\(\int (a+b \sqrt [3]{x})^{10} \, dx\) [2328]

Optimal result

Integrand size = 11, antiderivative size = 59 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}-\frac {a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3} \]

[Out]

3/11*a^2*(a+b*x^(1/3))^11/b^3-1/2*a*(a+b*x^(1/3))^12/b^3+3/13*(a+b*x^(1/3))^13/b^3

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}-\frac {a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3} \]

[In]

Int[(a + b*x^(1/3))^10,x]

[Out]

(3*a^2*(a + b*x^(1/3))^11)/(11*b^3) - (a*(a + b*x^(1/3))^12)/(2*b^3) + (3*(a + b*x^(1/3))^13)/(13*b^3)

Rule 45

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 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^{10}}{b^2}-\frac {2 a (a+b x)^{11}}{b^2}+\frac {(a+b x)^{12}}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}-\frac {a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(59)=118\).

Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.17 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {1}{286} \left (286 a^{10} x+2145 a^9 b x^{4/3}+7722 a^8 b^2 x^{5/3}+17160 a^7 b^3 x^2+25740 a^6 b^4 x^{7/3}+27027 a^5 b^5 x^{8/3}+20020 a^4 b^6 x^3+10296 a^3 b^7 x^{10/3}+3510 a^2 b^8 x^{11/3}+715 a b^9 x^4+66 b^{10} x^{13/3}\right ) \]

[In]

Integrate[(a + b*x^(1/3))^10,x]

[Out]

(286*a^10*x + 2145*a^9*b*x^(4/3) + 7722*a^8*b^2*x^(5/3) + 17160*a^7*b^3*x^2 + 25740*a^6*b^4*x^(7/3) + 27027*a^
5*b^5*x^(8/3) + 20020*a^4*b^6*x^3 + 10296*a^3*b^7*x^(10/3) + 3510*a^2*b^8*x^(11/3) + 715*a*b^9*x^4 + 66*b^10*x
^(13/3))/286

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(47)=94\).

Time = 3.68 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.86

method	result	size
derivativedivides	\(\frac {3 b^{10} x^{\frac {13}{3}}}{13}+\frac {5 a \,b^{9} x^{4}}{2}+\frac {135 a^{2} b^{8} x^{\frac {11}{3}}}{11}+36 a^{3} b^{7} x^{\frac {10}{3}}+70 a^{4} b^{6} x^{3}+\frac {189 a^{5} b^{5} x^{\frac {8}{3}}}{2}+90 a^{6} b^{4} x^{\frac {7}{3}}+60 a^{7} b^{3} x^{2}+27 a^{8} b^{2} x^{\frac {5}{3}}+\frac {15 a^{9} b \,x^{\frac {4}{3}}}{2}+a^{10} x\)	\(110\)
default	\(\frac {3 b^{10} x^{\frac {13}{3}}}{13}+\frac {5 a \,b^{9} x^{4}}{2}+\frac {135 a^{2} b^{8} x^{\frac {11}{3}}}{11}+36 a^{3} b^{7} x^{\frac {10}{3}}+70 a^{4} b^{6} x^{3}+\frac {189 a^{5} b^{5} x^{\frac {8}{3}}}{2}+90 a^{6} b^{4} x^{\frac {7}{3}}+60 a^{7} b^{3} x^{2}+27 a^{8} b^{2} x^{\frac {5}{3}}+\frac {15 a^{9} b \,x^{\frac {4}{3}}}{2}+a^{10} x\)	\(110\)
trager	\(\frac {a \left (5 b^{9} x^{3}+140 a^{3} b^{6} x^{2}+5 b^{9} x^{2}+120 x \,a^{6} b^{3}+140 a^{3} b^{6} x +5 x \,b^{9}+2 a^{9}+120 a^{6} b^{3}+140 a^{3} b^{6}+5 b^{9}\right ) \left (-1+x \right )}{2}+\frac {3 b \,x^{\frac {4}{3}} \left (2 b^{9} x^{3}+312 a^{3} b^{6} x^{2}+780 x \,a^{6} b^{3}+65 a^{9}\right )}{26}+\frac {27 a^{2} b^{2} x^{\frac {5}{3}} \left (10 b^{6} x^{2}+77 a^{3} b^{3} x +22 a^{6}\right )}{22}\)	\(160\)

[In]

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

[Out]

3/13*b^10*x^(13/3)+5/2*a*b^9*x^4+135/11*a^2*b^8*x^(11/3)+36*a^3*b^7*x^(10/3)+70*a^4*b^6*x^3+189/2*a^5*b^5*x^(8
/3)+90*a^6*b^4*x^(7/3)+60*a^7*b^3*x^2+27*a^8*b^2*x^(5/3)+15/2*a^9*b*x^(4/3)+a^10*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (47) = 94\).

Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.98 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {5}{2} \, a b^{9} x^{4} + 70 \, a^{4} b^{6} x^{3} + 60 \, a^{7} b^{3} x^{2} + a^{10} x + \frac {27}{22} \, {\left (10 \, a^{2} b^{8} x^{3} + 77 \, a^{5} b^{5} x^{2} + 22 \, a^{8} b^{2} x\right )} x^{\frac {2}{3}} + \frac {3}{26} \, {\left (2 \, b^{10} x^{4} + 312 \, a^{3} b^{7} x^{3} + 780 \, a^{6} b^{4} x^{2} + 65 \, a^{9} b x\right )} x^{\frac {1}{3}} \]

[In]

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

[Out]

5/2*a*b^9*x^4 + 70*a^4*b^6*x^3 + 60*a^7*b^3*x^2 + a^10*x + 27/22*(10*a^2*b^8*x^3 + 77*a^5*b^5*x^2 + 22*a^8*b^2
*x)*x^(2/3) + 3/26*(2*b^10*x^4 + 312*a^3*b^7*x^3 + 780*a^6*b^4*x^2 + 65*a^9*b*x)*x^(1/3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (53) = 106\).

Time = 0.48 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.31 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=a^{10} x + \frac {15 a^{9} b x^{\frac {4}{3}}}{2} + 27 a^{8} b^{2} x^{\frac {5}{3}} + 60 a^{7} b^{3} x^{2} + 90 a^{6} b^{4} x^{\frac {7}{3}} + \frac {189 a^{5} b^{5} x^{\frac {8}{3}}}{2} + 70 a^{4} b^{6} x^{3} + 36 a^{3} b^{7} x^{\frac {10}{3}} + \frac {135 a^{2} b^{8} x^{\frac {11}{3}}}{11} + \frac {5 a b^{9} x^{4}}{2} + \frac {3 b^{10} x^{\frac {13}{3}}}{13} \]

[In]

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

[Out]

a**10*x + 15*a**9*b*x**(4/3)/2 + 27*a**8*b**2*x**(5/3) + 60*a**7*b**3*x**2 + 90*a**6*b**4*x**(7/3) + 189*a**5*
b**5*x**(8/3)/2 + 70*a**4*b**6*x**3 + 36*a**3*b**7*x**(10/3) + 135*a**2*b**8*x**(11/3)/11 + 5*a*b**9*x**4/2 +
3*b**10*x**(13/3)/13

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13}}{13 \, b^{3}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{12} a}{2 \, b^{3}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{2}}{11 \, b^{3}} \]

[In]

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

[Out]

3/13*(b*x^(1/3) + a)^13/b^3 - 1/2*(b*x^(1/3) + a)^12*a/b^3 + 3/11*(b*x^(1/3) + a)^11*a^2/b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (47) = 94\).

Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=\frac {3}{13} \, b^{10} x^{\frac {13}{3}} + \frac {5}{2} \, a b^{9} x^{4} + \frac {135}{11} \, a^{2} b^{8} x^{\frac {11}{3}} + 36 \, a^{3} b^{7} x^{\frac {10}{3}} + 70 \, a^{4} b^{6} x^{3} + \frac {189}{2} \, a^{5} b^{5} x^{\frac {8}{3}} + 90 \, a^{6} b^{4} x^{\frac {7}{3}} + 60 \, a^{7} b^{3} x^{2} + 27 \, a^{8} b^{2} x^{\frac {5}{3}} + \frac {15}{2} \, a^{9} b x^{\frac {4}{3}} + a^{10} x \]

[In]

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

[Out]

3/13*b^10*x^(13/3) + 5/2*a*b^9*x^4 + 135/11*a^2*b^8*x^(11/3) + 36*a^3*b^7*x^(10/3) + 70*a^4*b^6*x^3 + 189/2*a^
5*b^5*x^(8/3) + 90*a^6*b^4*x^(7/3) + 60*a^7*b^3*x^2 + 27*a^8*b^2*x^(5/3) + 15/2*a^9*b*x^(4/3) + a^10*x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} \, dx=a^{10}\,x+\frac {3\,b^{10}\,x^{13/3}}{13}+\frac {5\,a\,b^9\,x^4}{2}+\frac {15\,a^9\,b\,x^{4/3}}{2}+60\,a^7\,b^3\,x^2+70\,a^4\,b^6\,x^3+27\,a^8\,b^2\,x^{5/3}+90\,a^6\,b^4\,x^{7/3}+\frac {189\,a^5\,b^5\,x^{8/3}}{2}+36\,a^3\,b^7\,x^{10/3}+\frac {135\,a^2\,b^8\,x^{11/3}}{11} \]

[In]

int((a + b*x^(1/3))^10,x)

[Out]

a^10*x + (3*b^10*x^(13/3))/13 + (5*a*b^9*x^4)/2 + (15*a^9*b*x^(4/3))/2 + 60*a^7*b^3*x^2 + 70*a^4*b^6*x^3 + 27*
a^8*b^2*x^(5/3) + 90*a^6*b^4*x^(7/3) + (189*a^5*b^5*x^(8/3))/2 + 36*a^3*b^7*x^(10/3) + (135*a^2*b^8*x^(11/3))/
11