Integrand size = 18, antiderivative size = 114 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=a^3 x+2 a^2 c x^{3/2}+\frac {9}{5} a^2 b x^{5/3}+\frac {3}{2} a c^2 x^2+\frac {36}{13} a b c x^{13/6}+\frac {9}{7} a b^2 x^{7/3}+\frac {2}{5} c^3 x^{5/2}+\frac {9}{8} b c^2 x^{8/3}+\frac {18}{17} b^2 c x^{17/6}+\frac {b^3 x^3}{3} \] Output:
a^3*x+2*a^2*c*x^(3/2)+9/5*a^2*b*x^(5/3)+3/2*a*c^2*x^2+36/13*a*b*c*x^(13/6) +9/7*a*b^2*x^(7/3)+2/5*c^3*x^(5/2)+9/8*b*c^2*x^(8/3)+18/17*b^2*c*x^(17/6)+ 1/3*b^3*x^3
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=\frac {185640 a^3 x+371280 a^2 c x^{3/2}+334152 a^2 b x^{5/3}+278460 a c^2 x^2+514080 a b c x^{13/6}+238680 a b^2 x^{7/3}+74256 c^3 x^{5/2}+208845 b c^2 x^{8/3}+196560 b^2 c x^{17/6}+61880 b^3 x^3}{185640} \] Input:
Integrate[(a + c*Sqrt[x] + b*x^(2/3))^3,x]
Output:
(185640*a^3*x + 371280*a^2*c*x^(3/2) + 334152*a^2*b*x^(5/3) + 278460*a*c^2 *x^2 + 514080*a*b*c*x^(13/6) + 238680*a*b^2*x^(7/3) + 74256*c^3*x^(5/2) + 208845*b*c^2*x^(8/3) + 196560*b^2*c*x^(17/6) + 61880*b^3*x^3)/185640
Time = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {7267, 2465, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b x^{2/3}+c \sqrt {x}\right )^3 \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 6 \int \left (a+\left (\sqrt [6]{x} b+c\right ) \sqrt {x}\right )^3 x^{5/6}d\sqrt [6]{x}\) |
\(\Big \downarrow \) 2465 |
\(\displaystyle 6 \int \left (x^{5/6} a^3+3 b x^{3/2} a^2+3 c x^{4/3} a^2+3 b^2 x^{13/6} a+6 b c x^2 a+3 c^2 x^{11/6} a+b^3 x^{17/6}+3 b^2 c x^{8/3}+3 b c^2 x^{5/2}+c^3 x^{7/3}\right )d\sqrt [6]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \left (\frac {a^3 x}{6}+\frac {3}{10} a^2 b x^{5/3}+\frac {1}{3} a^2 c x^{3/2}+\frac {3}{14} a b^2 x^{7/3}+\frac {6}{13} a b c x^{13/6}+\frac {1}{4} a c^2 x^2+\frac {b^3 x^3}{18}+\frac {3}{17} b^2 c x^{17/6}+\frac {3}{16} b c^2 x^{8/3}+\frac {1}{15} c^3 x^{5/2}\right )\) |
Input:
Int[(a + c*Sqrt[x] + b*x^(2/3))^3,x]
Output:
6*((a^3*x)/6 + (a^2*c*x^(3/2))/3 + (3*a^2*b*x^(5/3))/10 + (a*c^2*x^2)/4 + (6*a*b*c*x^(13/6))/13 + (3*a*b^2*x^(7/3))/14 + (c^3*x^(5/2))/15 + (3*b*c^2 *x^(8/3))/16 + (3*b^2*c*x^(17/6))/17 + (b^3*x^3)/18)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ [Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x ] && IGtQ[p, 0]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 4.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(a^{3} x +2 a^{2} c \,x^{\frac {3}{2}}+\frac {9 a^{2} b \,x^{\frac {5}{3}}}{5}+\frac {3 a \,c^{2} x^{2}}{2}+\frac {36 a b c \,x^{\frac {13}{6}}}{13}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {2 c^{3} x^{\frac {5}{2}}}{5}+\frac {9 b \,c^{2} x^{\frac {8}{3}}}{8}+\frac {18 b^{2} c \,x^{\frac {17}{6}}}{17}+\frac {b^{3} x^{3}}{3}\) | \(85\) |
