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

Optimal. Leaf size=50 \[ 6 a^2 b^2 x-\frac{4 a^3 b}{x}-\frac{a^4}{3 x^3}+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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Rubi [A]  time = 0.0211109, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 270} \[ 6 a^2 b^2 x-\frac{4 a^3 b}{x}-\frac{a^4}{3 x^3}+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a/x + b*x)^4,x]

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0060251, size = 50, normalized size = 1. \[ 6 a^2 b^2 x-\frac{4 a^3 b}{x}-\frac{a^4}{3 x^3}+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a/x + b*x)^4,x]

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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Maple [A]  time = 0.006, size = 45, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{3\,{x}^{3}}}-4\,{\frac{{a}^{3}b}{x}}+6\,{a}^{2}{b}^{2}x+{\frac{4\,a{b}^{3}{x}^{3}}{3}}+{\frac{{b}^{4}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5

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Maxima [A]  time = 1.17249, size = 59, normalized size = 1.18 \begin{align*} \frac{1}{5} \, b^{4} x^{5} + \frac{4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac{4 \, a^{3} b}{x} - \frac{a^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 4*a^3*b/x - 1/3*a^4/x^3

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Fricas [A]  time = 0.757692, size = 104, normalized size = 2.08 \begin{align*} \frac{3 \, b^{4} x^{8} + 20 \, a b^{3} x^{6} + 90 \, a^{2} b^{2} x^{4} - 60 \, a^{3} b x^{2} - 5 \, a^{4}}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*b^4*x^8 + 20*a*b^3*x^6 + 90*a^2*b^2*x^4 - 60*a^3*b*x^2 - 5*a^4)/x^3

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Sympy [A]  time = 0.350488, size = 48, normalized size = 0.96 \begin{align*} 6 a^{2} b^{2} x + \frac{4 a b^{3} x^{3}}{3} + \frac{b^{4} x^{5}}{5} - \frac{a^{4} + 12 a^{3} b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*a**2*b**2*x + 4*a*b**3*x**3/3 + b**4*x**5/5 - (a**4 + 12*a**3*b*x**2)/(3*x**3)

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Giac [A]  time = 1.11432, size = 61, normalized size = 1.22 \begin{align*} \frac{1}{5} \, b^{4} x^{5} + \frac{4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac{12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3