3.2336 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^8} \, dx\)

Optimal. Leaf size=144 \[ -\frac{135 a^8 b^2}{19 x^{19/3}}-\frac{20 a^7 b^3}{x^6}-\frac{630 a^6 b^4}{17 x^{17/3}}-\frac{189 a^5 b^5}{4 x^{16/3}}-\frac{42 a^4 b^6}{x^5}-\frac{180 a^3 b^7}{7 x^{14/3}}-\frac{135 a^2 b^8}{13 x^{13/3}}-\frac{3 a^9 b}{2 x^{20/3}}-\frac{a^{10}}{7 x^7}-\frac{5 a b^9}{2 x^4}-\frac{3 b^{10}}{11 x^{11/3}} \]

[Out]

-a^10/(7*x^7) - (3*a^9*b)/(2*x^(20/3)) - (135*a^8*b^2)/(19*x^(19/3)) - (20*a^7*b^3)/x^6 - (630*a^6*b^4)/(17*x^
(17/3)) - (189*a^5*b^5)/(4*x^(16/3)) - (42*a^4*b^6)/x^5 - (180*a^3*b^7)/(7*x^(14/3)) - (135*a^2*b^8)/(13*x^(13
/3)) - (5*a*b^9)/(2*x^4) - (3*b^10)/(11*x^(11/3))

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Rubi [A]  time = 0.0675585, antiderivative size = 144, 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} \[ -\frac{135 a^8 b^2}{19 x^{19/3}}-\frac{20 a^7 b^3}{x^6}-\frac{630 a^6 b^4}{17 x^{17/3}}-\frac{189 a^5 b^5}{4 x^{16/3}}-\frac{42 a^4 b^6}{x^5}-\frac{180 a^3 b^7}{7 x^{14/3}}-\frac{135 a^2 b^8}{13 x^{13/3}}-\frac{3 a^9 b}{2 x^{20/3}}-\frac{a^{10}}{7 x^7}-\frac{5 a b^9}{2 x^4}-\frac{3 b^{10}}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^8,x]

[Out]

-a^10/(7*x^7) - (3*a^9*b)/(2*x^(20/3)) - (135*a^8*b^2)/(19*x^(19/3)) - (20*a^7*b^3)/x^6 - (630*a^6*b^4)/(17*x^
(17/3)) - (189*a^5*b^5)/(4*x^(16/3)) - (42*a^4*b^6)/x^5 - (180*a^3*b^7)/(7*x^(14/3)) - (135*a^2*b^8)/(13*x^(13
/3)) - (5*a*b^9)/(2*x^4) - (3*b^10)/(11*x^(11/3))

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 )^{10}}{x^8} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{22}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{22}}+\frac{10 a^9 b}{x^{21}}+\frac{45 a^8 b^2}{x^{20}}+\frac{120 a^7 b^3}{x^{19}}+\frac{210 a^6 b^4}{x^{18}}+\frac{252 a^5 b^5}{x^{17}}+\frac{210 a^4 b^6}{x^{16}}+\frac{120 a^3 b^7}{x^{15}}+\frac{45 a^2 b^8}{x^{14}}+\frac{10 a b^9}{x^{13}}+\frac{b^{10}}{x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{7 x^7}-\frac{3 a^9 b}{2 x^{20/3}}-\frac{135 a^8 b^2}{19 x^{19/3}}-\frac{20 a^7 b^3}{x^6}-\frac{630 a^6 b^4}{17 x^{17/3}}-\frac{189 a^5 b^5}{4 x^{16/3}}-\frac{42 a^4 b^6}{x^5}-\frac{180 a^3 b^7}{7 x^{14/3}}-\frac{135 a^2 b^8}{13 x^{13/3}}-\frac{5 a b^9}{2 x^4}-\frac{3 b^{10}}{11 x^{11/3}}\\ \end{align*}