default | \(\frac {2 c^{3} x^{\frac {5}{2}}}{5}+3 c^{2} \left (\frac {3 b \,x^{\frac {8}{3}}}{8}+\frac {a \,x^{2}}{2}\right )+3 c \left (\frac {6 b^{2} x^{\frac {17}{6}}}{17}+\frac {12 a b \,x^{\frac {13}{6}}}{13}+\frac {2 a^{2} x^{\frac {3}{2}}}{3}\right )+a^{3} x +\frac {b^{3} x^{3}}{3}+\frac {9 a^{2} b \,x^{\frac {5}{3}}}{5}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}\) | \(86\) |
orering | \(\text {Expression too large to display}\) | \(1457\) |
Input:
int((a+c*x^(1/2)+b*x^(2/3))^3,x,method=_RETURNVERBOSE)
Output:
a^3*x+2*a^2*c*x^(3/2)+9/5*a^2*b*x^(5/3)+3/2*a*c^2*x^2+36/13*a*b*c*x^(13/6) +9/7*a*b^2*x^(7/3)+2/5*c^3*x^(5/2)+9/8*b*c^2*x^(8/3)+18/17*b^2*c*x^(17/6)+ 1/3*b^3*x^3
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.80 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {36}{13} \, a b c x^{\frac {13}{6}} + \frac {3}{2} \, a c^{2} x^{2} + a^{3} x + \frac {9}{40} \, {\left (5 \, b c^{2} x^{2} + 8 \, a^{2} b x\right )} x^{\frac {2}{3}} + \frac {2}{5} \, {\left (c^{3} x^{2} + 5 \, a^{2} c x\right )} \sqrt {x} \] Input:
integrate((a+c*x^(1/2)+b*x^(2/3))^3,x, algorithm="fricas")
Output:
1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/7*a*b^2*x^(7/3) + 36/13*a*b*c*x^(13 /6) + 3/2*a*c^2*x^2 + a^3*x + 9/40*(5*b*c^2*x^2 + 8*a^2*b*x)*x^(2/3) + 2/5 *(c^3*x^2 + 5*a^2*c*x)*sqrt(x)
Time = 0.92 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=a^{3} x + \frac {9 a^{2} b x^{\frac {5}{3}}}{5} + 2 a^{2} c x^{\frac {3}{2}} + \frac {9 a b^{2} x^{\frac {7}{3}}}{7} + \frac {36 a b c x^{\frac {13}{6}}}{13} + \frac {3 a c^{2} x^{2}}{2} + \frac {b^{3} x^{3}}{3} + \frac {18 b^{2} c x^{\frac {17}{6}}}{17} + \frac {9 b c^{2} x^{\frac {8}{3}}}{8} + \frac {2 c^{3} x^{\frac {5}{2}}}{5} \] Input:
integrate((a+c*x**(1/2)+b*x**(2/3))**3,x)
Output:
a**3*x + 9*a**2*b*x**(5/3)/5 + 2*a**2*c*x**(3/2) + 9*a*b**2*x**(7/3)/7 + 3 6*a*b*c*x**(13/6)/13 + 3*a*c**2*x**2/2 + b**3*x**3/3 + 18*b**2*c*x**(17/6) /17 + 9*b*c**2*x**(8/3)/8 + 2*c**3*x**(5/2)/5
Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{8} \, b c^{2} x^{\frac {8}{3}} + \frac {2}{5} \, c^{3} x^{\frac {5}{2}} + a^{3} x + \frac {1}{5} \, {\left (9 \, b x^{\frac {5}{3}} + 10 \, c x^{\frac {3}{2}}\right )} a^{2} + \frac {3}{182} \, {\left (78 \, b^{2} x^{\frac {7}{3}} + 168 \, b c x^{\frac {13}{6}} + 91 \, c^{2} x^{2}\right )} a \] Input:
integrate((a+c*x^(1/2)+b*x^(2/3))^3,x, algorithm="maxima")
Output:
1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/8*b*c^2*x^(8/3) + 2/5*c^3*x^(5/2) + a^3*x + 1/5*(9*b*x^(5/3) + 10*c*x^(3/2))*a^2 + 3/182*(78*b^2*x^(7/3) + 16 8*b*c*x^(13/6) + 91*c^2*x^2)*a
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{8} \, b c^{2} x^{\frac {8}{3}} + \frac {2}{5} \, c^{3} x^{\frac {5}{2}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {36}{13} \, a b c x^{\frac {13}{6}} + \frac {3}{2} \, a c^{2} x^{2} + \frac {9}{5} \, a^{2} b x^{\frac {5}{3}} + 2 \, a^{2} c x^{\frac {3}{2}} + a^{3} x \] Input:
integrate((a+c*x^(1/2)+b*x^(2/3))^3,x, algorithm="giac")
Output:
1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/8*b*c^2*x^(8/3) + 2/5*c^3*x^(5/2) + 9/7*a*b^2*x^(7/3) + 36/13*a*b*c*x^(13/6) + 3/2*a*c^2*x^2 + 9/5*a^2*b*x^(5 /3) + 2*a^2*c*x^(3/2) + a^3*x
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=a^3\,x+\frac {b^3\,x^3}{3}+\frac {2\,c^3\,x^{5/2}}{5}+\frac {9\,a^2\,b\,x^{5/3}}{5}+\frac {9\,a\,b^2\,x^{7/3}}{7}+\frac {3\,a\,c^2\,x^2}{2}+2\,a^2\,c\,x^{3/2}+\frac {9\,b\,c^2\,x^{8/3}}{8}+\frac {18\,b^2\,c\,x^{17/6}}{17}+\frac {36\,a\,b\,c\,x^{13/6}}{13} \] Input:
int((a + b*x^(2/3) + c*x^(1/2))^3,x)
Output:
a^3*x + (b^3*x^3)/3 + (2*c^3*x^(5/2))/5 + (9*a^2*b*x^(5/3))/5 + (9*a*b^2*x ^(7/3))/7 + (3*a*c^2*x^2)/2 + 2*a^2*c*x^(3/2) + (9*b*c^2*x^(8/3))/8 + (18* b^2*c*x^(17/6))/17 + (36*a*b*c*x^(13/6))/13
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \left (a+c \sqrt {x}+b x^{2/3}\right )^3 \, dx=\frac {x \left (196560 x^{\frac {11}{6}} b^{2} c +514080 x^{\frac {7}{6}} a b c +334152 x^{\frac {2}{3}} a^{2} b +208845 x^{\frac {5}{3}} b \,c^{2}+238680 x^{\frac {4}{3}} a \,b^{2}+371280 \sqrt {x}\, a^{2} c +74256 \sqrt {x}\, c^{3} x +185640 a^{3}+278460 a \,c^{2} x +61880 b^{3} x^{2}\right )}{185640} \] Input:
int((a+c*x^(1/2)+b*x^(2/3))^3,x)
Output:
(x*(196560*x**(5/6)*b**2*c*x + 514080*x**(1/6)*a*b*c*x + 334152*x**(2/3)*a **2*b + 208845*x**(2/3)*b*c**2*x + 238680*x**(1/3)*a*b**2*x + 371280*sqrt( x)*a**2*c + 74256*sqrt(x)*c**3*x + 185640*a**3 + 278460*a*c**2*x + 61880*b **3*x**2))/185640