3.310 \(\int x (a+b x^3)^8 \, dx\)

Optimal. Leaf size=106 \[ \frac{7}{5} a^2 b^6 x^{20}+\frac{56}{17} a^3 b^5 x^{17}+5 a^4 b^4 x^{14}+\frac{56}{11} a^5 b^3 x^{11}+\frac{7}{2} a^6 b^2 x^8+\frac{8}{5} a^7 b x^5+\frac{a^8 x^2}{2}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{26}}{26} \]

[Out]

(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/
17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^23)/23 + (b^8*x^26)/26

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Rubi [A]  time = 0.038971, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{7}{5} a^2 b^6 x^{20}+\frac{56}{17} a^3 b^5 x^{17}+5 a^4 b^4 x^{14}+\frac{56}{11} a^5 b^3 x^{11}+\frac{7}{2} a^6 b^2 x^8+\frac{8}{5} a^7 b x^5+\frac{a^8 x^2}{2}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{26}}{26} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^8,x]

[Out]

(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/
17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^23)/23 + (b^8*x^26)/26

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (a+b x^3\right )^8 \, dx &=\int \left (a^8 x+8 a^7 b x^4+28 a^6 b^2 x^7+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{13}+56 a^3 b^5 x^{16}+28 a^2 b^6 x^{19}+8 a b^7 x^{22}+b^8 x^{25}\right ) \, dx\\ &=\frac{a^8 x^2}{2}+\frac{8}{5} a^7 b x^5+\frac{7}{2} a^6 b^2 x^8+\frac{56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac{56}{17} a^3 b^5 x^{17}+\frac{7}{5} a^2 b^6 x^{20}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{26}}{26}\\ \end{align*}

Mathematica [A]  time = 0.0026069, size = 106, normalized size = 1. \[ \frac{7}{5} a^2 b^6 x^{20}+\frac{56}{17} a^3 b^5 x^{17}+5 a^4 b^4 x^{14}+\frac{56}{11} a^5 b^3 x^{11}+\frac{7}{2} a^6 b^2 x^8+\frac{8}{5} a^7 b x^5+\frac{a^8 x^2}{2}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{26}}{26} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^8,x]

[Out]

(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/
17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^23)/23 + (b^8*x^26)/26

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Maple [A]  time = 0.001, size = 91, normalized size = 0.9 \begin{align*}{\frac{{a}^{8}{x}^{2}}{2}}+{\frac{8\,{a}^{7}b{x}^{5}}{5}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{8}}{2}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{11}}{11}}+5\,{a}^{4}{b}^{4}{x}^{14}+{\frac{56\,{a}^{3}{b}^{5}{x}^{17}}{17}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{20}}{5}}+{\frac{8\,a{b}^{7}{x}^{23}}{23}}+{\frac{{b}^{8}{x}^{26}}{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a)^8,x)

[Out]

1/2*a^8*x^2+8/5*a^7*b*x^5+7/2*a^6*b^2*x^8+56/11*a^5*b^3*x^11+5*a^4*b^4*x^14+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^2
0+8/23*a*b^7*x^23+1/26*b^8*x^26

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Maxima [A]  time = 0.964522, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{26} \, b^{8} x^{26} + \frac{8}{23} \, a b^{7} x^{23} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac{56}{11} \, a^{5} b^{3} x^{11} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{8}{5} \, a^{7} b x^{5} + \frac{1}{2} \, a^{8} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11
+ 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/2*a^8*x^2

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Fricas [A]  time = 1.69437, size = 215, normalized size = 2.03 \begin{align*} \frac{1}{26} x^{26} b^{8} + \frac{8}{23} x^{23} b^{7} a + \frac{7}{5} x^{20} b^{6} a^{2} + \frac{56}{17} x^{17} b^{5} a^{3} + 5 x^{14} b^{4} a^{4} + \frac{56}{11} x^{11} b^{3} a^{5} + \frac{7}{2} x^{8} b^{2} a^{6} + \frac{8}{5} x^{5} b a^{7} + \frac{1}{2} x^{2} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/26*x^26*b^8 + 8/23*x^23*b^7*a + 7/5*x^20*b^6*a^2 + 56/17*x^17*b^5*a^3 + 5*x^14*b^4*a^4 + 56/11*x^11*b^3*a^5
+ 7/2*x^8*b^2*a^6 + 8/5*x^5*b*a^7 + 1/2*x^2*a^8

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Sympy [A]  time = 0.096183, size = 105, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{2}}{2} + \frac{8 a^{7} b x^{5}}{5} + \frac{7 a^{6} b^{2} x^{8}}{2} + \frac{56 a^{5} b^{3} x^{11}}{11} + 5 a^{4} b^{4} x^{14} + \frac{56 a^{3} b^{5} x^{17}}{17} + \frac{7 a^{2} b^{6} x^{20}}{5} + \frac{8 a b^{7} x^{23}}{23} + \frac{b^{8} x^{26}}{26} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**3+a)**8,x)

[Out]

a**8*x**2/2 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**8/2 + 56*a**5*b**3*x**11/11 + 5*a**4*b**4*x**14 + 56*a**3*b**5*
x**17/17 + 7*a**2*b**6*x**20/5 + 8*a*b**7*x**23/23 + b**8*x**26/26

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Giac [A]  time = 1.3652, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{26} \, b^{8} x^{26} + \frac{8}{23} \, a b^{7} x^{23} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac{56}{11} \, a^{5} b^{3} x^{11} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{8}{5} \, a^{7} b x^{5} + \frac{1}{2} \, a^{8} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11
+ 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/2*a^8*x^2