3.2337 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^9} \, dx\)

Optimal. Leaf size=144 \[ -\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{a^{10}}{8 x^8}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^7*b^3)/(7*x^7) - (63*a^6*b^4)/
(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(1
6*x^(16/3)) - (2*a*b^9)/x^5 - (3*b^10)/(14*x^(14/3))

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Rubi [A]  time = 0.0706039, 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} \[ -\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{a^{10}}{8 x^8}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^9,x]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^7*b^3)/(7*x^7) - (63*a^6*b^4)/
(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(1
6*x^(16/3)) - (2*a*b^9)/x^5 - (3*b^10)/(14*x^(14/3))

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^9} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{25}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{25}}+\frac{10 a^9 b}{x^{24}}+\frac{45 a^8 b^2}{x^{23}}+\frac{120 a^7 b^3}{x^{22}}+\frac{210 a^6 b^4}{x^{21}}+\frac{252 a^5 b^5}{x^{20}}+\frac{210 a^4 b^6}{x^{19}}+\frac{120 a^3 b^7}{x^{18}}+\frac{45 a^2 b^8}{x^{17}}+\frac{10 a b^9}{x^{16}}+\frac{b^{10}}{x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{8 x^8}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}}\\ \end{align*}

Mathematica [A]  time = 0.0575085, size = 144, normalized size = 1. \[ -\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{a^{10}}{8 x^8}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^9,x]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^7*b^3)/(7*x^7) - (63*a^6*b^4)/
(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(1
6*x^(16/3)) - (2*a*b^9)/x^5 - (3*b^10)/(14*x^(14/3))

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Maple [A]  time = 0.008, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{8\,{x}^{8}}}-{\frac{30\,{a}^{9}b}{23}{x}^{-{\frac{23}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{22}{x}^{-{\frac{22}{3}}}}-{\frac{120\,{a}^{7}{b}^{3}}{7\,{x}^{7}}}-{\frac{63\,{a}^{6}{b}^{4}}{2}{x}^{-{\frac{20}{3}}}}-{\frac{756\,{a}^{5}{b}^{5}}{19}{x}^{-{\frac{19}{3}}}}-35\,{\frac{{a}^{4}{b}^{6}}{{x}^{6}}}-{\frac{360\,{a}^{3}{b}^{7}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{16}{x}^{-{\frac{16}{3}}}}-2\,{\frac{a{b}^{9}}{{x}^{5}}}-{\frac{3\,{b}^{10}}{14}{x}^{-{\frac{14}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^9,x)

[Out]

-1/8*a^10/x^8-30/23*a^9*b/x^(23/3)-135/22*a^8*b^2/x^(22/3)-120/7*a^7*b^3/x^7-63/2*a^6*b^4/x^(20/3)-756/19*a^5*
b^5/x^(19/3)-35*a^4*b^6/x^6-360/17*a^3*b^7/x^(17/3)-135/16*a^2*b^8/x^(16/3)-2*a*b^9/x^5-3/14*b^10/x^(14/3)

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Maxima [A]  time = 0.977002, size = 151, normalized size = 1.05 \begin{align*} -\frac{1961256 \, b^{10} x^{\frac{10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac{8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac{7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac{5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac{4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac{2}{3}} + 11938080 \, a^{9} b x^{\frac{1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^9,x, algorithm="maxima")

[Out]

-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8*x^(8/3) + 193818240*a^3*b^7*x^(7/3)
+ 320338480*a^4*b^6*x^2 + 364174272*a^5*b^5*x^(5/3) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 561632
40*a^8*b^2*x^(2/3) + 11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8

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Fricas [A]  time = 1.42656, size = 331, normalized size = 2.3 \begin{align*} -\frac{18305056 \, a b^{9} x^{3} + 320338480 \, a^{4} b^{6} x^{2} + 156900480 \, a^{7} b^{3} x + 1144066 \, a^{10} + 73899 \,{\left (1045 \, a^{2} b^{8} x^{2} + 4928 \, a^{5} b^{5} x + 760 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 5016 \,{\left (391 \, b^{10} x^{3} + 38640 \, a^{3} b^{7} x^{2} + 57477 \, a^{6} b^{4} x + 2380 \, a^{9} b\right )} x^{\frac{1}{3}}}{9152528 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^9,x, algorithm="fricas")

[Out]

-1/9152528*(18305056*a*b^9*x^3 + 320338480*a^4*b^6*x^2 + 156900480*a^7*b^3*x + 1144066*a^10 + 73899*(1045*a^2*
b^8*x^2 + 4928*a^5*b^5*x + 760*a^8*b^2)*x^(2/3) + 5016*(391*b^10*x^3 + 38640*a^3*b^7*x^2 + 57477*a^6*b^4*x + 2
380*a^9*b)*x^(1/3))/x^8

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Sympy [A]  time = 29.53, size = 146, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{8 x^{8}} - \frac{30 a^{9} b}{23 x^{\frac{23}{3}}} - \frac{135 a^{8} b^{2}}{22 x^{\frac{22}{3}}} - \frac{120 a^{7} b^{3}}{7 x^{7}} - \frac{63 a^{6} b^{4}}{2 x^{\frac{20}{3}}} - \frac{756 a^{5} b^{5}}{19 x^{\frac{19}{3}}} - \frac{35 a^{4} b^{6}}{x^{6}} - \frac{360 a^{3} b^{7}}{17 x^{\frac{17}{3}}} - \frac{135 a^{2} b^{8}}{16 x^{\frac{16}{3}}} - \frac{2 a b^{9}}{x^{5}} - \frac{3 b^{10}}{14 x^{\frac{14}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**9,x)

[Out]

-a**10/(8*x**8) - 30*a**9*b/(23*x**(23/3)) - 135*a**8*b**2/(22*x**(22/3)) - 120*a**7*b**3/(7*x**7) - 63*a**6*b
**4/(2*x**(20/3)) - 756*a**5*b**5/(19*x**(19/3)) - 35*a**4*b**6/x**6 - 360*a**3*b**7/(17*x**(17/3)) - 135*a**2
*b**8/(16*x**(16/3)) - 2*a*b**9/x**5 - 3*b**10/(14*x**(14/3))

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Giac [A]  time = 1.18835, size = 151, normalized size = 1.05 \begin{align*} -\frac{1961256 \, b^{10} x^{\frac{10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac{8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac{7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac{5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac{4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac{2}{3}} + 11938080 \, a^{9} b x^{\frac{1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^9,x, algorithm="giac")

[Out]

-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8*x^(8/3) + 193818240*a^3*b^7*x^(7/3)
+ 320338480*a^4*b^6*x^2 + 364174272*a^5*b^5*x^(5/3) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 561632
40*a^8*b^2*x^(2/3) + 11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8