Mathematica [A]  time = 0.0594525, size = 144, normalized size = 1. \[ -\frac{135 a^8 b^2}{19 x^{19/3}}-\frac{20 a^7 b^3}{x^6}-\frac{630 a^6 b^4}{17 x^{17/3}}-\frac{189 a^5 b^5}{4 x^{16/3}}-\frac{42 a^4 b^6}{x^5}-\frac{180 a^3 b^7}{7 x^{14/3}}-\frac{135 a^2 b^8}{13 x^{13/3}}-\frac{3 a^9 b}{2 x^{20/3}}-\frac{a^{10}}{7 x^7}-\frac{5 a b^9}{2 x^4}-\frac{3 b^{10}}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^8,x]

[Out]

-a^10/(7*x^7) - (3*a^9*b)/(2*x^(20/3)) - (135*a^8*b^2)/(19*x^(19/3)) - (20*a^7*b^3)/x^6 - (630*a^6*b^4)/(17*x^
(17/3)) - (189*a^5*b^5)/(4*x^(16/3)) - (42*a^4*b^6)/x^5 - (180*a^3*b^7)/(7*x^(14/3)) - (135*a^2*b^8)/(13*x^(13
/3)) - (5*a*b^9)/(2*x^4) - (3*b^10)/(11*x^(11/3))

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Maple [A]  time = 0.009, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{7\,{x}^{7}}}-{\frac{3\,{a}^{9}b}{2}{x}^{-{\frac{20}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{19}{x}^{-{\frac{19}{3}}}}-20\,{\frac{{a}^{7}{b}^{3}}{{x}^{6}}}-{\frac{630\,{a}^{6}{b}^{4}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{189\,{a}^{5}{b}^{5}}{4}{x}^{-{\frac{16}{3}}}}-42\,{\frac{{a}^{4}{b}^{6}}{{x}^{5}}}-{\frac{180\,{a}^{3}{b}^{7}}{7}{x}^{-{\frac{14}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{13}{x}^{-{\frac{13}{3}}}}-{\frac{5\,a{b}^{9}}{2\,{x}^{4}}}-{\frac{3\,{b}^{10}}{11}{x}^{-{\frac{11}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^8,x)

[Out]

-1/7*a^10/x^7-3/2*a^9*b/x^(20/3)-135/19*a^8*b^2/x^(19/3)-20*a^7*b^3/x^6-630/17*a^6*b^4/x^(17/3)-189/4*a^5*b^5/
x^(16/3)-42*a^4*b^6/x^5-180/7*a^3*b^7/x^(14/3)-135/13*a^2*b^8/x^(13/3)-5/2*a*b^9/x^4-3/11*b^10/x^(11/3)

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Maxima [A]  time = 0.978934, size = 151, normalized size = 1.05 \begin{align*} -\frac{352716 \, b^{10} x^{\frac{10}{3}} + 3233230 \, a b^{9} x^{3} + 13430340 \, a^{2} b^{8} x^{\frac{8}{3}} + 33256080 \, a^{3} b^{7} x^{\frac{7}{3}} + 54318264 \, a^{4} b^{6} x^{2} + 61108047 \, a^{5} b^{5} x^{\frac{5}{3}} + 47927880 \, a^{6} b^{4} x^{\frac{4}{3}} + 25865840 \, a^{7} b^{3} x + 9189180 \, a^{8} b^{2} x^{\frac{2}{3}} + 1939938 \, a^{9} b x^{\frac{1}{3}} + 184756 \, a^{10}}{1293292 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="maxima")

[Out]

-1/1293292*(352716*b^10*x^(10/3) + 3233230*a*b^9*x^3 + 13430340*a^2*b^8*x^(8/3) + 33256080*a^3*b^7*x^(7/3) + 5
4318264*a^4*b^6*x^2 + 61108047*a^5*b^5*x^(5/3) + 47927880*a^6*b^4*x^(4/3) + 25865840*a^7*b^3*x + 9189180*a^8*b
^2*x^(2/3) + 1939938*a^9*b*x^(1/3) + 184756*a^10)/x^7

