3.128 \(\int \frac{x^{-1+n} \left (b+2 c x^n\right )}{\left (b x^n+c x^{2 n}\right )^8} \, dx\)

Optimal. Leaf size=21 \[ -\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]

[Out]

-1/(7*n*x^(7*n)*(b + c*x^n)^7)

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Rubi [A]  time = 0.0627039, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n))^8,x]

[Out]

-1/(7*n*x^(7*n)*(b + c*x^n)^7)

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Rubi in Sympy [A]  time = 11.0252, size = 19, normalized size = 0.9 \[ - \frac{x^{- 7 n}}{7 n \left (b + c x^{n}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)*(b+2*c*x**n)/(b*x**n+c*x**(2*n))**8,x)

[Out]

-x**(-7*n)/(7*n*(b + c*x**n)**7)

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Mathematica [B]  time = 0.0788099, size = 127, normalized size = 6.05 \[ -\frac{x^{-7 n} \left (b^{14}+1716 b^7 c^7 x^{7 n}+12012 b^6 c^8 x^{8 n}+36036 b^5 c^9 x^{9 n}+60060 b^4 c^{10} x^{10 n}+60060 b^3 c^{11} x^{11 n}+36036 b^2 c^{12} x^{12 n}+12012 b c^{13} x^{13 n}+1716 c^{14} x^{14 n}\right )}{7 b^{14} n \left (b+c x^n\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n))^8,x]

[Out]

-(b^14 + 1716*b^7*c^7*x^(7*n) + 12012*b^6*c^8*x^(8*n) + 36036*b^5*c^9*x^(9*n) +
60060*b^4*c^10*x^(10*n) + 60060*b^3*c^11*x^(11*n) + 36036*b^2*c^12*x^(12*n) + 12
012*b*c^13*x^(13*n) + 1716*c^14*x^(14*n))/(7*b^14*n*x^(7*n)*(b + c*x^n)^7)

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Maple [B]  time = 0.086, size = 203, normalized size = 9.7 \[ -132\,{\frac{{c}^{6}}{{b}^{13}n{x}^{n}}}+66\,{\frac{{c}^{5}}{{b}^{12}n \left ({x}^{n} \right ) ^{2}}}-30\,{\frac{{c}^{4}}{{b}^{11}n \left ({x}^{n} \right ) ^{3}}}+12\,{\frac{{c}^{3}}{{b}^{10}n \left ({x}^{n} \right ) ^{4}}}-4\,{\frac{{c}^{2}}{{b}^{9}n \left ({x}^{n} \right ) ^{5}}}+{\frac{c}{{b}^{8}n \left ({x}^{n} \right ) ^{6}}}-{\frac{1}{7\,{b}^{7}n \left ({x}^{n} \right ) ^{7}}}+{\frac{{c}^{7} \left ( 924\, \left ({x}^{n} \right ) ^{6}{c}^{6}+6006\,b{c}^{5} \left ({x}^{n} \right ) ^{5}+16380\,{b}^{2}{c}^{4} \left ({x}^{n} \right ) ^{4}+24024\,{b}^{3}{c}^{3} \left ({x}^{n} \right ) ^{3}+20020\,{b}^{4}{c}^{2} \left ({x}^{n} \right ) ^{2}+9009\,{b}^{5}c{x}^{n}+1716\,{b}^{6} \right ) }{7\,{b}^{13}n \left ( b+c{x}^{n} \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(b+2*c*x^n)/(b*x^n+c*x^(2*n))^8,x)

[Out]

-132/b^13*c^6/n/(x^n)+66/b^12*c^5/n/(x^n)^2-30/b^11*c^4/n/(x^n)^3+12/b^10*c^3/n/
(x^n)^4-4/b^9*c^2/n/(x^n)^5+1/b^8*c/n/(x^n)^6-1/7/b^7/n/(x^n)^7+1/7*c^7*(924*(x^
n)^6*c^6+6006*b*c^5*(x^n)^5+16380*b^2*c^4*(x^n)^4+24024*b^3*c^3*(x^n)^3+20020*b^
4*c^2*(x^n)^2+9009*b^5*c*x^n+1716*b^6)/b^13/n/(b+c*x^n)^7

