3.283 \(\int \frac{(a+b x^3)^5}{x^8} \, dx\)

Optimal. Leaf size=62 \[ 5 a^2 b^3 x^2-\frac{10 a^3 b^2}{x}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{7 x^7}+a b^4 x^5+\frac{b^5 x^8}{8} \]

[Out]

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

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Rubi [A]  time = 0.0231554, antiderivative size = 62, 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} \[ 5 a^2 b^3 x^2-\frac{10 a^3 b^2}{x}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{7 x^7}+a b^4 x^5+\frac{b^5 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0040037, size = 62, normalized size = 1. \[ 5 a^2 b^3 x^2-\frac{10 a^3 b^2}{x}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{7 x^7}+a b^4 x^5+\frac{b^5 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^8,x]

[Out]

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

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Maple [A]  time = 0.004, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{7\,{x}^{7}}}-{\frac{5\,{a}^{4}b}{4\,{x}^{4}}}-10\,{\frac{{a}^{3}{b}^{2}}{x}}+5\,{a}^{2}{b}^{3}{x}^{2}+a{b}^{4}{x}^{5}+{\frac{{b}^{5}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^8,x)

[Out]

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

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Maxima [A]  time = 0.968058, size = 80, normalized size = 1.29 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + a b^{4} x^{5} + 5 \, a^{2} b^{3} x^{2} - \frac{280 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 4 \, a^{5}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^8,x, algorithm="maxima")

[Out]

1/8*b^5*x^8 + a*b^4*x^5 + 5*a^2*b^3*x^2 - 1/28*(280*a^3*b^2*x^6 + 35*a^4*b*x^3 + 4*a^5)/x^7

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Fricas [A]  time = 1.55032, size = 132, normalized size = 2.13 \begin{align*} \frac{7 \, b^{5} x^{15} + 56 \, a b^{4} x^{12} + 280 \, a^{2} b^{3} x^{9} - 560 \, a^{3} b^{2} x^{6} - 70 \, a^{4} b x^{3} - 8 \, a^{5}}{56 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^8,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.508487, size = 60, normalized size = 0.97 \begin{align*} 5 a^{2} b^{3} x^{2} + a b^{4} x^{5} + \frac{b^{5} x^{8}}{8} - \frac{4 a^{5} + 35 a^{4} b x^{3} + 280 a^{3} b^{2} x^{6}}{28 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**8,x)

[Out]

5*a**2*b**3*x**2 + a*b**4*x**5 + b**5*x**8/8 - (4*a**5 + 35*a**4*b*x**3 + 280*a**3*b**2*x**6)/(28*x**7)

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Giac [A]  time = 1.1007, size = 80, normalized size = 1.29 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + a b^{4} x^{5} + 5 \, a^{2} b^{3} x^{2} - \frac{280 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 4 \, a^{5}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^8,x, algorithm="giac")

[Out]

1/8*b^5*x^8 + a*b^4*x^5 + 5*a^2*b^3*x^2 - 1/28*(280*a^3*b^2*x^6 + 35*a^4*b*x^3 + 4*a^5)/x^7