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

Optimal. Leaf size=63 \[ 2 a^2 b^3 x^5+5 a^3 b^2 x^2-\frac{5 a^4 b}{x}-\frac{a^5}{4 x^4}+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (b^5*x^11)/11

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Rubi [A]  time = 0.0234461, antiderivative size = 63, 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} \[ 2 a^2 b^3 x^5+5 a^3 b^2 x^2-\frac{5 a^4 b}{x}-\frac{a^5}{4 x^4}+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (b^5*x^11)/11

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

Mathematica [A]  time = 0.0059232, size = 63, normalized size = 1. \[ 2 a^2 b^3 x^5+5 a^3 b^2 x^2-\frac{5 a^4 b}{x}-\frac{a^5}{4 x^4}+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (b^5*x^11)/11

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{4\,{x}^{4}}}-5\,{\frac{{a}^{4}b}{x}}+5\,{a}^{3}{b}^{2}{x}^{2}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^5/x^4-5*a^4*b/x+5*a^3*b^2*x^2+2*a^2*b^3*x^5+5/8*a*b^4*x^8+1/11*b^5*x^11

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Maxima [A]  time = 0.945505, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac{20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^3 + a^5)/x^4

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Fricas [A]  time = 1.68098, size = 135, normalized size = 2.14 \begin{align*} \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/88*(8*b^5*x^15 + 55*a*b^4*x^12 + 176*a^2*b^3*x^9 + 440*a^3*b^2*x^6 - 440*a^4*b*x^3 - 22*a^5)/x^4

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Sympy [A]  time = 0.430648, size = 61, normalized size = 0.97 \begin{align*} 5 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{11}}{11} - \frac{a^{5} + 20 a^{4} b x^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**3*b**2*x**2 + 2*a**2*b**3*x**5 + 5*a*b**4*x**8/8 + b**5*x**11/11 - (a**5 + 20*a**4*b*x**3)/(4*x**4)

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Giac [A]  time = 1.14624, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac{20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^3 + a^5)/x^4