Integrand size = 20, antiderivative size = 103 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{5} a^3 x^{5/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{13} a \left (b^2+a c\right ) x^{13/2}+\frac {2}{17} b \left (b^2+6 a c\right ) x^{17/2}+\frac {2}{7} c \left (b^2+a c\right ) x^{21/2}+\frac {6}{25} b c^2 x^{25/2}+\frac {2}{29} c^3 x^{29/2} \] Output:
2/5*a^3*x^(5/2)+2/3*a^2*b*x^(9/2)+6/13*a*(a*c+b^2)*x^(13/2)+2/17*b*(6*a*c+ b^2)*x^(17/2)+2/7*c*(a*c+b^2)*x^(21/2)+6/25*b*c^2*x^(25/2)+2/29*c^3*x^(29/ 2)
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 \left (672945 a^3 x^{5/2}+1121575 a^2 b x^{9/2}+776475 a b^2 x^{13/2}+776475 a^2 c x^{13/2}+197925 b^3 x^{17/2}+1187550 a b c x^{17/2}+480675 b^2 c x^{21/2}+480675 a c^2 x^{21/2}+403767 b c^2 x^{25/2}+116025 c^3 x^{29/2}\right )}{3364725} \] Input:
Integrate[x^(3/2)*(a + b*x^2 + c*x^4)^3,x]
Output:
(2*(672945*a^3*x^(5/2) + 1121575*a^2*b*x^(9/2) + 776475*a*b^2*x^(13/2) + 7 76475*a^2*c*x^(13/2) + 197925*b^3*x^(17/2) + 1187550*a*b*c*x^(17/2) + 4806 75*b^2*c*x^(21/2) + 480675*a*c^2*x^(21/2) + 403767*b*c^2*x^(25/2) + 116025 *c^3*x^(29/2)))/3364725
Time = 0.38 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1433, 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.
\(\displaystyle \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \int \left (a^3 x^{3/2}+3 a^2 b x^{7/2}+3 c x^{19/2} \left (a c+b^2\right )+b x^{15/2} \left (6 a c+b^2\right )+3 a x^{11/2} \left (a c+b^2\right )+3 b c^2 x^{23/2}+c^3 x^{27/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{5} a^3 x^{5/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {2}{7} c x^{21/2} \left (a c+b^2\right )+\frac {2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac {6}{13} a x^{13/2} \left (a c+b^2\right )+\frac {6}{25} b c^2 x^{25/2}+\frac {2}{29} c^3 x^{29/2}\) |
Input:
Int[x^(3/2)*(a + b*x^2 + c*x^4)^3,x]
Output:
(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*(b^2 + a*c)*x^(13/2))/13 + (2*b*(b^2 + 6*a*c)*x^(17/2))/17 + (2*c*(b^2 + a*c)*x^(21/2))/7 + (6*b*c^2* x^(25/2))/25 + (2*c^3*x^(29/2))/29
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 a \,c^{2} x^{8}+480675 b^{2} c \,x^{8}+1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 a^{2} c \,x^{4}+776475 b^{2} x^{4} a +1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 a \,c^{2} x^{8}+480675 b^{2} c \,x^{8}+1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 a^{2} c \,x^{4}+776475 b^{2} x^{4} a +1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 a \,c^{2} x^{8}+480675 b^{2} c \,x^{8}+1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 a^{2} c \,x^{4}+776475 b^{2} x^{4} a +1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
orering | \(\frac {2 x^{\frac {5}{2}} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 a \,c^{2} x^{8}+480675 b^{2} c \,x^{8}+1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 a^{2} c \,x^{4}+776475 b^{2} x^{4} a +1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725}\) | \(90\) |
derivativedivides | \(\frac {2 c^{3} x^{\frac {29}{2}}}{29}+\frac {6 b \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {2 a^{3} x^{\frac {5}{2}}}{5}\) | \(111\) |
default | \(\frac {2 c^{3} x^{\frac {29}{2}}}{29}+\frac {6 b \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {2 a^{3} x^{\frac {5}{2}}}{5}\) | \(111\) |
Input:
int(x^(3/2)*(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
2/3364725*x^(5/2)*(116025*c^3*x^12+403767*b*c^2*x^10+480675*a*c^2*x^8+4806 75*b^2*c*x^8+1187550*a*b*c*x^6+197925*b^3*x^6+776475*a^2*c*x^4+776475*a*b^ 2*x^4+1121575*a^2*b*x^2+672945*a^3)
