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

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

[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

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Rubi [A]  time = 0.23, antiderivative size = 226, 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} \[ -\frac {165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac {30 b^3}{a^{11} \left (a+b x^2\right )^2}-\frac {14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac {7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac {7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac {5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac {5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac {b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac {b^3}{18 a^4 \left (a+b x^2\right )^9}-\frac {55 b^2}{2 a^{12} x^2}+\frac {110 b^3 \log \left (a+b x^2\right )}{a^{13}}-\frac {220 b^3 \log (x)}{a^{13}}+\frac {5 b}{2 a^{11} x^4}-\frac {1}{6 a^{10} x^6} \]

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 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^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*}

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Mathematica [A]  time = 0.13, size = 162, normalized size = 0.72 \[ -\frac {-27720 b^3 \log \left (a+b x^2\right )+\frac {a \left (42 a^{11}-252 a^{10} b x^2+2772 a^9 b^2 x^4+78419 a^8 b^3 x^6+456291 a^7 b^4 x^8+1326204 a^6 b^5 x^{10}+2318316 a^5 b^6 x^{12}+2604294 a^4 b^7 x^{14}+1905750 a^3 b^8 x^{16}+882420 a^2 b^9 x^{18}+235620 a b^{10} x^{20}+27720 b^{11} x^{22}\right )}{x^6 \left (a+b x^2\right )^9}+55440 b^3 \log (x)}{252 a^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^10),x]

[Out]

-1/252*((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])/a^13

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fricas [B]  time = 1.11, size = 453, normalized size = 2.00 \[ -\frac {27720 \, a b^{11} x^{22} + 235620 \, a^{2} b^{10} x^{20} + 882420 \, a^{3} b^{9} x^{18} + 1905750 \, a^{4} b^{8} x^{16} + 2604294 \, a^{5} b^{7} x^{14} + 2318316 \, a^{6} b^{6} x^{12} + 1326204 \, a^{7} b^{5} x^{10} + 456291 \, a^{8} b^{4} x^{8} + 78419 \, a^{9} b^{3} x^{6} + 2772 \, a^{10} b^{2} x^{4} - 252 \, a^{11} b x^{2} + 42 \, a^{12} - 27720 \, {\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 55440 \, {\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \relax (x)}{252 \, {\left (a^{13} b^{9} x^{24} + 9 \, a^{14} b^{8} x^{22} + 36 \, a^{15} b^{7} x^{20} + 84 \, a^{16} b^{6} x^{18} + 126 \, a^{17} b^{5} x^{16} + 126 \, a^{18} b^{4} x^{14} + 84 \, a^{19} b^{3} x^{12} + 36 \, a^{20} b^{2} x^{10} + 9 \, a^{21} b x^{8} + a^{22} x^{6}\right )}} \]

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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giac [A]  time = 0.64, size = 187, normalized size = 0.83 \[ -\frac {110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} + \frac {110 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{13}} + \frac {1210 \, b^{3} x^{6} - 165 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - a^{3}}{6 \, a^{13} x^{6}} - \frac {78419 \, b^{12} x^{18} + 726561 \, a b^{11} x^{16} + 2996964 \, a^{2} b^{10} x^{14} + 7225764 \, a^{3} b^{9} x^{12} + 11226726 \, a^{4} b^{8} x^{10} + 11663316 \, a^{5} b^{7} x^{8} + 8108184 \, a^{6} b^{6} x^{6} + 3641256 \, a^{7} b^{5} x^{4} + 960210 \, a^{8} b^{4} x^{2} + 113620 \, a^{9} b^{3}}{252 \, {\left (b x^{2} + a\right )}^{9} a^{13}} \]

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)

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maple [A]  time = 0.02, size = 209, normalized size = 0.92 \[ -\frac {b^{3}}{18 \left (b \,x^{2}+a \right )^{9} a^{4}}-\frac {b^{3}}{4 \left (b \,x^{2}+a \right )^{8} a^{5}}-\frac {5 b^{3}}{7 \left (b \,x^{2}+a \right )^{7} a^{6}}-\frac {5 b^{3}}{3 \left (b \,x^{2}+a \right )^{6} a^{7}}-\frac {7 b^{3}}{2 \left (b \,x^{2}+a \right )^{5} a^{8}}-\frac {7 b^{3}}{\left (b \,x^{2}+a \right )^{4} a^{9}}-\frac {14 b^{3}}{\left (b \,x^{2}+a \right )^{3} a^{10}}-\frac {30 b^{3}}{\left (b \,x^{2}+a \right )^{2} a^{11}}-\frac {165 b^{3}}{2 \left (b \,x^{2}+a \right ) a^{12}}-\frac {220 b^{3} \ln \relax (x )}{a^{13}}+\frac {110 b^{3} \ln \left (b \,x^{2}+a \right )}{a^{13}}-\frac {55 b^{2}}{2 a^{12} x^{2}}+\frac {5 b}{2 a^{11} x^{4}}-\frac {1}{6 a^{10} x^{6}} \]

