3.2565 \(\int x^{-1+3 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=62 \[ \frac{a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac{\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac{a \left (a+b x^n\right )^{10}}{5 b^3 n} \]

[Out]

(a^2*(a + b*x^n)^9)/(9*b^3*n) - (a*(a + b*x^n)^10)/(5*b^3*n) + (a + b*x^n)^11/(1
1*b^3*n)

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Rubi [A]  time = 0.105918, antiderivative size = 62, 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 \[ \frac{a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac{\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac{a \left (a+b x^n\right )^{10}}{5 b^3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)*(a + b*x^n)^8,x]

[Out]

(a^2*(a + b*x^n)^9)/(9*b^3*n) - (a*(a + b*x^n)^10)/(5*b^3*n) + (a + b*x^n)^11/(1
1*b^3*n)

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Rubi in Sympy [A]  time = 17.5499, size = 49, normalized size = 0.79 \[ \frac{a^{2} \left (a + b x^{n}\right )^{9}}{9 b^{3} n} - \frac{a \left (a + b x^{n}\right )^{10}}{5 b^{3} n} + \frac{\left (a + b x^{n}\right )^{11}}{11 b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)**8,x)

[Out]

a**2*(a + b*x**n)**9/(9*b**3*n) - a*(a + b*x**n)**10/(5*b**3*n) + (a + b*x**n)**
11/(11*b**3*n)

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Mathematica [A]  time = 0.0411991, size = 113, normalized size = 1.82 \[ \frac{x^{3 n} \left (165 a^8+990 a^7 b x^n+2772 a^6 b^2 x^{2 n}+4620 a^5 b^3 x^{3 n}+4950 a^4 b^4 x^{4 n}+3465 a^3 b^5 x^{5 n}+1540 a^2 b^6 x^{6 n}+396 a b^7 x^{7 n}+45 b^8 x^{8 n}\right )}{495 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)*(a + b*x^n)^8,x]

[Out]

(x^(3*n)*(165*a^8 + 990*a^7*b*x^n + 2772*a^6*b^2*x^(2*n) + 4620*a^5*b^3*x^(3*n)
+ 4950*a^4*b^4*x^(4*n) + 3465*a^3*b^5*x^(5*n) + 1540*a^2*b^6*x^(6*n) + 396*a*b^7
*x^(7*n) + 45*b^8*x^(8*n)))/(495*n)

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Maple [B]  time = 0.04, size = 136, normalized size = 2.2 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{4\,a{b}^{7} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{9}}{9\,n}}+7\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{8}}{n}}+10\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{28\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{5}}{5\,n}}+2\,{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{3}}{3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)^8,x)

[Out]

1/11*b^8/n*(x^n)^11+4/5*a*b^7/n*(x^n)^10+28/9*a^2*b^6/n*(x^n)^9+7*a^3*b^5/n*(x^n
)^8+10*a^4*b^4/n*(x^n)^7+28/3*a^5*b^3/n*(x^n)^6+28/5*a^6*b^2/n*(x^n)^5+2*a^7*b/n
*(x^n)^4+1/3*a^8/n*(x^n)^3

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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^(3*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227273, size = 153, normalized size = 2.47 \[ \frac{45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(3*n - 1),x, algorithm="fricas")

[Out]

1/495*(45*b^8*x^(11*n) + 396*a*b^7*x^(10*n) + 1540*a^2*b^6*x^(9*n) + 3465*a^3*b^
5*x^(8*n) + 4950*a^4*b^4*x^(7*n) + 4620*a^5*b^3*x^(6*n) + 2772*a^6*b^2*x^(5*n) +
990*a^7*b*x^(4*n) + 165*a^8*x^(3*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+3*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{3 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(3*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(3*n - 1), x)