3.2563 \(\int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=106 \[ \frac{a^4 \left (a+b x^n\right )^9}{9 b^5 n}-\frac{2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac{6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}+\frac{\left (a+b x^n\right )^{13}}{13 b^5 n}-\frac{a \left (a+b x^n\right )^{12}}{3 b^5 n} \]

[Out]

(a^4*(a + b*x^n)^9)/(9*b^5*n) - (2*a^3*(a + b*x^n)^10)/(5*b^5*n) + (6*a^2*(a + b
*x^n)^11)/(11*b^5*n) - (a*(a + b*x^n)^12)/(3*b^5*n) + (a + b*x^n)^13/(13*b^5*n)

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Rubi [A]  time = 0.150641, antiderivative size = 106, 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 \[ \frac{a^4 \left (a+b x^n\right )^9}{9 b^5 n}-\frac{2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac{6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}+\frac{\left (a+b x^n\right )^{13}}{13 b^5 n}-\frac{a \left (a+b x^n\right )^{12}}{3 b^5 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

(a^4*(a + b*x^n)^9)/(9*b^5*n) - (2*a^3*(a + b*x^n)^10)/(5*b^5*n) + (6*a^2*(a + b
*x^n)^11)/(11*b^5*n) - (a*(a + b*x^n)^12)/(3*b^5*n) + (a + b*x^n)^13/(13*b^5*n)

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Rubi in Sympy [A]  time = 25.8409, size = 90, normalized size = 0.85 \[ \frac{a^{4} \left (a + b x^{n}\right )^{9}}{9 b^{5} n} - \frac{2 a^{3} \left (a + b x^{n}\right )^{10}}{5 b^{5} n} + \frac{6 a^{2} \left (a + b x^{n}\right )^{11}}{11 b^{5} n} - \frac{a \left (a + b x^{n}\right )^{12}}{3 b^{5} n} + \frac{\left (a + b x^{n}\right )^{13}}{13 b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+5*n)*(a+b*x**n)**8,x)

[Out]

a**4*(a + b*x**n)**9/(9*b**5*n) - 2*a**3*(a + b*x**n)**10/(5*b**5*n) + 6*a**2*(a
+ b*x**n)**11/(11*b**5*n) - a*(a + b*x**n)**12/(3*b**5*n) + (a + b*x**n)**13/(1
3*b**5*n)

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Mathematica [A]  time = 0.0426221, size = 113, normalized size = 1.07 \[ \frac{x^{5 n} \left (1287 a^8+8580 a^7 b x^n+25740 a^6 b^2 x^{2 n}+45045 a^5 b^3 x^{3 n}+50050 a^4 b^4 x^{4 n}+36036 a^3 b^5 x^{5 n}+16380 a^2 b^6 x^{6 n}+4290 a b^7 x^{7 n}+495 b^8 x^{8 n}\right )}{6435 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

(x^(5*n)*(1287*a^8 + 8580*a^7*b*x^n + 25740*a^6*b^2*x^(2*n) + 45045*a^5*b^3*x^(3
*n) + 50050*a^4*b^4*x^(4*n) + 36036*a^3*b^5*x^(5*n) + 16380*a^2*b^6*x^(6*n) + 42
90*a*b^7*x^(7*n) + 495*b^8*x^(8*n)))/(6435*n)

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Maple [A]  time = 0.043, size = 136, normalized size = 1.3 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{2\,a{b}^{7} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{28\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{9}}{9\,n}}+7\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{8}}{n}}+4\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{4\,b{a}^{7} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{5}}{5\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+5*n)*(a+b*x^n)^8,x)

[Out]

1/13*b^8/n*(x^n)^13+2/3*a*b^7/n*(x^n)^12+28/11*a^2*b^6/n*(x^n)^11+28/5*a^3*b^5/n
*(x^n)^10+70/9*a^4*b^4/n*(x^n)^9+7*a^5*b^3/n*(x^n)^8+4*a^6*b^2/n*(x^n)^7+4/3*a^7
*b/n*(x^n)^6+1/5*a^8/n*(x^n)^5

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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^(5*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226501, size = 153, normalized size = 1.44 \[ \frac{495 \, b^{8} x^{13 \, n} + 4290 \, a b^{7} x^{12 \, n} + 16380 \, a^{2} b^{6} x^{11 \, n} + 36036 \, a^{3} b^{5} x^{10 \, n} + 50050 \, a^{4} b^{4} x^{9 \, n} + 45045 \, a^{5} b^{3} x^{8 \, n} + 25740 \, a^{6} b^{2} x^{7 \, n} + 8580 \, a^{7} b x^{6 \, n} + 1287 \, a^{8} x^{5 \, n}}{6435 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(5*n - 1),x, algorithm="fricas")

[Out]

1/6435*(495*b^8*x^(13*n) + 4290*a*b^7*x^(12*n) + 16380*a^2*b^6*x^(11*n) + 36036*
a^3*b^5*x^(10*n) + 50050*a^4*b^4*x^(9*n) + 45045*a^5*b^3*x^(8*n) + 25740*a^6*b^2
*x^(7*n) + 8580*a^7*b*x^(6*n) + 1287*a^8*x^(5*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+5*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{5 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(5*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(5*n - 1), x)