3.275 \(\int x^4 (a+b x^3)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{5}{7} a^2 b^3 x^{14}+\frac{10}{11} a^3 b^2 x^{11}+\frac{5}{8} a^4 b x^8+\frac{a^5 x^5}{5}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{20}}{20} \]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^11)/11 + (5*a^2*b^3*x^14)/7 + (5*a*b^4*x^17)/17 + (b^5*x^20)/20

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Rubi [A]  time = 0.024613, antiderivative size = 69, 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} \[ \frac{5}{7} a^2 b^3 x^{14}+\frac{10}{11} a^3 b^2 x^{11}+\frac{5}{8} a^4 b x^8+\frac{a^5 x^5}{5}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^11)/11 + (5*a^2*b^3*x^14)/7 + (5*a*b^4*x^17)/17 + (b^5*x^20)/20

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 )^5 \, dx &=\int \left (a^5 x^4+5 a^4 b x^7+10 a^3 b^2 x^{10}+10 a^2 b^3 x^{13}+5 a b^4 x^{16}+b^5 x^{19}\right ) \, dx\\ &=\frac{a^5 x^5}{5}+\frac{5}{8} a^4 b x^8+\frac{10}{11} a^3 b^2 x^{11}+\frac{5}{7} a^2 b^3 x^{14}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{20}}{20}\\ \end{align*}

Mathematica [A]  time = 0.0020652, size = 69, normalized size = 1. \[ \frac{5}{7} a^2 b^3 x^{14}+\frac{10}{11} a^3 b^2 x^{11}+\frac{5}{8} a^4 b x^8+\frac{a^5 x^5}{5}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^11)/11 + (5*a^2*b^3*x^14)/7 + (5*a*b^4*x^17)/17 + (b^5*x^20)/20

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Maple [A]  time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{11}}{11}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{14}}{7}}+{\frac{5\,a{b}^{4}{x}^{17}}{17}}+{\frac{{b}^{5}{x}^{20}}{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^3+a)^5,x)

[Out]

1/5*a^5*x^5+5/8*a^4*b*x^8+10/11*a^3*b^2*x^11+5/7*a^2*b^3*x^14+5/17*a*b^4*x^17+1/20*b^5*x^20

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Maxima [A]  time = 0.999348, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/20*b^5*x^20 + 5/17*a*b^4*x^17 + 5/7*a^2*b^3*x^14 + 10/11*a^3*b^2*x^11 + 5/8*a^4*b*x^8 + 1/5*a^5*x^5

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Fricas [A]  time = 1.49457, size = 139, normalized size = 2.01 \begin{align*} \frac{1}{20} x^{20} b^{5} + \frac{5}{17} x^{17} b^{4} a + \frac{5}{7} x^{14} b^{3} a^{2} + \frac{10}{11} x^{11} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{5} x^{5} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/20*x^20*b^5 + 5/17*x^17*b^4*a + 5/7*x^14*b^3*a^2 + 10/11*x^11*b^2*a^3 + 5/8*x^8*b*a^4 + 1/5*x^5*a^5

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Sympy [A]  time = 0.099232, size = 66, normalized size = 0.96 \begin{align*} \frac{a^{5} x^{5}}{5} + \frac{5 a^{4} b x^{8}}{8} + \frac{10 a^{3} b^{2} x^{11}}{11} + \frac{5 a^{2} b^{3} x^{14}}{7} + \frac{5 a b^{4} x^{17}}{17} + \frac{b^{5} x^{20}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**3+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**8/8 + 10*a**3*b**2*x**11/11 + 5*a**2*b**3*x**14/7 + 5*a*b**4*x**17/17 + b**5*x**20/2
0

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Giac [A]  time = 1.10621, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/20*b^5*x^20 + 5/17*a*b^4*x^17 + 5/7*a^2*b^3*x^14 + 10/11*a^3*b^2*x^11 + 5/8*a^4*b*x^8 + 1/5*a^5*x^5