3.2571 \(\int x^{-1+4 n} (a+b x^n)^8 \, dx\)

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]

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

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Rubi [A]  time = 0.05, antiderivative size = 84, 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} \[ \frac {3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac {a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac {\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac {3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]

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)

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+4 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^8}{b^3}+\frac {3 a^2 (a+b x)^9}{b^3}-\frac {3 a (a+b x)^{10}}{b^3}+\frac {(a+b x)^{11}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\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}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 53, normalized size = 0.63 \[ -\frac {\left (a+b x^n\right )^9 \left (a^3-9 a^2 b x^n+45 a b^2 x^{2 n}-165 b^3 x^{3 n}\right )}{1980 b^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1980*((a + b*x^n)^9*(a^3 - 9*a^2*b*x^n + 45*a*b^2*x^(2*n) - 165*b^3*x^(3*n)))/(b^4*n)

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fricas [A]  time = 0.72, size = 113, normalized size = 1.35 \[ \frac {165 \, b^{8} x^{12 \, n} + 1440 \, a b^{7} x^{11 \, n} + 5544 \, a^{2} b^{6} x^{10 \, n} + 12320 \, a^{3} b^{5} x^{9 \, n} + 17325 \, a^{4} b^{4} x^{8 \, n} + 15840 \, a^{5} b^{3} x^{7 \, n} + 9240 \, a^{6} b^{2} x^{6 \, n} + 3168 \, a^{7} b x^{5 \, n} + 495 \, a^{8} x^{4 \, n}}{1980 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^8,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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{8} x^{4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 136, normalized size = 1.62 \[ \frac {a^{8} x^{4 n}}{4 n}+\frac {8 a^{7} b \,x^{5 n}}{5 n}+\frac {14 a^{6} b^{2} x^{6 n}}{3 n}+\frac {8 a^{5} b^{3} x^{7 n}}{n}+\frac {35 a^{4} b^{4} x^{8 n}}{4 n}+\frac {56 a^{3} b^{5} x^{9 n}}{9 n}+\frac {14 a^{2} b^{6} x^{10 n}}{5 n}+\frac {8 a \,b^{7} x^{11 n}}{11 n}+\frac {b^{8} x^{12 n}}{12 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+4*n)*(b*x^n+a)^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 [A]  time = 0.47, size = 135, normalized size = 1.61 \[ \frac {b^{8} x^{12 \, n}}{12 \, n} + \frac {8 \, a b^{7} x^{11 \, n}}{11 \, n} + \frac {14 \, a^{2} b^{6} x^{10 \, n}}{5 \, n} + \frac {56 \, a^{3} b^{5} x^{9 \, n}}{9 \, n} + \frac {35 \, a^{4} b^{4} x^{8 \, n}}{4 \, n} + \frac {8 \, a^{5} b^{3} x^{7 \, n}}{n} + \frac {14 \, a^{6} b^{2} x^{6 \, n}}{3 \, n} + \frac {8 \, a^{7} b x^{5 \, n}}{5 \, n} + \frac {a^{8} x^{4 \, n}}{4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.46, size = 135, normalized size = 1.61 \[ \frac {a^8\,x^{4\,n}}{4\,n}+\frac {b^8\,x^{12\,n}}{12\,n}+\frac {14\,a^6\,b^2\,x^{6\,n}}{3\,n}+\frac {8\,a^5\,b^3\,x^{7\,n}}{n}+\frac {35\,a^4\,b^4\,x^{8\,n}}{4\,n}+\frac {56\,a^3\,b^5\,x^{9\,n}}{9\,n}+\frac {14\,a^2\,b^6\,x^{10\,n}}{5\,n}+\frac {8\,a^7\,b\,x^{5\,n}}{5\,n}+\frac {8\,a\,b^7\,x^{11\,n}}{11\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(4*n - 1)*(a + b*x^n)^8,x)

[Out]

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

[Out]

Timed out

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