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Fricas [A]  time = 1.49638, size = 324, normalized size = 2.25 \begin{align*} -\frac{3233230 \, a b^{9} x^{3} + 54318264 \, a^{4} b^{6} x^{2} + 25865840 \, a^{7} b^{3} x + 184756 \, a^{10} + 35343 \,{\left (380 \, a^{2} b^{8} x^{2} + 1729 \, a^{5} b^{5} x + 260 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 1482 \,{\left (238 \, b^{10} x^{3} + 22440 \, a^{3} b^{7} x^{2} + 32340 \, a^{6} b^{4} x + 1309 \, a^{9} b\right )} x^{\frac{1}{3}}}{1293292 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="fricas")

[Out]

-1/1293292*(3233230*a*b^9*x^3 + 54318264*a^4*b^6*x^2 + 25865840*a^7*b^3*x + 184756*a^10 + 35343*(380*a^2*b^8*x
^2 + 1729*a^5*b^5*x + 260*a^8*b^2)*x^(2/3) + 1482*(238*b^10*x^3 + 22440*a^3*b^7*x^2 + 32340*a^6*b^4*x + 1309*a
^9*b)*x^(1/3))/x^7

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Sympy [A]  time = 24.7602, size = 146, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{7 x^{7}} - \frac{3 a^{9} b}{2 x^{\frac{20}{3}}} - \frac{135 a^{8} b^{2}}{19 x^{\frac{19}{3}}} - \frac{20 a^{7} b^{3}}{x^{6}} - \frac{630 a^{6} b^{4}}{17 x^{\frac{17}{3}}} - \frac{189 a^{5} b^{5}}{4 x^{\frac{16}{3}}} - \frac{42 a^{4} b^{6}}{x^{5}} - \frac{180 a^{3} b^{7}}{7 x^{\frac{14}{3}}} - \frac{135 a^{2} b^{8}}{13 x^{\frac{13}{3}}} - \frac{5 a b^{9}}{2 x^{4}} - \frac{3 b^{10}}{11 x^{\frac{11}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**8,x)

[Out]

-a**10/(7*x**7) - 3*a**9*b/(2*x**(20/3)) - 135*a**8*b**2/(19*x**(19/3)) - 20*a**7*b**3/x**6 - 630*a**6*b**4/(1
7*x**(17/3)) - 189*a**5*b**5/(4*x**(16/3)) - 42*a**4*b**6/x**5 - 180*a**3*b**7/(7*x**(14/3)) - 135*a**2*b**8/(
13*x**(13/3)) - 5*a*b**9/(2*x**4) - 3*b**10/(11*x**(11/3))

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Giac [A]  time = 1.25184, size = 151, normalized size = 1.05 \begin{align*} -\frac{352716 \, b^{10} x^{\frac{10}{3}} + 3233230 \, a b^{9} x^{3} + 13430340 \, a^{2} b^{8} x^{\frac{8}{3}} + 33256080 \, a^{3} b^{7} x^{\frac{7}{3}} + 54318264 \, a^{4} b^{6} x^{2} + 61108047 \, a^{5} b^{5} x^{\frac{5}{3}} + 47927880 \, a^{6} b^{4} x^{\frac{4}{3}} + 25865840 \, a^{7} b^{3} x + 9189180 \, a^{8} b^{2} x^{\frac{2}{3}} + 1939938 \, a^{9} b x^{\frac{1}{3}} + 184756 \, a^{10}}{1293292 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="giac")

[Out]

-1/1293292*(352716*b^10*x^(10/3) + 3233230*a*b^9*x^3 + 13430340*a^2*b^8*x^(8/3) + 33256080*a^3*b^7*x^(7/3) + 5
4318264*a^4*b^6*x^2 + 61108047*a^5*b^5*x^(5/3) + 47927880*a^6*b^4*x^(4/3) + 25865840*a^7*b^3*x + 9189180*a^8*b
^2*x^(2/3) + 1939938*a^9*b*x^(1/3) + 184756*a^10)/x^7