3.2561 \(\int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=150 \[ \frac{a^6 \left (a+b x^n\right )^9}{9 b^7 n}-\frac{3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac{15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac{5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac{15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}+\frac{\left (a+b x^n\right )^{15}}{15 b^7 n}-\frac{3 a \left (a+b x^n\right )^{14}}{7 b^7 n} \]

[Out]

(a^6*(a + b*x^n)^9)/(9*b^7*n) - (3*a^5*(a + b*x^n)^10)/(5*b^7*n) + (15*a^4*(a +
b*x^n)^11)/(11*b^7*n) - (5*a^3*(a + b*x^n)^12)/(3*b^7*n) + (15*a^2*(a + b*x^n)^1
3)/(13*b^7*n) - (3*a*(a + b*x^n)^14)/(7*b^7*n) + (a + b*x^n)^15/(15*b^7*n)

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Rubi [A]  time = 0.194401, antiderivative size = 150, 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 \[ \frac{a^6 \left (a+b x^n\right )^9}{9 b^7 n}-\frac{3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac{15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac{5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac{15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}+\frac{\left (a+b x^n\right )^{15}}{15 b^7 n}-\frac{3 a \left (a+b x^n\right )^{14}}{7 b^7 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 7*n)*(a + b*x^n)^8,x]

[Out]

(a^6*(a + b*x^n)^9)/(9*b^7*n) - (3*a^5*(a + b*x^n)^10)/(5*b^7*n) + (15*a^4*(a +
b*x^n)^11)/(11*b^7*n) - (5*a^3*(a + b*x^n)^12)/(3*b^7*n) + (15*a^2*(a + b*x^n)^1
3)/(13*b^7*n) - (3*a*(a + b*x^n)^14)/(7*b^7*n) + (a + b*x^n)^15/(15*b^7*n)

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Rubi in Sympy [A]  time = 30.3127, size = 134, normalized size = 0.89 \[ \frac{a^{8} x^{7 n}}{7 n} + \frac{a^{7} b x^{8 n}}{n} + \frac{28 a^{6} b^{2} x^{9 n}}{9 n} + \frac{28 a^{5} b^{3} x^{10 n}}{5 n} + \frac{70 a^{4} b^{4} x^{11 n}}{11 n} + \frac{14 a^{3} b^{5} x^{12 n}}{3 n} + \frac{28 a^{2} b^{6} x^{13 n}}{13 n} + \frac{4 a b^{7} x^{14 n}}{7 n} + \frac{b^{8} x^{15 n}}{15 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+7*n)*(a+b*x**n)**8,x)

[Out]

a**8*x**(7*n)/(7*n) + a**7*b*x**(8*n)/n + 28*a**6*b**2*x**(9*n)/(9*n) + 28*a**5*
b**3*x**(10*n)/(5*n) + 70*a**4*b**4*x**(11*n)/(11*n) + 14*a**3*b**5*x**(12*n)/(3
*n) + 28*a**2*b**6*x**(13*n)/(13*n) + 4*a*b**7*x**(14*n)/(7*n) + b**8*x**(15*n)/
(15*n)

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Mathematica [A]  time = 0.0420576, size = 113, normalized size = 0.75 \[ \frac{x^{7 n} \left (6435 a^8+45045 a^7 b x^n+140140 a^6 b^2 x^{2 n}+252252 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+210210 a^3 b^5 x^{5 n}+97020 a^2 b^6 x^{6 n}+25740 a b^7 x^{7 n}+3003 b^8 x^{8 n}\right )}{45045 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 7*n)*(a + b*x^n)^8,x]

[Out]

(x^(7*n)*(6435*a^8 + 45045*a^7*b*x^n + 140140*a^6*b^2*x^(2*n) + 252252*a^5*b^3*x
^(3*n) + 286650*a^4*b^4*x^(4*n) + 210210*a^3*b^5*x^(5*n) + 97020*a^2*b^6*x^(6*n)
+ 25740*a*b^7*x^(7*n) + 3003*b^8*x^(8*n)))/(45045*n)

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Maple [A]  time = 0.041, size = 135, normalized size = 0.9 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{15}}{15\,n}}+{\frac{4\,a{b}^{7} \left ({x}^{n} \right ) ^{14}}{7\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{14\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{28\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{8}}{n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{7}}{7\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+7*n)*(a+b*x^n)^8,x)

[Out]

1/15*b^8/n*(x^n)^15+4/7*a*b^7/n*(x^n)^14+28/13*a^2*b^6/n*(x^n)^13+14/3*a^3*b^5/n
*(x^n)^12+70/11*a^4*b^4/n*(x^n)^11+28/5*a^5*b^3/n*(x^n)^10+28/9*a^6*b^2/n*(x^n)^
9+a^7*b/n*(x^n)^8+1/7*a^8/n*(x^n)^7

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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^(7*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239124, size = 153, normalized size = 1.02 \[ \frac{3003 \, b^{8} x^{15 \, n} + 25740 \, a b^{7} x^{14 \, n} + 97020 \, a^{2} b^{6} x^{13 \, n} + 210210 \, a^{3} b^{5} x^{12 \, n} + 286650 \, a^{4} b^{4} x^{11 \, n} + 252252 \, a^{5} b^{3} x^{10 \, n} + 140140 \, a^{6} b^{2} x^{9 \, n} + 45045 \, a^{7} b x^{8 \, n} + 6435 \, a^{8} x^{7 \, n}}{45045 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(7*n - 1),x, algorithm="fricas")

[Out]

1/45045*(3003*b^8*x^(15*n) + 25740*a*b^7*x^(14*n) + 97020*a^2*b^6*x^(13*n) + 210
210*a^3*b^5*x^(12*n) + 286650*a^4*b^4*x^(11*n) + 252252*a^5*b^3*x^(10*n) + 14014
0*a^6*b^2*x^(9*n) + 45045*a^7*b*x^(8*n) + 6435*a^8*x^(7*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+7*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{7 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(7*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(7*n - 1), x)