3.641 \(\int \frac{(a+b x^4)^3}{x^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{2 x^2}+\frac{1}{2} a b^2 x^6+\frac{b^3 x^{10}}{10} \]

[Out]

-a^3/(2*x^2) + (3*a^2*b*x^2)/2 + (a*b^2*x^6)/2 + (b^3*x^10)/10

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Rubi [A]  time = 0.0135031, antiderivative size = 43, 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} \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{2 x^2}+\frac{1}{2} a b^2 x^6+\frac{b^3 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^3/x^3,x]

[Out]

-a^3/(2*x^2) + (3*a^2*b*x^2)/2 + (a*b^2*x^6)/2 + (b^3*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^4\right )^3}{x^3} \, dx &=\int \left (\frac{a^3}{x^3}+3 a^2 b x+3 a b^2 x^5+b^3 x^9\right ) \, dx\\ &=-\frac{a^3}{2 x^2}+\frac{3}{2} a^2 b x^2+\frac{1}{2} a b^2 x^6+\frac{b^3 x^{10}}{10}\\ \end{align*}

Mathematica [A]  time = 0.0061005, size = 43, normalized size = 1. \[ \frac{3}{2} a^2 b x^2-\frac{a^3}{2 x^2}+\frac{1}{2} a b^2 x^6+\frac{b^3 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^3/x^3,x]

[Out]

-a^3/(2*x^2) + (3*a^2*b*x^2)/2 + (a*b^2*x^6)/2 + (b^3*x^10)/10

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{a{b}^{2}{x}^{6}}{2}}+{\frac{{b}^{3}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^3/x^3,x)

[Out]

-1/2*a^3/x^2+3/2*a^2*b*x^2+1/2*a*b^2*x^6+1/10*b^3*x^10

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Maxima [A]  time = 0.959947, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{10} \, b^{3} x^{10} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{2} \, a^{2} b x^{2} - \frac{a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="maxima")

[Out]

1/10*b^3*x^10 + 1/2*a*b^2*x^6 + 3/2*a^2*b*x^2 - 1/2*a^3/x^2

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Fricas [A]  time = 1.3749, size = 78, normalized size = 1.81 \begin{align*} \frac{b^{3} x^{12} + 5 \, a b^{2} x^{8} + 15 \, a^{2} b x^{4} - 5 \, a^{3}}{10 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="fricas")

[Out]

1/10*(b^3*x^12 + 5*a*b^2*x^8 + 15*a^2*b*x^4 - 5*a^3)/x^2

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Sympy [A]  time = 0.344856, size = 37, normalized size = 0.86 \begin{align*} - \frac{a^{3}}{2 x^{2}} + \frac{3 a^{2} b x^{2}}{2} + \frac{a b^{2} x^{6}}{2} + \frac{b^{3} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**3/x**3,x)

[Out]

-a**3/(2*x**2) + 3*a**2*b*x**2/2 + a*b**2*x**6/2 + b**3*x**10/10

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Giac [A]  time = 1.10822, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{10} \, b^{3} x^{10} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{2} \, a^{2} b x^{2} - \frac{a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="giac")

[Out]

1/10*b^3*x^10 + 1/2*a*b^2*x^6 + 3/2*a^2*b*x^2 - 1/2*a^3/x^2