3.2321 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^5} \, dx\)

Optimal. Leaf size=75 \[ -\frac{3 a^3 b^2}{x^{10/3}}-\frac{10 a^2 b^3}{3 x^3}-\frac{15 a^4 b}{11 x^{11/3}}-\frac{a^5}{4 x^4}-\frac{15 a b^4}{8 x^{8/3}}-\frac{3 b^5}{7 x^{7/3}} \]

[Out]

-a^5/(4*x^4) - (15*a^4*b)/(11*x^(11/3)) - (3*a^3*b^2)/x^(10/3) - (10*a^2*b^3)/(3*x^3) - (15*a*b^4)/(8*x^(8/3))
- (3*b^5)/(7*x^(7/3))

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Rubi [A]  time = 0.0334341, antiderivative size = 75, 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} \[ -\frac{3 a^3 b^2}{x^{10/3}}-\frac{10 a^2 b^3}{3 x^3}-\frac{15 a^4 b}{11 x^{11/3}}-\frac{a^5}{4 x^4}-\frac{15 a b^4}{8 x^{8/3}}-\frac{3 b^5}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^5,x]

[Out]

-a^5/(4*x^4) - (15*a^4*b)/(11*x^(11/3)) - (3*a^3*b^2)/x^(10/3) - (10*a^2*b^3)/(3*x^3) - (15*a*b^4)/(8*x^(8/3))
- (3*b^5)/(7*x^(7/3))

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^5} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{13}}+\frac{5 a^4 b}{x^{12}}+\frac{10 a^3 b^2}{x^{11}}+\frac{10 a^2 b^3}{x^{10}}+\frac{5 a b^4}{x^9}+\frac{b^5}{x^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^5}{4 x^4}-\frac{15 a^4 b}{11 x^{11/3}}-\frac{3 a^3 b^2}{x^{10/3}}-\frac{10 a^2 b^3}{3 x^3}-\frac{15 a b^4}{8 x^{8/3}}-\frac{3 b^5}{7 x^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.0268084, size = 67, normalized size = 0.89 \[ -\frac{5544 a^3 b^2 x^{2/3}+6160 a^2 b^3 x+2520 a^4 b \sqrt [3]{x}+462 a^5+3465 a b^4 x^{4/3}+792 b^5 x^{5/3}}{1848 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^5,x]

[Out]

-(462*a^5 + 2520*a^4*b*x^(1/3) + 5544*a^3*b^2*x^(2/3) + 6160*a^2*b^3*x + 3465*a*b^4*x^(4/3) + 792*b^5*x^(5/3))
/(1848*x^4)

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Maple [A]  time = 0.005, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{15\,{a}^{4}b}{11}{x}^{-{\frac{11}{3}}}}-3\,{\frac{{a}^{3}{b}^{2}}{{x}^{10/3}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}-{\frac{15\,a{b}^{4}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{b}^{5}}{7}{x}^{-{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^5/x^4-15/11*a^4*b/x^(11/3)-3*a^3*b^2/x^(10/3)-10/3*a^2*b^3/x^3-15/8*a*b^4/x^(8/3)-3/7*b^5/x^(7/3)

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Maxima [A]  time = 0.972887, size = 77, normalized size = 1.03 \begin{align*} -\frac{792 \, b^{5} x^{\frac{5}{3}} + 3465 \, a b^{4} x^{\frac{4}{3}} + 6160 \, a^{2} b^{3} x + 5544 \, a^{3} b^{2} x^{\frac{2}{3}} + 2520 \, a^{4} b x^{\frac{1}{3}} + 462 \, a^{5}}{1848 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1848*(792*b^5*x^(5/3) + 3465*a*b^4*x^(4/3) + 6160*a^2*b^3*x + 5544*a^3*b^2*x^(2/3) + 2520*a^4*b*x^(1/3) + 4
62*a^5)/x^4

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Fricas [A]  time = 1.43486, size = 150, normalized size = 2. \begin{align*} -\frac{6160 \, a^{2} b^{3} x + 462 \, a^{5} + 792 \,{\left (b^{5} x + 7 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 315 \,{\left (11 \, a b^{4} x + 8 \, a^{4} b\right )} x^{\frac{1}{3}}}{1848 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1848*(6160*a^2*b^3*x + 462*a^5 + 792*(b^5*x + 7*a^3*b^2)*x^(2/3) + 315*(11*a*b^4*x + 8*a^4*b)*x^(1/3))/x^4

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Sympy [A]  time = 4.79161, size = 75, normalized size = 1. \begin{align*} - \frac{a^{5}}{4 x^{4}} - \frac{15 a^{4} b}{11 x^{\frac{11}{3}}} - \frac{3 a^{3} b^{2}}{x^{\frac{10}{3}}} - \frac{10 a^{2} b^{3}}{3 x^{3}} - \frac{15 a b^{4}}{8 x^{\frac{8}{3}}} - \frac{3 b^{5}}{7 x^{\frac{7}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5/(4*x**4) - 15*a**4*b/(11*x**(11/3)) - 3*a**3*b**2/x**(10/3) - 10*a**2*b**3/(3*x**3) - 15*a*b**4/(8*x**(8
/3)) - 3*b**5/(7*x**(7/3))

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Giac [A]  time = 1.19275, size = 77, normalized size = 1.03 \begin{align*} -\frac{792 \, b^{5} x^{\frac{5}{3}} + 3465 \, a b^{4} x^{\frac{4}{3}} + 6160 \, a^{2} b^{3} x + 5544 \, a^{3} b^{2} x^{\frac{2}{3}} + 2520 \, a^{4} b x^{\frac{1}{3}} + 462 \, a^{5}}{1848 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1848*(792*b^5*x^(5/3) + 3465*a*b^4*x^(4/3) + 6160*a^2*b^3*x + 5544*a^3*b^2*x^(2/3) + 2520*a^4*b*x^(1/3) + 4
62*a^5)/x^4