3.308 \(\int x^4 (a+b x^3)^8 \, dx\)

Optimal. Leaf size=103 \[ \frac{28}{23} a^2 b^6 x^{23}+\frac{14}{5} a^3 b^5 x^{20}+\frac{70}{17} a^4 b^4 x^{17}+4 a^5 b^3 x^{14}+\frac{28}{11} a^6 b^2 x^{11}+a^7 b x^8+\frac{a^8 x^5}{5}+\frac{4}{13} a b^7 x^{26}+\frac{b^8 x^{29}}{29} \]

[Out]

(a^8*x^5)/5 + a^7*b*x^8 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 +
(28*a^2*b^6*x^23)/23 + (4*a*b^7*x^26)/13 + (b^8*x^29)/29

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Rubi [A]  time = 0.0449214, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{28}{23} a^2 b^6 x^{23}+\frac{14}{5} a^3 b^5 x^{20}+\frac{70}{17} a^4 b^4 x^{17}+4 a^5 b^3 x^{14}+\frac{28}{11} a^6 b^2 x^{11}+a^7 b x^8+\frac{a^8 x^5}{5}+\frac{4}{13} a b^7 x^{26}+\frac{b^8 x^{29}}{29} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^8,x]

[Out]

(a^8*x^5)/5 + a^7*b*x^8 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 +
(28*a^2*b^6*x^23)/23 + (4*a*b^7*x^26)/13 + (b^8*x^29)/29

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^4 \left (a+b x^3\right )^8 \, dx &=\int \left (a^8 x^4+8 a^7 b x^7+28 a^6 b^2 x^{10}+56 a^5 b^3 x^{13}+70 a^4 b^4 x^{16}+56 a^3 b^5 x^{19}+28 a^2 b^6 x^{22}+8 a b^7 x^{25}+b^8 x^{28}\right ) \, dx\\ &=\frac{a^8 x^5}{5}+a^7 b x^8+\frac{28}{11} a^6 b^2 x^{11}+4 a^5 b^3 x^{14}+\frac{70}{17} a^4 b^4 x^{17}+\frac{14}{5} a^3 b^5 x^{20}+\frac{28}{23} a^2 b^6 x^{23}+\frac{4}{13} a b^7 x^{26}+\frac{b^8 x^{29}}{29}\\ \end{align*}

Mathematica [A]  time = 0.0029223, size = 103, normalized size = 1. \[ \frac{28}{23} a^2 b^6 x^{23}+\frac{14}{5} a^3 b^5 x^{20}+\frac{70}{17} a^4 b^4 x^{17}+4 a^5 b^3 x^{14}+\frac{28}{11} a^6 b^2 x^{11}+a^7 b x^8+\frac{a^8 x^5}{5}+\frac{4}{13} a b^7 x^{26}+\frac{b^8 x^{29}}{29} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^8,x]

[Out]

(a^8*x^5)/5 + a^7*b*x^8 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 +
(28*a^2*b^6*x^23)/23 + (4*a*b^7*x^26)/13 + (b^8*x^29)/29

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Maple [A]  time = 0., size = 90, normalized size = 0.9 \begin{align*}{\frac{{a}^{8}{x}^{5}}{5}}+{a}^{7}b{x}^{8}+{\frac{28\,{a}^{6}{b}^{2}{x}^{11}}{11}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{70\,{a}^{4}{b}^{4}{x}^{17}}{17}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{20}}{5}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{23}}{23}}+{\frac{4\,a{b}^{7}{x}^{26}}{13}}+{\frac{{b}^{8}{x}^{29}}{29}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^3+a)^8,x)

[Out]

1/5*a^8*x^5+a^7*b*x^8+28/11*a^6*b^2*x^11+4*a^5*b^3*x^14+70/17*a^4*b^4*x^17+14/5*a^3*b^5*x^20+28/23*a^2*b^6*x^2
3+4/13*a*b^7*x^26+1/29*b^8*x^29

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Maxima [A]  time = 0.985057, size = 120, normalized size = 1.17 \begin{align*} \frac{1}{29} \, b^{8} x^{29} + \frac{4}{13} \, a b^{7} x^{26} + \frac{28}{23} \, a^{2} b^{6} x^{23} + \frac{14}{5} \, a^{3} b^{5} x^{20} + \frac{70}{17} \, a^{4} b^{4} x^{17} + 4 \, a^{5} b^{3} x^{14} + \frac{28}{11} \, a^{6} b^{2} x^{11} + a^{7} b x^{8} + \frac{1}{5} \, a^{8} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/29*b^8*x^29 + 4/13*a*b^7*x^26 + 28/23*a^2*b^6*x^23 + 14/5*a^3*b^5*x^20 + 70/17*a^4*b^4*x^17 + 4*a^5*b^3*x^14
+ 28/11*a^6*b^2*x^11 + a^7*b*x^8 + 1/5*a^8*x^5

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Fricas [A]  time = 1.56245, size = 215, normalized size = 2.09 \begin{align*} \frac{1}{29} x^{29} b^{8} + \frac{4}{13} x^{26} b^{7} a + \frac{28}{23} x^{23} b^{6} a^{2} + \frac{14}{5} x^{20} b^{5} a^{3} + \frac{70}{17} x^{17} b^{4} a^{4} + 4 x^{14} b^{3} a^{5} + \frac{28}{11} x^{11} b^{2} a^{6} + x^{8} b a^{7} + \frac{1}{5} x^{5} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/29*x^29*b^8 + 4/13*x^26*b^7*a + 28/23*x^23*b^6*a^2 + 14/5*x^20*b^5*a^3 + 70/17*x^17*b^4*a^4 + 4*x^14*b^3*a^5
+ 28/11*x^11*b^2*a^6 + x^8*b*a^7 + 1/5*x^5*a^8

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Sympy [A]  time = 0.09773, size = 102, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{5}}{5} + a^{7} b x^{8} + \frac{28 a^{6} b^{2} x^{11}}{11} + 4 a^{5} b^{3} x^{14} + \frac{70 a^{4} b^{4} x^{17}}{17} + \frac{14 a^{3} b^{5} x^{20}}{5} + \frac{28 a^{2} b^{6} x^{23}}{23} + \frac{4 a b^{7} x^{26}}{13} + \frac{b^{8} x^{29}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**3+a)**8,x)

[Out]

a**8*x**5/5 + a**7*b*x**8 + 28*a**6*b**2*x**11/11 + 4*a**5*b**3*x**14 + 70*a**4*b**4*x**17/17 + 14*a**3*b**5*x
**20/5 + 28*a**2*b**6*x**23/23 + 4*a*b**7*x**26/13 + b**8*x**29/29

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Giac [A]  time = 1.12662, size = 120, normalized size = 1.17 \begin{align*} \frac{1}{29} \, b^{8} x^{29} + \frac{4}{13} \, a b^{7} x^{26} + \frac{28}{23} \, a^{2} b^{6} x^{23} + \frac{14}{5} \, a^{3} b^{5} x^{20} + \frac{70}{17} \, a^{4} b^{4} x^{17} + 4 \, a^{5} b^{3} x^{14} + \frac{28}{11} \, a^{6} b^{2} x^{11} + a^{7} b x^{8} + \frac{1}{5} \, a^{8} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/29*b^8*x^29 + 4/13*a*b^7*x^26 + 28/23*a^2*b^6*x^23 + 14/5*a^3*b^5*x^20 + 70/17*a^4*b^4*x^17 + 4*a^5*b^3*x^14
+ 28/11*a^6*b^2*x^11 + a^7*b*x^8 + 1/5*a^8*x^5