 + 36*a^2*b^10*x^20 + 84*a^3*b^9*x^18 + 126*a^4*b^8*x^16 + 126*a^5*b^7*x^
14 + 84*a^6*b^6*x^12 + 36*a^7*b^5*x^10 + 9*a^8*b^4*x^8 + a^9*b^3*x^6)*log(x))/(a^13*b^9*x^24 + 9*a^14*b^8*x^22
+ 36*a^15*b^7*x^20 + 84*a^16*b^6*x^18 + 126*a^17*b^5*x^16 + 126*a^18*b^4*x^14 + 84*a^19*b^3*x^12 + 36*a^20*b^
2*x^10 + 9*a^21*b*x^8 + a^22*x^6)

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.02488, size = 252, normalized size = 1.12 \begin{align*} -\frac{110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} + \frac{110 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{13}} + \frac{1210 \, b^{3} x^{6} - 165 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - a^{3}}{6 \, a^{13} x^{6}} - \frac{78419 \, b^{12} x^{18} + 726561 \, a b^{11} x^{16} + 2996964 \, a^{2} b^{10} x^{14} + 7225764 \, a^{3} b^{9} x^{12} + 11226726 \, a^{4} b^{8} x^{10} + 11663316 \, a^{5} b^{7} x^{8} + 8108184 \, a^{6} b^{6} x^{6} + 3641256 \, a^{7} b^{5} x^{4} + 960210 \, a^{8} b^{4} x^{2} + 113620 \, a^{9} b^{3}}{252 \,{\left (b x^{2} + a\right )}^{9} a^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-110*b^3*log(x^2)/a^13 + 110*b^3*log(abs(b*x^2 + a))/a^13 + 1/6*(1210*b^3*x^6 - 165*a*b^2*x^4 + 15*a^2*b*x^2 -
a^3)/(a^13*x^6) - 1/252*(78419*b^12*x^18 + 726561*a*b^11*x^16 + 2996964*a^2*b^10*x^14 + 7225764*a^3*b^9*x^12
+ 11226726*a^4*b^8*x^10 + 11663316*a^5*b^7*x^8 + 8108184*a^6*b^6*x^6 + 3641256*a^7*b^5*x^4 + 960210*a^8*b^4*x^
2 + 113620*a^9*b^3)/((b*x^2 + a)^9*a^13)