3.2319 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

[Out]

-(a + b*x^(1/3))^6/(2*a*x^2)

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Rubi [A]  time = 0.0033234, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^3,x]

[Out]

-(a + b*x^(1/3))^6/(2*a*x^2)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.00401, size = 21, normalized size = 1. \[ -\frac{\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^3,x]

[Out]

-(a + b*x^(1/3))^6/(2*a*x^2)

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Maple [B]  time = 0.007, size = 58, normalized size = 2.8 \begin{align*} -10\,{\frac{{a}^{2}{b}^{3}}{x}}-3\,{\frac{{a}^{4}b}{{x}^{5/3}}}-{\frac{15\,{a}^{3}{b}^{2}}{2}{x}^{-{\frac{4}{3}}}}-{\frac{15\,a{b}^{4}}{2}{x}^{-{\frac{2}{3}}}}-{\frac{{a}^{5}}{2\,{x}^{2}}}-3\,{\frac{{b}^{5}}{\sqrt [3]{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5/x^3,x)

[Out]

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

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Maxima [B]  time = 1.03246, size = 74, normalized size = 3.52 \begin{align*} -\frac{6 \, b^{5} x^{\frac{5}{3}} + 15 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 6 \, a^{4} b x^{\frac{1}{3}} + a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.49597, size = 134, normalized size = 6.38 \begin{align*} -\frac{20 \, a^{2} b^{3} x + a^{5} + 3 \,{\left (2 \, b^{5} x + 5 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (5 \, a b^{4} x + 2 \, a^{4} b\right )} x^{\frac{1}{3}}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.31962, size = 70, normalized size = 3.33 \begin{align*} - \frac{a^{5}}{2 x^{2}} - \frac{3 a^{4} b}{x^{\frac{5}{3}}} - \frac{15 a^{3} b^{2}}{2 x^{\frac{4}{3}}} - \frac{10 a^{2} b^{3}}{x} - \frac{15 a b^{4}}{2 x^{\frac{2}{3}}} - \frac{3 b^{5}}{\sqrt [3]{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x**3,x)

[Out]

-a**5/(2*x**2) - 3*a**4*b/x**(5/3) - 15*a**3*b**2/(2*x**(4/3)) - 10*a**2*b**3/x - 15*a*b**4/(2*x**(2/3)) - 3*b
**5/x**(1/3)

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Giac [B]  time = 1.23521, size = 74, normalized size = 3.52 \begin{align*} -\frac{6 \, b^{5} x^{\frac{5}{3}} + 15 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 6 \, a^{4} b x^{\frac{1}{3}} + a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^3,x, algorithm="giac")

[Out]

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