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{3364725} \, {\left (116025 \, c^{3} x^{14} + 403767 \, b c^{2} x^{12} + 480675 \, {\left (b^{2} c + a c^{2}\right )} x^{10} + 197925 \, {\left (b^{3} + 6 \, a b c\right )} x^{8} + 1121575 \, a^{2} b x^{4} + 776475 \, {\left (a b^{2} + a^{2} c\right )} x^{6} + 672945 \, a^{3} x^{2}\right )} \sqrt {x} \] Input:
integrate(x^(3/2)*(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
2/3364725*(116025*c^3*x^14 + 403767*b*c^2*x^12 + 480675*(b^2*c + a*c^2)*x^ 10 + 197925*(b^3 + 6*a*b*c)*x^8 + 1121575*a^2*b*x^4 + 776475*(a*b^2 + a^2* c)*x^6 + 672945*a^3*x^2)*sqrt(x)
Time = 1.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 a^{3} x^{\frac {5}{2}}}{5} + \frac {2 a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 a b^{2} x^{\frac {13}{2}}}{13} + \frac {12 a b c x^{\frac {17}{2}}}{17} + \frac {2 a c^{2} x^{\frac {21}{2}}}{7} + \frac {2 b^{3} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c x^{\frac {21}{2}}}{7} + \frac {6 b c^{2} x^{\frac {25}{2}}}{25} + \frac {2 c^{3} x^{\frac {29}{2}}}{29} \] Input:
integrate(x**(3/2)*(c*x**4+b*x**2+a)**3,x)
Output:
2*a**3*x**(5/2)/5 + 2*a**2*b*x**(9/2)/3 + 6*a**2*c*x**(13/2)/13 + 6*a*b**2 *x**(13/2)/13 + 12*a*b*c*x**(17/2)/17 + 2*a*c**2*x**(21/2)/7 + 2*b**3*x**( 17/2)/17 + 2*b**2*c*x**(21/2)/7 + 6*b*c**2*x**(25/2)/25 + 2*c**3*x**(29/2) /29
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {6}{25} \, b c^{2} x^{\frac {25}{2}} + \frac {2}{7} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {6}{13} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \] Input:
integrate(x^(3/2)*(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*(b^2*c + a*c^2)*x^(21/2) + 2 /17*(b^3 + 6*a*b*c)*x^(17/2) + 2/3*a^2*b*x^(9/2) + 6/13*(a*b^2 + a^2*c)*x^ (13/2) + 2/5*a^3*x^(5/2)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {6}{25} \, b c^{2} x^{\frac {25}{2}} + \frac {2}{7} \, b^{2} c x^{\frac {21}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {2}{17} \, b^{3} x^{\frac {17}{2}} + \frac {12}{17} \, a b c x^{\frac {17}{2}} + \frac {6}{13} \, a b^{2} x^{\frac {13}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \] Input:
integrate(x^(3/2)*(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*b^2*c*x^(21/2) + 2/7*a*c^2*x ^(21/2) + 2/17*b^3*x^(17/2) + 12/17*a*b*c*x^(17/2) + 6/13*a*b^2*x^(13/2) + 6/13*a^2*c*x^(13/2) + 2/3*a^2*b*x^(9/2) + 2/5*a^3*x^(5/2)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=x^{17/2}\,\left (\frac {2\,b^3}{17}+\frac {12\,a\,c\,b}{17}\right )+\frac {2\,a^3\,x^{5/2}}{5}+\frac {2\,c^3\,x^{29/2}}{29}+\frac {2\,a^2\,b\,x^{9/2}}{3}+\frac {6\,b\,c^2\,x^{25/2}}{25}+\frac {6\,a\,x^{13/2}\,\left (b^2+a\,c\right )}{13}+\frac {2\,c\,x^{21/2}\,\left (b^2+a\,c\right )}{7} \] Input:
int(x^(3/2)*(a + b*x^2 + c*x^4)^3,x)
Output:
x^(17/2)*((2*b^3)/17 + (12*a*b*c)/17) + (2*a^3*x^(5/2))/5 + (2*c^3*x^(29/2 ))/29 + (2*a^2*b*x^(9/2))/3 + (6*b*c^2*x^(25/2))/25 + (6*a*x^(13/2)*(a*c + b^2))/13 + (2*c*x^(21/2)*(a*c + b^2))/7
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 \sqrt {x}\, x^{2} \left (116025 c^{3} x^{12}+403767 b \,c^{2} x^{10}+480675 a \,c^{2} x^{8}+480675 b^{2} c \,x^{8}+1187550 a b c \,x^{6}+197925 b^{3} x^{6}+776475 a^{2} c \,x^{4}+776475 a \,b^{2} x^{4}+1121575 a^{2} b \,x^{2}+672945 a^{3}\right )}{3364725} \] Input:
int(x^(3/2)*(c*x^4+b*x^2+a)^3,x)
Output:
(2*sqrt(x)*x**2*(672945*a**3 + 1121575*a**2*b*x**2 + 776475*a**2*c*x**4 + 776475*a*b**2*x**4 + 1187550*a*b*c*x**6 + 480675*a*c**2*x**8 + 197925*b**3 *x**6 + 480675*b**2*c*x**8 + 403767*b*c**2*x**10 + 116025*c**3*x**12))/336 4725