3.282 \(\int \frac{(a+b x^3)^5}{x^6} \, dx\)

Optimal. Leaf size=65 \[ \frac{5}{2} a^2 b^3 x^4+10 a^3 b^2 x-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{5 x^5}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.0213671, antiderivative size = 65, 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} \[ \frac{5}{2} a^2 b^3 x^4+10 a^3 b^2 x-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{5 x^5}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0039056, size = 65, normalized size = 1. \[ \frac{5}{2} a^2 b^3 x^4+10 a^3 b^2 x-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{5 x^5}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^6,x]

[Out]

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

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Maple [A]  time = 0.007, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}+10\,{a}^{3}{b}^{2}x+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{{b}^{5}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^6,x)

[Out]

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

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Maxima [A]  time = 0.982891, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac{25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="maxima")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*x^3 + 2*a^5)/x^5

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Fricas [A]  time = 1.61169, size = 135, normalized size = 2.08 \begin{align*} \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="fricas")

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b*x^3 - 14*a^5)/x^5

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Sympy [A]  time = 0.440089, size = 63, normalized size = 0.97 \begin{align*} 10 a^{3} b^{2} x + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{10}}{10} - \frac{2 a^{5} + 25 a^{4} b x^{3}}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**6,x)

[Out]

10*a**3*b**2*x + 5*a**2*b**3*x**4/2 + 5*a*b**4*x**7/7 + b**5*x**10/10 - (2*a**5 + 25*a**4*b*x**3)/(10*x**5)

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Giac [A]  time = 1.10208, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac{25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="giac")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*x^3 + 2*a^5)/x^5