3.2564 \(\int x^{-1+4 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]

[Out]

-(a^3*(a + b*x^n)^9)/(9*b^4*n) + (3*a^2*(a + b*x^n)^10)/(10*b^4*n) - (3*a*(a + b
*x^n)^11)/(11*b^4*n) + (a + b*x^n)^12/(12*b^4*n)

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Rubi [A]  time = 0.126183, antiderivative size = 84, 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 \[ -\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)*(a + b*x^n)^8,x]

[Out]

-(a^3*(a + b*x^n)^9)/(9*b^4*n) + (3*a^2*(a + b*x^n)^10)/(10*b^4*n) - (3*a*(a + b
*x^n)^11)/(11*b^4*n) + (a + b*x^n)^12/(12*b^4*n)

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Rubi in Sympy [A]  time = 22.1104, size = 71, normalized size = 0.85 \[ - \frac{a^{3} \left (a + b x^{n}\right )^{9}}{9 b^{4} n} + \frac{3 a^{2} \left (a + b x^{n}\right )^{10}}{10 b^{4} n} - \frac{3 a \left (a + b x^{n}\right )^{11}}{11 b^{4} n} + \frac{\left (a + b x^{n}\right )^{12}}{12 b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)*(a+b*x**n)**8,x)

[Out]

-a**3*(a + b*x**n)**9/(9*b**4*n) + 3*a**2*(a + b*x**n)**10/(10*b**4*n) - 3*a*(a
+ b*x**n)**11/(11*b**4*n) + (a + b*x**n)**12/(12*b**4*n)

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Mathematica [A]  time = 0.0462353, size = 113, normalized size = 1.35 \[ \frac{x^{4 n} \left (495 a^8+3168 a^7 b x^n+9240 a^6 b^2 x^{2 n}+15840 a^5 b^3 x^{3 n}+17325 a^4 b^4 x^{4 n}+12320 a^3 b^5 x^{5 n}+5544 a^2 b^6 x^{6 n}+1440 a b^7 x^{7 n}+165 b^8 x^{8 n}\right )}{1980 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)*(a + b*x^n)^8,x]

[Out]

(x^(4*n)*(495*a^8 + 3168*a^7*b*x^n + 9240*a^6*b^2*x^(2*n) + 15840*a^5*b^3*x^(3*n
) + 17325*a^4*b^4*x^(4*n) + 12320*a^3*b^5*x^(5*n) + 5544*a^2*b^6*x^(6*n) + 1440*
a*b^7*x^(7*n) + 165*b^8*x^(8*n)))/(1980*n)

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Maple [A]  time = 0.04, size = 136, normalized size = 1.6 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{12}}{12\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{14\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{8}}{4\,n}}+8\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{14\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{4}}{4\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)*(a+b*x^n)^8,x)

[Out]

1/12*b^8/n*(x^n)^12+8/11*a*b^7/n*(x^n)^11+14/5*a^2*b^6/n*(x^n)^10+56/9*a^3*b^5/n
*(x^n)^9+35/4*a^4*b^4/n*(x^n)^8+8*a^5*b^3/n*(x^n)^7+14/3*a^6*b^2/n*(x^n)^6+8/5*a
^7*b/n*(x^n)^5+1/4*a^8/n*(x^n)^4

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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^(4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22563, size = 153, normalized size = 1.82 \[ \frac{165 \, b^{8} x^{12 \, n} + 1440 \, a b^{7} x^{11 \, n} + 5544 \, a^{2} b^{6} x^{10 \, n} + 12320 \, a^{3} b^{5} x^{9 \, n} + 17325 \, a^{4} b^{4} x^{8 \, n} + 15840 \, a^{5} b^{3} x^{7 \, n} + 9240 \, a^{6} b^{2} x^{6 \, n} + 3168 \, a^{7} b x^{5 \, n} + 495 \, a^{8} x^{4 \, n}}{1980 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(4*n - 1),x, algorithm="fricas")

[Out]

1/1980*(165*b^8*x^(12*n) + 1440*a*b^7*x^(11*n) + 5544*a^2*b^6*x^(10*n) + 12320*a
^3*b^5*x^(9*n) + 17325*a^4*b^4*x^(8*n) + 15840*a^5*b^3*x^(7*n) + 9240*a^6*b^2*x^
(6*n) + 3168*a^7*b*x^(5*n) + 495*a^8*x^(4*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+4*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(4*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(4*n - 1), x)