3.317 \(\int \frac{(a+b x^3)^8}{x^9} \, dx\)

Optimal. Leaf size=100 \[ \frac{14}{5} a^2 b^6 x^{10}+8 a^3 b^5 x^7+\frac{35}{2} a^4 b^4 x^4-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 x-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{8 x^8}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{16}}{16} \]

[Out]

-a^8/(8*x^8) - (8*a^7*b)/(5*x^5) - (14*a^6*b^2)/x^2 + 56*a^5*b^3*x + (35*a^4*b^4*x^4)/2 + 8*a^3*b^5*x^7 + (14*
a^2*b^6*x^10)/5 + (8*a*b^7*x^13)/13 + (b^8*x^16)/16

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Rubi [A]  time = 0.0358057, antiderivative size = 100, 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} \[ \frac{14}{5} a^2 b^6 x^{10}+8 a^3 b^5 x^7+\frac{35}{2} a^4 b^4 x^4-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 x-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{8 x^8}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^9,x]

[Out]

-a^8/(8*x^8) - (8*a^7*b)/(5*x^5) - (14*a^6*b^2)/x^2 + 56*a^5*b^3*x + (35*a^4*b^4*x^4)/2 + 8*a^3*b^5*x^7 + (14*
a^2*b^6*x^10)/5 + (8*a*b^7*x^13)/13 + (b^8*x^16)/16

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 \frac{\left (a+b x^3\right )^8}{x^9} \, dx &=\int \left (56 a^5 b^3+\frac{a^8}{x^9}+\frac{8 a^7 b}{x^6}+\frac{28 a^6 b^2}{x^3}+70 a^4 b^4 x^3+56 a^3 b^5 x^6+28 a^2 b^6 x^9+8 a b^7 x^{12}+b^8 x^{15}\right ) \, dx\\ &=-\frac{a^8}{8 x^8}-\frac{8 a^7 b}{5 x^5}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 x+\frac{35}{2} a^4 b^4 x^4+8 a^3 b^5 x^7+\frac{14}{5} a^2 b^6 x^{10}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{16}}{16}\\ \end{align*}

Mathematica [A]  time = 0.0045352, size = 100, normalized size = 1. \[ \frac{14}{5} a^2 b^6 x^{10}+8 a^3 b^5 x^7+\frac{35}{2} a^4 b^4 x^4-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 x-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{8 x^8}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^9,x]

[Out]

-a^8/(8*x^8) - (8*a^7*b)/(5*x^5) - (14*a^6*b^2)/x^2 + 56*a^5*b^3*x + (35*a^4*b^4*x^4)/2 + 8*a^3*b^5*x^7 + (14*
a^2*b^6*x^10)/5 + (8*a*b^7*x^13)/13 + (b^8*x^16)/16

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{8\,{x}^{8}}}-{\frac{8\,{a}^{7}b}{5\,{x}^{5}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}+56\,{a}^{5}{b}^{3}x+{\frac{35\,{a}^{4}{b}^{4}{x}^{4}}{2}}+8\,{a}^{3}{b}^{5}{x}^{7}+{\frac{14\,{a}^{2}{b}^{6}{x}^{10}}{5}}+{\frac{8\,a{b}^{7}{x}^{13}}{13}}+{\frac{{b}^{8}{x}^{16}}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^9,x)

[Out]

-1/8*a^8/x^8-8/5*a^7*b/x^5-14*a^6*b^2/x^2+56*a^5*b^3*x+35/2*a^4*b^4*x^4+8*a^3*b^5*x^7+14/5*a^2*b^6*x^10+8/13*a
*b^7*x^13+1/16*b^8*x^16

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Maxima [A]  time = 0.953487, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{8}{13} \, a b^{7} x^{13} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 8 \, a^{3} b^{5} x^{7} + \frac{35}{2} \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x - \frac{560 \, a^{6} b^{2} x^{6} + 64 \, a^{7} b x^{3} + 5 \, a^{8}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="maxima")

[Out]

1/16*b^8*x^16 + 8/13*a*b^7*x^13 + 14/5*a^2*b^6*x^10 + 8*a^3*b^5*x^7 + 35/2*a^4*b^4*x^4 + 56*a^5*b^3*x - 1/40*(
560*a^6*b^2*x^6 + 64*a^7*b*x^3 + 5*a^8)/x^8

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Fricas [A]  time = 1.56741, size = 231, normalized size = 2.31 \begin{align*} \frac{65 \, b^{8} x^{24} + 640 \, a b^{7} x^{21} + 2912 \, a^{2} b^{6} x^{18} + 8320 \, a^{3} b^{5} x^{15} + 18200 \, a^{4} b^{4} x^{12} + 58240 \, a^{5} b^{3} x^{9} - 14560 \, a^{6} b^{2} x^{6} - 1664 \, a^{7} b x^{3} - 130 \, a^{8}}{1040 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="fricas")

[Out]

1/1040*(65*b^8*x^24 + 640*a*b^7*x^21 + 2912*a^2*b^6*x^18 + 8320*a^3*b^5*x^15 + 18200*a^4*b^4*x^12 + 58240*a^5*
b^3*x^9 - 14560*a^6*b^2*x^6 - 1664*a^7*b*x^3 - 130*a^8)/x^8

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Sympy [A]  time = 0.582506, size = 100, normalized size = 1. \begin{align*} 56 a^{5} b^{3} x + \frac{35 a^{4} b^{4} x^{4}}{2} + 8 a^{3} b^{5} x^{7} + \frac{14 a^{2} b^{6} x^{10}}{5} + \frac{8 a b^{7} x^{13}}{13} + \frac{b^{8} x^{16}}{16} - \frac{5 a^{8} + 64 a^{7} b x^{3} + 560 a^{6} b^{2} x^{6}}{40 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**9,x)

[Out]

56*a**5*b**3*x + 35*a**4*b**4*x**4/2 + 8*a**3*b**5*x**7 + 14*a**2*b**6*x**10/5 + 8*a*b**7*x**13/13 + b**8*x**1
6/16 - (5*a**8 + 64*a**7*b*x**3 + 560*a**6*b**2*x**6)/(40*x**8)

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Giac [A]  time = 1.11585, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{8}{13} \, a b^{7} x^{13} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 8 \, a^{3} b^{5} x^{7} + \frac{35}{2} \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x - \frac{560 \, a^{6} b^{2} x^{6} + 64 \, a^{7} b x^{3} + 5 \, a^{8}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="giac")

[Out]

1/16*b^8*x^16 + 8/13*a*b^7*x^13 + 14/5*a^2*b^6*x^10 + 8*a^3*b^5*x^7 + 35/2*a^4*b^4*x^4 + 56*a^5*b^3*x - 1/40*(
560*a^6*b^2*x^6 + 64*a^7*b*x^3 + 5*a^8)/x^8