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

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*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 = 0.525568, 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 \[ \frac{110 b^3 \log \left (a+b x^2\right )}{a^{13}}-\frac{220 b^3 \log (x)}{a^{13}}-\frac{165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac{55 b^2}{2 a^{12} x^2}-\frac{30 b^3}{a^{11} \left (a+b x^2\right )^2}+\frac{5 b}{2 a^{11} x^4}-\frac{14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac{1}{6 a^{10} x^6}-\frac{7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac{7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac{5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac{5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac{b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac{b^3}{18 a^4 \left (a+b x^2\right )^9} \]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**2+a)**10,x)

[Out]

Timed out

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Mathematica [A]  time = 0.248067, 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]

-((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 = 0.03, size = 209, normalized size = 0.9 \[ -{\frac{1}{6\,{a}^{10}{x}^{6}}}+{\frac{5\,b}{2\,{a}^{11}{x}^{4}}}-{\frac{55\,{b}^{2}}{2\,{a}^{12}{x}^{2}}}-{\frac{{b}^{3}}{18\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{{b}^{3}}{4\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{5\,{b}^{3}}{7\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{5\,{b}^{3}}{3\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{7\,{b}^{3}}{2\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-7\,{\frac{{b}^{3}}{{a}^{9} \left ( b{x}^{2}+a \right ) ^{4}}}-14\,{\frac{{b}^{3}}{{a}^{10} \left ( b{x}^{2}+a \right ) ^{3}}}-30\,{\frac{{b}^{3}}{{a}^{11} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{165\,{b}^{3}}{2\,{a}^{12} \left ( b{x}^{2}+a \right ) }}-220\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{13}}}+110\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{{a}^{13}}} \]

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)^2-16
5/2*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 = 1.39422, size = 347, normalized size = 1.54 \[ -\frac{27720 \, b^{11} x^{22} + 235620 \, a b^{10} x^{20} + 882420 \, a^{2} b^{9} x^{18} + 1905750 \, a^{3} b^{8} x^{16} + 2604294 \, a^{4} b^{7} x^{14} + 2318316 \, a^{5} b^{6} x^{12} + 1326204 \, a^{6} b^{5} x^{10} + 456291 \, a^{7} b^{4} x^{8} + 78419 \, a^{8} b^{3} x^{6} + 2772 \, a^{9} b^{2} x^{4} - 252 \, a^{10} b x^{2} + 42 \, a^{11}}{252 \,{\left (a^{12} b^{9} x^{24} + 9 \, a^{13} b^{8} x^{22} + 36 \, a^{14} b^{7} x^{20} + 84 \, a^{15} b^{6} x^{18} + 126 \, a^{16} b^{5} x^{16} + 126 \, a^{17} b^{4} x^{14} + 84 \, a^{18} b^{3} x^{12} + 36 \, a^{19} b^{2} x^{10} + 9 \, a^{20} b x^{8} + a^{21} x^{6}\right )}} + \frac{110 \, b^{3} \log \left (b x^{2} + a\right )}{a^{13}} - \frac{110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^10*x^7),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 [A]  time = 0.234776, size = 612, normalized size = 2.71 \[ -\frac{27720 \, a b^{11} x^{22} + 235620 \, a^{2} b^{10} x^{20} + 882420 \, a^{3} b^{9} x^{18} + 1905750 \, a^{4} b^{8} x^{16} + 2604294 \, a^{5} b^{7} x^{14} + 2318316 \, a^{6} b^{6} x^{12} + 1326204 \, a^{7} b^{5} x^{10} + 456291 \, a^{8} b^{4} x^{8} + 78419 \, a^{9} b^{3} x^{6} + 2772 \, a^{10} b^{2} x^{4} - 252 \, a^{11} b x^{2} + 42 \, a^{12} - 27720 \,{\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 55440 \,{\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (x\right )}{252 \,{\left (a^{13} b^{9} x^{24} + 9 \, a^{14} b^{8} x^{22} + 36 \, a^{15} b^{7} x^{20} + 84 \, a^{16} b^{6} x^{18} + 126 \, a^{17} b^{5} x^{16} + 126 \, a^{18} b^{4} x^{14} + 84 \, a^{19} b^{3} x^{12} + 36 \, a^{20} b^{2} x^{10} + 9 \, a^{21} b x^{8} + a^{22} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^10*x^7),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213004, size = 252, normalized size = 1.12 \[ -\frac{110 \, b^{3}{\rm ln}\left (x^{2}\right )}{a^{13}} + \frac{110 \, b^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{13}} + \frac{1210 \, b^{3} x^{6} - 165 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - a^{3}}{6 \, a^{13} x^{6}} - \frac{78419 \, b^{12} x^{18} + 726561 \, a b^{11} x^{16} + 2996964 \, a^{2} b^{10} x^{14} + 7225764 \, a^{3} b^{9} x^{12} + 11226726 \, a^{4} b^{8} x^{10} + 11663316 \, a^{5} b^{7} x^{8} + 8108184 \, a^{6} b^{6} x^{6} + 3641256 \, a^{7} b^{5} x^{4} + 960210 \, a^{8} b^{4} x^{2} + 113620 \, a^{9} b^{3}}{252 \,{\left (b x^{2} + a\right )}^{9} a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-110*b^3*ln(x^2)/a^13 + 110*b^3*ln(abs(b*x^2 + a))/a^13 + 1/6*(1210*b^3*x^6 - 16
5*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^1
0 + 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)