3.2568 \(\int x^{-1+7 n} (a+b x^n)^8 \, dx\)

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]

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

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Rubi [A]  time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}-\frac {5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac {15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac {3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac {a^6 \left (a+b x^n\right )^9}{9 b^7 n}+\frac {\left (a+b x^n\right )^{15}}{15 b^7 n}-\frac {3 a \left (a+b x^n\right )^{14}}{7 b^7 n} \]

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)^13)/(13*b^7*n) - (3*a*(a + b*x^n)^14)/(7*b^7*n) + (a + b*x^n)
^15/(15*b^7*n)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^6 (a+b x)^8}{b^6}-\frac {6 a^5 (a+b x)^9}{b^6}+\frac {15 a^4 (a+b x)^{10}}{b^6}-\frac {20 a^3 (a+b x)^{11}}{b^6}+\frac {15 a^2 (a+b x)^{12}}{b^6}-\frac {6 a (a+b x)^{13}}{b^6}+\frac {(a+b x)^{14}}{b^6}\right ) \, dx,x,x^n\right )}{n}\\ &=\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 {3 a \left (a+b x^n\right )^{14}}{7 b^7 n}+\frac {\left (a+b x^n\right )^{15}}{15 b^7 n}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 127, normalized size = 0.85 \[ \frac {\frac {1}{7} a^8 x^{7 n}+a^7 b x^{8 n}+\frac {28}{9} a^6 b^2 x^{9 n}+\frac {28}{5} a^5 b^3 x^{10 n}+\frac {70}{11} a^4 b^4 x^{11 n}+\frac {14}{3} a^3 b^5 x^{12 n}+\frac {28}{13} a^2 b^6 x^{13 n}+\frac {4}{7} a b^7 x^{14 n}+\frac {1}{15} b^8 x^{15 n}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 113, normalized size = 0.75 \[ \frac {3003 \, b^{8} x^{15 \, n} + 25740 \, a b^{7} x^{14 \, n} + 97020 \, a^{2} b^{6} x^{13 \, n} + 210210 \, a^{3} b^{5} x^{12 \, n} + 286650 \, a^{4} b^{4} x^{11 \, n} + 252252 \, a^{5} b^{3} x^{10 \, n} + 140140 \, a^{6} b^{2} x^{9 \, n} + 45045 \, a^{7} b x^{8 \, n} + 6435 \, a^{8} x^{7 \, n}}{45045 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{8} x^{7 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 135, normalized size = 0.90 \[ \frac {a^{8} x^{7 n}}{7 n}+\frac {a^{7} b \,x^{8 n}}{n}+\frac {28 a^{6} b^{2} x^{9 n}}{9 n}+\frac {28 a^{5} b^{3} x^{10 n}}{5 n}+\frac {70 a^{4} b^{4} x^{11 n}}{11 n}+\frac {14 a^{3} b^{5} x^{12 n}}{3 n}+\frac {28 a^{2} b^{6} x^{13 n}}{13 n}+\frac {4 a \,b^{7} x^{14 n}}{7 n}+\frac {b^{8} x^{15 n}}{15 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+7*n)*(b*x^n+a)^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 [A]  time = 0.67, size = 134, normalized size = 0.89 \[ \frac {b^{8} x^{15 \, n}}{15 \, n} + \frac {4 \, a b^{7} x^{14 \, n}}{7 \, n} + \frac {28 \, a^{2} b^{6} x^{13 \, n}}{13 \, n} + \frac {14 \, a^{3} b^{5} x^{12 \, n}}{3 \, n} + \frac {70 \, a^{4} b^{4} x^{11 \, n}}{11 \, n} + \frac {28 \, a^{5} b^{3} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{6} b^{2} x^{9 \, n}}{9 \, n} + \frac {a^{7} b x^{8 \, n}}{n} + \frac {a^{8} x^{7 \, n}}{7 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+7*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.47, size = 134, normalized size = 0.89 \[ \frac {a^8\,x^{7\,n}}{7\,n}+\frac {b^8\,x^{15\,n}}{15\,n}+\frac {28\,a^6\,b^2\,x^{9\,n}}{9\,n}+\frac {28\,a^5\,b^3\,x^{10\,n}}{5\,n}+\frac {70\,a^4\,b^4\,x^{11\,n}}{11\,n}+\frac {14\,a^3\,b^5\,x^{12\,n}}{3\,n}+\frac {28\,a^2\,b^6\,x^{13\,n}}{13\,n}+\frac {a^7\,b\,x^{8\,n}}{n}+\frac {4\,a\,b^7\,x^{14\,n}}{7\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7*n - 1)*(a + b*x^n)^8,x)

[Out]

(a^8*x^(7*n))/(7*n) + (b^8*x^(15*n))/(15*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) + (a^7*b*x^(8*n))/n + (4*
a*b^7*x^(14*n))/(7*n)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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