3.2566 \(\int x^{-1+2 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0599271, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.266, size = 31, normalized size = 0.78 \[ - \frac{a \left (a + b x^{n}\right )^{9}}{9 b^{2} n} + \frac{\left (a + b x^{n}\right )^{10}}{10 b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0412804, size = 113, normalized size = 2.82 \[ \frac{x^{2 n} \left (45 a^8+240 a^7 b x^n+630 a^6 b^2 x^{2 n}+1008 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+720 a^3 b^5 x^{5 n}+315 a^2 b^6 x^{6 n}+80 a b^7 x^{7 n}+9 b^8 x^{8 n}\right )}{90 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.04, size = 136, normalized size = 3.4 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{10}}{10\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{7\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{8}}{2\,n}}+8\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{2}}{2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^8,x)

[Out]

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

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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^(2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226286, size = 153, normalized size = 3.82 \[ \frac{9 \, b^{8} x^{10 \, n} + 80 \, a b^{7} x^{9 \, n} + 315 \, a^{2} b^{6} x^{8 \, n} + 720 \, a^{3} b^{5} x^{7 \, n} + 1050 \, a^{4} b^{4} x^{6 \, n} + 1008 \, a^{5} b^{3} x^{5 \, n} + 630 \, a^{6} b^{2} x^{4 \, n} + 240 \, a^{7} b x^{3 \, n} + 45 \, a^{8} x^{2 \, n}}{90 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^8*x^(2*n - 1), x)