3.2320 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^4} \, dx\)

Optimal. Leaf size=73 \[ -\frac{30 a^3 b^2}{7 x^{7/3}}-\frac{5 a^2 b^3}{x^2}-\frac{15 a^4 b}{8 x^{8/3}}-\frac{a^5}{3 x^3}-\frac{3 a b^4}{x^{5/3}}-\frac{3 b^5}{4 x^{4/3}} \]

[Out]

-a^5/(3*x^3) - (15*a^4*b)/(8*x^(8/3)) - (30*a^3*b^2)/(7*x^(7/3)) - (5*a^2*b^3)/x^2 - (3*a*b^4)/x^(5/3) - (3*b^
5)/(4*x^(4/3))

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Rubi [A]  time = 0.0332598, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{30 a^3 b^2}{7 x^{7/3}}-\frac{5 a^2 b^3}{x^2}-\frac{15 a^4 b}{8 x^{8/3}}-\frac{a^5}{3 x^3}-\frac{3 a b^4}{x^{5/3}}-\frac{3 b^5}{4 x^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^4,x]

[Out]

-a^5/(3*x^3) - (15*a^4*b)/(8*x^(8/3)) - (30*a^3*b^2)/(7*x^(7/3)) - (5*a^2*b^3)/x^2 - (3*a*b^4)/x^(5/3) - (3*b^
5)/(4*x^(4/3))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{10}}+\frac{5 a^4 b}{x^9}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^7}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^5}{3 x^3}-\frac{15 a^4 b}{8 x^{8/3}}-\frac{30 a^3 b^2}{7 x^{7/3}}-\frac{5 a^2 b^3}{x^2}-\frac{3 a b^4}{x^{5/3}}-\frac{3 b^5}{4 x^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.0254497, size = 67, normalized size = 0.92 \[ -\frac{720 a^3 b^2 x^{2/3}+840 a^2 b^3 x+315 a^4 b \sqrt [3]{x}+56 a^5+504 a b^4 x^{4/3}+126 b^5 x^{5/3}}{168 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^4,x]

[Out]

-(56*a^5 + 315*a^4*b*x^(1/3) + 720*a^3*b^2*x^(2/3) + 840*a^2*b^3*x + 504*a*b^4*x^(4/3) + 126*b^5*x^(5/3))/(168
*x^3)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{3\,{x}^{3}}}-{\frac{15\,{a}^{4}b}{8}{x}^{-{\frac{8}{3}}}}-{\frac{30\,{a}^{3}{b}^{2}}{7}{x}^{-{\frac{7}{3}}}}-5\,{\frac{{a}^{2}{b}^{3}}{{x}^{2}}}-3\,{\frac{a{b}^{4}}{{x}^{5/3}}}-{\frac{3\,{b}^{5}}{4}{x}^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5/x^4,x)

[Out]

-1/3*a^5/x^3-15/8*a^4*b/x^(8/3)-30/7*a^3*b^2/x^(7/3)-5*a^2*b^3/x^2-3*a*b^4/x^(5/3)-3/4*b^5/x^(4/3)

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Maxima [A]  time = 1.02281, size = 77, normalized size = 1.05 \begin{align*} -\frac{126 \, b^{5} x^{\frac{5}{3}} + 504 \, a b^{4} x^{\frac{4}{3}} + 840 \, a^{2} b^{3} x + 720 \, a^{3} b^{2} x^{\frac{2}{3}} + 315 \, a^{4} b x^{\frac{1}{3}} + 56 \, a^{5}}{168 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^4,x, algorithm="maxima")

[Out]

-1/168*(126*b^5*x^(5/3) + 504*a*b^4*x^(4/3) + 840*a^2*b^3*x + 720*a^3*b^2*x^(2/3) + 315*a^4*b*x^(1/3) + 56*a^5
)/x^3

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Fricas [A]  time = 1.41697, size = 146, normalized size = 2. \begin{align*} -\frac{840 \, a^{2} b^{3} x + 56 \, a^{5} + 18 \,{\left (7 \, b^{5} x + 40 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 63 \,{\left (8 \, a b^{4} x + 5 \, a^{4} b\right )} x^{\frac{1}{3}}}{168 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^4,x, algorithm="fricas")

[Out]

-1/168*(840*a^2*b^3*x + 56*a^5 + 18*(7*b^5*x + 40*a^3*b^2)*x^(2/3) + 63*(8*a*b^4*x + 5*a^4*b)*x^(1/3))/x^3

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Sympy [A]  time = 2.46796, size = 73, normalized size = 1. \begin{align*} - \frac{a^{5}}{3 x^{3}} - \frac{15 a^{4} b}{8 x^{\frac{8}{3}}} - \frac{30 a^{3} b^{2}}{7 x^{\frac{7}{3}}} - \frac{5 a^{2} b^{3}}{x^{2}} - \frac{3 a b^{4}}{x^{\frac{5}{3}}} - \frac{3 b^{5}}{4 x^{\frac{4}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x**4,x)

[Out]

-a**5/(3*x**3) - 15*a**4*b/(8*x**(8/3)) - 30*a**3*b**2/(7*x**(7/3)) - 5*a**2*b**3/x**2 - 3*a*b**4/x**(5/3) - 3
*b**5/(4*x**(4/3))

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Giac [A]  time = 1.19878, size = 77, normalized size = 1.05 \begin{align*} -\frac{126 \, b^{5} x^{\frac{5}{3}} + 504 \, a b^{4} x^{\frac{4}{3}} + 840 \, a^{2} b^{3} x + 720 \, a^{3} b^{2} x^{\frac{2}{3}} + 315 \, a^{4} b x^{\frac{1}{3}} + 56 \, a^{5}}{168 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^4,x, algorithm="giac")

[Out]

-1/168*(126*b^5*x^(5/3) + 504*a*b^4*x^(4/3) + 840*a^2*b^3*x + 720*a^3*b^2*x^(2/3) + 315*a^4*b*x^(1/3) + 56*a^5
)/x^3