3.2322 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^6} \, dx\)

Optimal. Leaf size=77 \[ -\frac{30 a^3 b^2}{13 x^{13/3}}-\frac{5 a^2 b^3}{2 x^4}-\frac{15 a^4 b}{14 x^{14/3}}-\frac{a^5}{5 x^5}-\frac{15 a b^4}{11 x^{11/3}}-\frac{3 b^5}{10 x^{10/3}} \]

[Out]

-a^5/(5*x^5) - (15*a^4*b)/(14*x^(14/3)) - (30*a^3*b^2)/(13*x^(13/3)) - (5*a^2*b^3)/(2*x^4) - (15*a*b^4)/(11*x^
(11/3)) - (3*b^5)/(10*x^(10/3))

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Rubi [A]  time = 0.0337491, antiderivative size = 77, 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} \[ -\frac{30 a^3 b^2}{13 x^{13/3}}-\frac{5 a^2 b^3}{2 x^4}-\frac{15 a^4 b}{14 x^{14/3}}-\frac{a^5}{5 x^5}-\frac{15 a b^4}{11 x^{11/3}}-\frac{3 b^5}{10 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^6,x]

[Out]

-a^5/(5*x^5) - (15*a^4*b)/(14*x^(14/3)) - (30*a^3*b^2)/(13*x^(13/3)) - (5*a^2*b^3)/(2*x^4) - (15*a*b^4)/(11*x^
(11/3)) - (3*b^5)/(10*x^(10/3))

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^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{16}}+\frac{5 a^4 b}{x^{15}}+\frac{10 a^3 b^2}{x^{14}}+\frac{10 a^2 b^3}{x^{13}}+\frac{5 a b^4}{x^{12}}+\frac{b^5}{x^{11}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^5}{5 x^5}-\frac{15 a^4 b}{14 x^{14/3}}-\frac{30 a^3 b^2}{13 x^{13/3}}-\frac{5 a^2 b^3}{2 x^4}-\frac{15 a b^4}{11 x^{11/3}}-\frac{3 b^5}{10 x^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.0264998, size = 67, normalized size = 0.87 \[ -\frac{23100 a^3 b^2 x^{2/3}+25025 a^2 b^3 x+10725 a^4 b \sqrt [3]{x}+2002 a^5+13650 a b^4 x^{4/3}+3003 b^5 x^{5/3}}{10010 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^6,x]

[Out]

-(2002*a^5 + 10725*a^4*b*x^(1/3) + 23100*a^3*b^2*x^(2/3) + 25025*a^2*b^3*x + 13650*a*b^4*x^(4/3) + 3003*b^5*x^
(5/3))/(10010*x^5)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{15\,{a}^{4}b}{14}{x}^{-{\frac{14}{3}}}}-{\frac{30\,{a}^{3}{b}^{2}}{13}{x}^{-{\frac{13}{3}}}}-{\frac{5\,{a}^{2}{b}^{3}}{2\,{x}^{4}}}-{\frac{15\,a{b}^{4}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{3\,{b}^{5}}{10}{x}^{-{\frac{10}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5/x^6,x)

[Out]

-1/5*a^5/x^5-15/14*a^4*b/x^(14/3)-30/13*a^3*b^2/x^(13/3)-5/2*a^2*b^3/x^4-15/11*a*b^4/x^(11/3)-3/10*b^5/x^(10/3
)

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Maxima [A]  time = 0.99201, size = 77, normalized size = 1. \begin{align*} -\frac{3003 \, b^{5} x^{\frac{5}{3}} + 13650 \, a b^{4} x^{\frac{4}{3}} + 25025 \, a^{2} b^{3} x + 23100 \, a^{3} b^{2} x^{\frac{2}{3}} + 10725 \, a^{4} b x^{\frac{1}{3}} + 2002 \, a^{5}}{10010 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^6,x, algorithm="maxima")

[Out]

-1/10010*(3003*b^5*x^(5/3) + 13650*a*b^4*x^(4/3) + 25025*a^2*b^3*x + 23100*a^3*b^2*x^(2/3) + 10725*a^4*b*x^(1/
3) + 2002*a^5)/x^5

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Fricas [A]  time = 1.61666, size = 162, normalized size = 2.1 \begin{align*} -\frac{25025 \, a^{2} b^{3} x + 2002 \, a^{5} + 231 \,{\left (13 \, b^{5} x + 100 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 975 \,{\left (14 \, a b^{4} x + 11 \, a^{4} b\right )} x^{\frac{1}{3}}}{10010 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^6,x, algorithm="fricas")

[Out]

-1/10010*(25025*a^2*b^3*x + 2002*a^5 + 231*(13*b^5*x + 100*a^3*b^2)*x^(2/3) + 975*(14*a*b^4*x + 11*a^4*b)*x^(1
/3))/x^5

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Sympy [A]  time = 8.99631, size = 76, normalized size = 0.99 \begin{align*} - \frac{a^{5}}{5 x^{5}} - \frac{15 a^{4} b}{14 x^{\frac{14}{3}}} - \frac{30 a^{3} b^{2}}{13 x^{\frac{13}{3}}} - \frac{5 a^{2} b^{3}}{2 x^{4}} - \frac{15 a b^{4}}{11 x^{\frac{11}{3}}} - \frac{3 b^{5}}{10 x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x**6,x)

[Out]

-a**5/(5*x**5) - 15*a**4*b/(14*x**(14/3)) - 30*a**3*b**2/(13*x**(13/3)) - 5*a**2*b**3/(2*x**4) - 15*a*b**4/(11
*x**(11/3)) - 3*b**5/(10*x**(10/3))

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Giac [A]  time = 1.15169, size = 77, normalized size = 1. \begin{align*} -\frac{3003 \, b^{5} x^{\frac{5}{3}} + 13650 \, a b^{4} x^{\frac{4}{3}} + 25025 \, a^{2} b^{3} x + 23100 \, a^{3} b^{2} x^{\frac{2}{3}} + 10725 \, a^{4} b x^{\frac{1}{3}} + 2002 \, a^{5}}{10010 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^6,x, algorithm="giac")

[Out]

-1/10010*(3003*b^5*x^(5/3) + 13650*a*b^4*x^(4/3) + 25025*a^2*b^3*x + 23100*a^3*b^2*x^(2/3) + 10725*a^4*b*x^(1/
3) + 2002*a^5)/x^5