3.2578 \(\int x^{-1-10 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=50 \[ \frac{b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]

[Out]

-(a + b*x^n)^9/(10*a*n*x^(10*n)) + (b*(a + b*x^n)^9)/(90*a^2*n*x^(9*n))

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Rubi [A]  time = 0.0599408, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 10*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(10*a*n*x^(10*n)) + (b*(a + b*x^n)^9)/(90*a^2*n*x^(9*n))

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Rubi in Sympy [A]  time = 6.86058, size = 39, normalized size = 0.78 \[ - \frac{x^{- 10 n} \left (a + b x^{n}\right )^{9}}{10 a n} + \frac{b x^{- 9 n} \left (a + b x^{n}\right )^{9}}{90 a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-10*n)*(a+b*x**n)**8,x)

[Out]

-x**(-10*n)*(a + b*x**n)**9/(10*a*n) + b*x**(-9*n)*(a + b*x**n)**9/(90*a**2*n)

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Mathematica [B]  time = 0.0519608, size = 113, normalized size = 2.26 \[ -\frac{x^{-10 n} \left (9 a^8+80 a^7 b x^n+315 a^6 b^2 x^{2 n}+720 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+1008 a^3 b^5 x^{5 n}+630 a^2 b^6 x^{6 n}+240 a b^7 x^{7 n}+45 b^8 x^{8 n}\right )}{90 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 10*n)*(a + b*x^n)^8,x]

[Out]

-(9*a^8 + 80*a^7*b*x^n + 315*a^6*b^2*x^(2*n) + 720*a^5*b^3*x^(3*n) + 1050*a^4*b^
4*x^(4*n) + 1008*a^3*b^5*x^(5*n) + 630*a^2*b^6*x^(6*n) + 240*a*b^7*x^(7*n) + 45*
b^8*x^(8*n))/(90*n*x^(10*n))

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Maple [B]  time = 0.042, size = 136, normalized size = 2.7 \[ -{\frac{{b}^{8}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{8\,a{b}^{7}}{3\,n \left ({x}^{n} \right ) ^{3}}}-7\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,n \left ({x}^{n} \right ) ^{6}}}-8\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{7\,{a}^{6}{b}^{2}}{2\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{8\,b{a}^{7}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{{a}^{8}}{10\,n \left ({x}^{n} \right ) ^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-10*n)*(a+b*x^n)^8,x)

[Out]

-1/2*b^8/n/(x^n)^2-8/3*a*b^7/n/(x^n)^3-7*a^2*b^6/n/(x^n)^4-56/5*a^3*b^5/n/(x^n)^
5-35/3*a^4*b^4/n/(x^n)^6-8*a^5*b^3/n/(x^n)^7-7/2*a^6*b^2/n/(x^n)^8-8/9*a^7*b/n/(
x^n)^9-1/10*a^8/n/(x^n)^10

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-10*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227039, size = 153, normalized size = 3.06 \[ -\frac{45 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} + 630 \, a^{2} b^{6} x^{6 \, n} + 1008 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 720 \, a^{5} b^{3} x^{3 \, n} + 315 \, a^{6} b^{2} x^{2 \, n} + 80 \, a^{7} b x^{n} + 9 \, a^{8}}{90 \, n x^{10 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-10*n - 1),x, algorithm="fricas")

[Out]

-1/90*(45*b^8*x^(8*n) + 240*a*b^7*x^(7*n) + 630*a^2*b^6*x^(6*n) + 1008*a^3*b^5*x
^(5*n) + 1050*a^4*b^4*x^(4*n) + 720*a^5*b^3*x^(3*n) + 315*a^6*b^2*x^(2*n) + 80*a
^7*b*x^n + 9*a^8)/(n*x^(10*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**(-1-10*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234951, size = 163, normalized size = 3.26 \[ -\frac{{\left (45 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 240 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 630 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 1008 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 720 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 315 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 80 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 9 \, a^{8}\right )} e^{\left (-10 \, n{\rm ln}\left (x\right )\right )}}{90 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-10*n - 1),x, algorithm="giac")

[Out]

-1/90*(45*b^8*e^(8*n*ln(x)) + 240*a*b^7*e^(7*n*ln(x)) + 630*a^2*b^6*e^(6*n*ln(x)
) + 1008*a^3*b^5*e^(5*n*ln(x)) + 1050*a^4*b^4*e^(4*n*ln(x)) + 720*a^5*b^3*e^(3*n
*ln(x)) + 315*a^6*b^2*e^(2*n*ln(x)) + 80*a^7*b*e^(n*ln(x)) + 9*a^8)*e^(-10*n*ln(
x))/n