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

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

[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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Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 0.87 \[ -\frac {2002 a^5+10725 a^4 b \sqrt [3]{x}+23100 a^3 b^2 x^{2/3}+25025 a^2 b^3 x+13650 a b^4 x^{4/3}+3003 b^5 x^{5/3}}{10010 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^6,x]

[Out]

-1/10010*(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) + 300
3*b^5*x^(5/3))/x^5

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fricas [A]  time = 0.67, size = 59, normalized size = 0.77 \[ -\frac {25025 \, a^{2} b^{3} x + 2002 \, a^{5} + 231 \, {\left (13 \, b^{5} x + 100 \, a^{3} b^{2}\right )} x^{\frac {2}{3}} + 975 \, {\left (14 \, a b^{4} x + 11 \, a^{4} b\right )} x^{\frac {1}{3}}}{10010 \, x^{5}} \]

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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giac [A]  time = 0.16, size = 57, normalized size = 0.74 \[ -\frac {3003 \, b^{5} x^{\frac {5}{3}} + 13650 \, a b^{4} x^{\frac {4}{3}} + 25025 \, a^{2} b^{3} x + 23100 \, a^{3} b^{2} x^{\frac {2}{3}} + 10725 \, a^{4} b x^{\frac {1}{3}} + 2002 \, a^{5}}{10010 \, x^{5}} \]

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

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maple [A]  time = 0.01, size = 58, normalized size = 0.75 \[ -\frac {3 b^{5}}{10 x^{\frac {10}{3}}}-\frac {15 a \,b^{4}}{11 x^{\frac {11}{3}}}-\frac {5 a^{2} b^{3}}{2 x^{4}}-\frac {30 a^{3} b^{2}}{13 x^{\frac {13}{3}}}-\frac {15 a^{4} b}{14 x^{\frac {14}{3}}}-\frac {a^{5}}{5 x^{5}} \]

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 = 0.82, size = 57, normalized size = 0.74 \[ -\frac {3003 \, b^{5} x^{\frac {5}{3}} + 13650 \, a b^{4} x^{\frac {4}{3}} + 25025 \, a^{2} b^{3} x + 23100 \, a^{3} b^{2} x^{\frac {2}{3}} + 10725 \, a^{4} b x^{\frac {1}{3}} + 2002 \, a^{5}}{10010 \, x^{5}} \]

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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mupad [B]  time = 0.04, size = 57, normalized size = 0.74 \[ -\frac {\frac {a^5}{5}+\frac {3\,b^5\,x^{5/3}}{10}+\frac {5\,a^2\,b^3\,x}{2}+\frac {15\,a^4\,b\,x^{1/3}}{14}+\frac {15\,a\,b^4\,x^{4/3}}{11}+\frac {30\,a^3\,b^2\,x^{2/3}}{13}}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^5/x^6,x)

[Out]

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

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sympy [A]  time = 5.35, size = 76, normalized size = 0.99 \[ - \frac {a^{5}}{5 x^{5}} - \frac {15 a^{4} b}{14 x^{\frac {14}{3}}} - \frac {30 a^{3} b^{2}}{13 x^{\frac {13}{3}}} - \frac {5 a^{2} b^{3}}{2 x^{4}} - \frac {15 a b^{4}}{11 x^{\frac {11}{3}}} - \frac {3 b^{5}}{10 x^{\frac {10}{3}}} \]

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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