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

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maxima [A]  time = 1.71, size = 257, normalized size = 1.14 \[ -\frac {27720 \, b^{11} x^{22} + 235620 \, a b^{10} x^{20} + 882420 \, a^{2} b^{9} x^{18} + 1905750 \, a^{3} b^{8} x^{16} + 2604294 \, a^{4} b^{7} x^{14} + 2318316 \, a^{5} b^{6} x^{12} + 1326204 \, a^{6} b^{5} x^{10} + 456291 \, a^{7} b^{4} x^{8} + 78419 \, a^{8} b^{3} x^{6} + 2772 \, a^{9} b^{2} x^{4} - 252 \, a^{10} b x^{2} + 42 \, a^{11}}{252 \, {\left (a^{12} b^{9} x^{24} + 9 \, a^{13} b^{8} x^{22} + 36 \, a^{14} b^{7} x^{20} + 84 \, a^{15} b^{6} x^{18} + 126 \, a^{16} b^{5} x^{16} + 126 \, a^{17} b^{4} x^{14} + 84 \, a^{18} b^{3} x^{12} + 36 \, a^{19} b^{2} x^{10} + 9 \, a^{20} b x^{8} + a^{21} x^{6}\right )}} + \frac {110 \, b^{3} \log \left (b x^{2} + a\right )}{a^{13}} - \frac {110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.09, size = 255, normalized size = 1.13 \[ \frac {110\,b^3\,\ln \left (b\,x^2+a\right )}{a^{13}}-\frac {\frac {1}{6\,a}-\frac {b\,x^2}{a^2}+\frac {11\,b^2\,x^4}{a^3}+\frac {78419\,b^3\,x^6}{252\,a^4}+\frac {50699\,b^4\,x^8}{28\,a^5}+\frac {36839\,b^5\,x^{10}}{7\,a^6}+\frac {27599\,b^6\,x^{12}}{3\,a^7}+\frac {20669\,b^7\,x^{14}}{2\,a^8}+\frac {15125\,b^8\,x^{16}}{2\,a^9}+\frac {10505\,b^9\,x^{18}}{3\,a^{10}}+\frac {935\,b^{10}\,x^{20}}{a^{11}}+\frac {110\,b^{11}\,x^{22}}{a^{12}}}{a^9\,x^6+9\,a^8\,b\,x^8+36\,a^7\,b^2\,x^{10}+84\,a^6\,b^3\,x^{12}+126\,a^5\,b^4\,x^{14}+126\,a^4\,b^5\,x^{16}+84\,a^3\,b^6\,x^{18}+36\,a^2\,b^7\,x^{20}+9\,a\,b^8\,x^{22}+b^9\,x^{24}}-\frac {220\,b^3\,\ln \relax (x)}{a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b*x^2)^10),x)

[Out]

(110*b^3*log(a + b*x^2))/a^13 - (1/(6*a) - (b*x^2)/a^2 + (11*b^2*x^4)/a^3 + (78419*b^3*x^6)/(252*a^4) + (50699
*b^4*x^8)/(28*a^5) + (36839*b^5*x^10)/(7*a^6) + (27599*b^6*x^12)/(3*a^7) + (20669*b^7*x^14)/(2*a^8) + (15125*b
^8*x^16)/(2*a^9) + (10505*b^9*x^18)/(3*a^10) + (935*b^10*x^20)/a^11 + (110*b^11*x^22)/a^12)/(a^9*x^6 + b^9*x^2
4 + 9*a^8*b*x^8 + 9*a*b^8*x^22 + 36*a^7*b^2*x^10 + 84*a^6*b^3*x^12 + 126*a^5*b^4*x^14 + 126*a^4*b^5*x^16 + 84*
a^3*b^6*x^18 + 36*a^2*b^7*x^20) - (220*b^3*log(x))/a^13

________________________________________________________________________________________

sympy [A]  time = 1.60, size = 270, normalized size = 1.19 \[ \frac {- 42 a^{11} + 252 a^{10} b x^{2} - 2772 a^{9} b^{2} x^{4} - 78419 a^{8} b^{3} x^{6} - 456291 a^{7} b^{4} x^{8} - 1326204 a^{6} b^{5} x^{10} - 2318316 a^{5} b^{6} x^{12} - 2604294 a^{4} b^{7} x^{14} - 1905750 a^{3} b^{8} x^{16} - 882420 a^{2} b^{9} x^{18} - 235620 a b^{10} x^{20} - 27720 b^{11} x^{22}}{252 a^{21} x^{6} + 2268 a^{20} b x^{8} + 9072 a^{19} b^{2} x^{10} + 21168 a^{18} b^{3} x^{12} + 31752 a^{17} b^{4} x^{14} + 31752 a^{16} b^{5} x^{16} + 21168 a^{15} b^{6} x^{18} + 9072 a^{14} b^{7} x^{20} + 2268 a^{13} b^{8} x^{22} + 252 a^{12} b^{9} x^{24}} - \frac {220 b^{3} \log {\relax (x )}}{a^{13}} + \frac {110 b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-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)/(252*a**21*x**6 + 2268*a**20*b*x**8 + 9072*a**19*b**2*x**1
0 + 21168*a**18*b**3*x**12 + 31752*a**17*b**4*x**14 + 31752*a**16*b**5*x**16 + 21168*a**15*b**6*x**18 + 9072*a
**14*b**7*x**20 + 2268*a**13*b**8*x**22 + 252*a**12*b**9*x**24) - 220*b**3*log(x)/a**13 + 110*b**3*log(a/b + x
**2)/a**13

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