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

Optimal. Leaf size=39 \[ 3 a^2 b x-\frac{a^3}{3 x^3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

________________________________________________________________________________________

Rubi [A]  time = 0.0139555, antiderivative size = 39, 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} \[ 3 a^2 b x-\frac{a^3}{3 x^3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^3/x^4,x]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

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

Mathematica [A]  time = 0.003504, size = 39, normalized size = 1. \[ 3 a^2 b x-\frac{a^3}{3 x^3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^3/x^4,x]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{3\,{x}^{3}}}+3\,{a}^{2}bx+{\frac{3\,{x}^{5}a{b}^{2}}{5}}+{\frac{{b}^{3}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^3/x^3+3*a^2*b*x+3/5*x^5*a*b^2+1/9*b^3*x^9

________________________________________________________________________________________

Maxima [A]  time = 0.969767, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{9} \, b^{3} x^{9} + \frac{3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac{a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*x^9 + 3/5*a*b^2*x^5 + 3*a^2*b*x - 1/3*a^3/x^3

________________________________________________________________________________________

Fricas [A]  time = 1.4271, size = 85, normalized size = 2.18 \begin{align*} \frac{5 \, b^{3} x^{12} + 27 \, a b^{2} x^{8} + 135 \, a^{2} b x^{4} - 15 \, a^{3}}{45 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(5*b^3*x^12 + 27*a*b^2*x^8 + 135*a^2*b*x^4 - 15*a^3)/x^3

________________________________________________________________________________________

Sympy [A]  time = 0.354084, size = 36, normalized size = 0.92 \begin{align*} - \frac{a^{3}}{3 x^{3}} + 3 a^{2} b x + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**3/(3*x**3) + 3*a**2*b*x + 3*a*b**2*x**5/5 + b**3*x**9/9

________________________________________________________________________________________

Giac [A]  time = 1.09886, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{9} \, b^{3} x^{9} + \frac{3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac{a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*x^9 + 3/5*a*b^2*x^5 + 3*a^2*b*x - 1/3*a^3/x^3