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Maxima [A]  time = 0.798319, size = 826, normalized size = 39.33 \[ -\frac{1}{105} \, b{\left (\frac{360360 \, c^{13} x^{13 \, n} + 2342340 \, b c^{12} x^{12 \, n} + 6426420 \, b^{2} c^{11} x^{11 \, n} + 9579570 \, b^{3} c^{10} x^{10 \, n} + 8270262 \, b^{4} c^{9} x^{9 \, n} + 4018014 \, b^{5} c^{8} x^{8 \, n} + 934362 \, b^{6} c^{7} x^{7 \, n} + 45045 \, b^{7} c^{6} x^{6 \, n} - 5005 \, b^{8} c^{5} x^{5 \, n} + 1001 \, b^{9} c^{4} x^{4 \, n} - 273 \, b^{10} c^{3} x^{3 \, n} + 91 \, b^{11} c^{2} x^{2 \, n} - 35 \, b^{12} c x^{n} + 15 \, b^{13}}{b^{14} c^{7} n x^{14 \, n} + 7 \, b^{15} c^{6} n x^{13 \, n} + 21 \, b^{16} c^{5} n x^{12 \, n} + 35 \, b^{17} c^{4} n x^{11 \, n} + 35 \, b^{18} c^{3} n x^{10 \, n} + 21 \, b^{19} c^{2} n x^{9 \, n} + 7 \, b^{20} c n x^{8 \, n} + b^{21} n x^{7 \, n}} + \frac{360360 \, c^{7} \log \left (x\right )}{b^{15}} - \frac{360360 \, c^{7} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{15} n}\right )} + \frac{1}{105} \, c{\left (\frac{360360 \, c^{12} x^{12 \, n} + 2342340 \, b c^{11} x^{11 \, n} + 6426420 \, b^{2} c^{10} x^{10 \, n} + 9579570 \, b^{3} c^{9} x^{9 \, n} + 8270262 \, b^{4} c^{8} x^{8 \, n} + 4018014 \, b^{5} c^{7} x^{7 \, n} + 934362 \, b^{6} c^{6} x^{6 \, n} + 45045 \, b^{7} c^{5} x^{5 \, n} - 5005 \, b^{8} c^{4} x^{4 \, n} + 1001 \, b^{9} c^{3} x^{3 \, n} - 273 \, b^{10} c^{2} x^{2 \, n} + 91 \, b^{11} c x^{n} - 35 \, b^{12}}{b^{13} c^{7} n x^{13 \, n} + 7 \, b^{14} c^{6} n x^{12 \, n} + 21 \, b^{15} c^{5} n x^{11 \, n} + 35 \, b^{16} c^{4} n x^{10 \, n} + 35 \, b^{17} c^{3} n x^{9 \, n} + 21 \, b^{18} c^{2} n x^{8 \, n} + 7 \, b^{19} c n x^{7 \, n} + b^{20} n x^{6 \, n}} + \frac{360360 \, c^{6} \log \left (x\right )}{b^{14}} - \frac{360360 \, c^{6} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{14} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n)^8,x, algorithm="maxima")

[Out]

-1/105*b*((360360*c^13*x^(13*n) + 2342340*b*c^12*x^(12*n) + 6426420*b^2*c^11*x^(
11*n) + 9579570*b^3*c^10*x^(10*n) + 8270262*b^4*c^9*x^(9*n) + 4018014*b^5*c^8*x^
(8*n) + 934362*b^6*c^7*x^(7*n) + 45045*b^7*c^6*x^(6*n) - 5005*b^8*c^5*x^(5*n) +
1001*b^9*c^4*x^(4*n) - 273*b^10*c^3*x^(3*n) + 91*b^11*c^2*x^(2*n) - 35*b^12*c*x^
n + 15*b^13)/(b^14*c^7*n*x^(14*n) + 7*b^15*c^6*n*x^(13*n) + 21*b^16*c^5*n*x^(12*
n) + 35*b^17*c^4*n*x^(11*n) + 35*b^18*c^3*n*x^(10*n) + 21*b^19*c^2*n*x^(9*n) + 7
*b^20*c*n*x^(8*n) + b^21*n*x^(7*n)) + 360360*c^7*log(x)/b^15 - 360360*c^7*log((c
*x^n + b)/c)/(b^15*n)) + 1/105*c*((360360*c^12*x^(12*n) + 2342340*b*c^11*x^(11*n
) + 6426420*b^2*c^10*x^(10*n) + 9579570*b^3*c^9*x^(9*n) + 8270262*b^4*c^8*x^(8*n
) + 4018014*b^5*c^7*x^(7*n) + 934362*b^6*c^6*x^(6*n) + 45045*b^7*c^5*x^(5*n) - 5
005*b^8*c^4*x^(4*n) + 1001*b^9*c^3*x^(3*n) - 273*b^10*c^2*x^(2*n) + 91*b^11*c*x^
n - 35*b^12)/(b^13*c^7*n*x^(13*n) + 7*b^14*c^6*n*x^(12*n) + 21*b^15*c^5*n*x^(11*
n) + 35*b^16*c^4*n*x^(10*n) + 35*b^17*c^3*n*x^(9*n) + 21*b^18*c^2*n*x^(8*n) + 7*
b^19*c*n*x^(7*n) + b^20*n*x^(6*n)) + 360360*c^6*log(x)/b^14 - 360360*c^6*log((c*
x^n + b)/c)/(b^14*n))

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Fricas [A]  time = 0.353914, size = 142, normalized size = 6.76 \[ -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 21 \, b^{2} c^{5} n x^{12 \, n} + 35 \, b^{3} c^{4} n x^{11 \, n} + 35 \, b^{4} c^{3} n x^{10 \, n} + 21 \, b^{5} c^{2} n x^{9 \, n} + 7 \, b^{6} c n x^{8 \, n} + b^{7} n x^{7 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n)^8,x, algorithm="fricas")

[Out]

-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 21*b^2*c^5*n*x^(12*n) + 35*b^3*c^4*n
*x^(11*n) + 35*b^4*c^3*n*x^(10*n) + 21*b^5*c^2*n*x^(9*n) + 7*b^6*c*n*x^(8*n) + b
^7*n*x^(7*n))

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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(x**(-1+n)*(b+2*c*x**n)/(b*x**n+c*x**(2*n))**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272335, size = 27, normalized size = 1.29 \[ -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n}\right )}^{7} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n)^8,x, algorithm="giac")

[Out]

-1/7/((c*x^(2*n) + b*x^n)^7*n)