3.2338 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^{10}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{a^{10}}{9 x^9}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b^3)/x^8 - (630*a^6*b^4)/(23*x^
(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3))
- (5*a*b^9)/(3*x^6) - (3*b^10)/(17*x^(17/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0693011, antiderivative size = 142, 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} \[ -\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{a^{10}}{9 x^9}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^10,x]

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b^3)/x^8 - (630*a^6*b^4)/(23*x^
(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3))
- (5*a*b^9)/(3*x^6) - (3*b^10)/(17*x^(17/3))

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^{10}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{28}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{28}}+\frac{10 a^9 b}{x^{27}}+\frac{45 a^8 b^2}{x^{26}}+\frac{120 a^7 b^3}{x^{25}}+\frac{210 a^6 b^4}{x^{24}}+\frac{252 a^5 b^5}{x^{23}}+\frac{210 a^4 b^6}{x^{22}}+\frac{120 a^3 b^7}{x^{21}}+\frac{45 a^2 b^8}{x^{20}}+\frac{10 a b^9}{x^{19}}+\frac{b^{10}}{x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{9 x^9}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}}\\ \end{align*}

Mathematica [A]  time = 0.0610957, size = 142, normalized size = 1. \[ -\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{a^{10}}{9 x^9}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^10,x]

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b^3)/x^8 - (630*a^6*b^4)/(23*x^
(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3))
- (5*a*b^9)/(3*x^6) - (3*b^10)/(17*x^(17/3))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{9\,{x}^{9}}}-{\frac{15\,{a}^{9}b}{13}{x}^{-{\frac{26}{3}}}}-{\frac{27\,{a}^{8}{b}^{2}}{5}{x}^{-{\frac{25}{3}}}}-15\,{\frac{{a}^{7}{b}^{3}}{{x}^{8}}}-{\frac{630\,{a}^{6}{b}^{4}}{23}{x}^{-{\frac{23}{3}}}}-{\frac{378\,{a}^{5}{b}^{5}}{11}{x}^{-{\frac{22}{3}}}}-30\,{\frac{{a}^{4}{b}^{6}}{{x}^{7}}}-18\,{{a}^{3}{b}^{7}{x}^{-{\frac{20}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{5\,a{b}^{9}}{3\,{x}^{6}}}-{\frac{3\,{b}^{10}}{17}{x}^{-{\frac{17}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^10/x^9-15/13*a^9*b/x^(26/3)-27/5*a^8*b^2/x^(25/3)-15*a^7*b^3/x^8-630/23*a^6*b^4/x^(23/3)-378/11*a^5*b^5
/x^(22/3)-30*a^4*b^6/x^7-18*a^3*b^7/x^(20/3)-135/19*a^2*b^8/x^(19/3)-5/3*a*b^9/x^6-3/17*b^10/x^(17/3)

________________________________________________________________________________________

Maxima [A]  time = 0.966345, size = 151, normalized size = 1.06 \begin{align*} -\frac{8436285 \, b^{10} x^{\frac{10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac{8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac{7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac{5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac{4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac{2}{3}} + 55160325 \, a^{9} b x^{\frac{1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^8*x^(8/3) + 860501070*a^3*b^7*x^(7/3
) + 1434168450*a^4*b^6*x^2 + 1642774770*a^5*b^5*x^(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 2
58150321*a^8*b^2*x^(2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.533, size = 336, normalized size = 2.37 \begin{align*} -\frac{79676025 \, a b^{9} x^{3} + 1434168450 \, a^{4} b^{6} x^{2} + 717084225 \, a^{7} b^{3} x + 5311735 \, a^{10} + 1235169 \,{\left (275 \, a^{2} b^{8} x^{2} + 1330 \, a^{5} b^{5} x + 209 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 28215 \,{\left (299 \, b^{10} x^{3} + 30498 \, a^{3} b^{7} x^{2} + 46410 \, a^{6} b^{4} x + 1955 \, a^{9} b\right )} x^{\frac{1}{3}}}{47805615 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/47805615*(79676025*a*b^9*x^3 + 1434168450*a^4*b^6*x^2 + 717084225*a^7*b^3*x + 5311735*a^10 + 1235169*(275*a
^2*b^8*x^2 + 1330*a^5*b^5*x + 209*a^8*b^2)*x^(2/3) + 28215*(299*b^10*x^3 + 30498*a^3*b^7*x^2 + 46410*a^6*b^4*x
+ 1955*a^9*b)*x^(1/3))/x^9

________________________________________________________________________________________

Sympy [A]  time = 43.9362, size = 144, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{9 x^{9}} - \frac{15 a^{9} b}{13 x^{\frac{26}{3}}} - \frac{27 a^{8} b^{2}}{5 x^{\frac{25}{3}}} - \frac{15 a^{7} b^{3}}{x^{8}} - \frac{630 a^{6} b^{4}}{23 x^{\frac{23}{3}}} - \frac{378 a^{5} b^{5}}{11 x^{\frac{22}{3}}} - \frac{30 a^{4} b^{6}}{x^{7}} - \frac{18 a^{3} b^{7}}{x^{\frac{20}{3}}} - \frac{135 a^{2} b^{8}}{19 x^{\frac{19}{3}}} - \frac{5 a b^{9}}{3 x^{6}} - \frac{3 b^{10}}{17 x^{\frac{17}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**10/(9*x**9) - 15*a**9*b/(13*x**(26/3)) - 27*a**8*b**2/(5*x**(25/3)) - 15*a**7*b**3/x**8 - 630*a**6*b**4/(2
3*x**(23/3)) - 378*a**5*b**5/(11*x**(22/3)) - 30*a**4*b**6/x**7 - 18*a**3*b**7/x**(20/3) - 135*a**2*b**8/(19*x
**(19/3)) - 5*a*b**9/(3*x**6) - 3*b**10/(17*x**(17/3))

________________________________________________________________________________________

Giac [A]  time = 1.15399, size = 151, normalized size = 1.06 \begin{align*} -\frac{8436285 \, b^{10} x^{\frac{10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac{8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac{7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac{5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac{4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac{2}{3}} + 55160325 \, a^{9} b x^{\frac{1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^8*x^(8/3) + 860501070*a^3*b^7*x^(7/3
) + 1434168450*a^4*b^6*x^2 + 1642774770*a^5*b^5*x^(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 2
58150321*a^8*b^2*x^(2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9