3.284 \(\int \frac{(a+b x^3)^5}{x^9} \, dx\)

Optimal. Leaf size=61 \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 x-\frac{a^4 b}{x^5}-\frac{a^5}{8 x^8}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^7}{7} \]

[Out]

-a^5/(8*x^8) - (a^4*b)/x^5 - (5*a^3*b^2)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4 + (b^5*x^7)/7

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Rubi [A]  time = 0.0224884, antiderivative size = 61, 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} \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 x-\frac{a^4 b}{x^5}-\frac{a^5}{8 x^8}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^9,x]

[Out]

-a^5/(8*x^8) - (a^4*b)/x^5 - (5*a^3*b^2)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4 + (b^5*x^7)/7

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 )^5}{x^9} \, dx &=\int \left (10 a^2 b^3+\frac{a^5}{x^9}+\frac{5 a^4 b}{x^6}+\frac{10 a^3 b^2}{x^3}+5 a b^4 x^3+b^5 x^6\right ) \, dx\\ &=-\frac{a^5}{8 x^8}-\frac{a^4 b}{x^5}-\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 x+\frac{5}{4} a b^4 x^4+\frac{b^5 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0039244, size = 61, normalized size = 1. \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 x-\frac{a^4 b}{x^5}-\frac{a^5}{8 x^8}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^9,x]

[Out]

-a^5/(8*x^8) - (a^4*b)/x^5 - (5*a^3*b^2)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4 + (b^5*x^7)/7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^9,x)

[Out]

-1/8*a^5/x^8-a^4*b/x^5-5*a^3*b^2/x^2+10*a^2*b^3*x+5/4*a*b^4*x^4+1/7*b^5*x^7

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Maxima [A]  time = 0.964053, size = 76, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + \frac{5}{4} \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x - \frac{40 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} + a^{5}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="maxima")

[Out]

1/7*b^5*x^7 + 5/4*a*b^4*x^4 + 10*a^2*b^3*x - 1/8*(40*a^3*b^2*x^6 + 8*a^4*b*x^3 + a^5)/x^8

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Fricas [A]  time = 1.56072, size = 132, normalized size = 2.16 \begin{align*} \frac{8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="fricas")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*x^3 - 7*a^5)/x^8

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Sympy [A]  time = 0.532178, size = 60, normalized size = 0.98 \begin{align*} 10 a^{2} b^{3} x + \frac{5 a b^{4} x^{4}}{4} + \frac{b^{5} x^{7}}{7} - \frac{a^{5} + 8 a^{4} b x^{3} + 40 a^{3} b^{2} x^{6}}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**9,x)

[Out]

10*a**2*b**3*x + 5*a*b**4*x**4/4 + b**5*x**7/7 - (a**5 + 8*a**4*b*x**3 + 40*a**3*b**2*x**6)/(8*x**8)

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Giac [A]  time = 1.10539, size = 76, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + \frac{5}{4} \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x - \frac{40 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} + a^{5}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="giac")

[Out]

1/7*b^5*x^7 + 5/4*a*b^4*x^4 + 10*a^2*b^3*x - 1/8*(40*a^3*b^2*x^6 + 8*a^4*b*x^3 + a^5)/x^8