Integrand size = 20, antiderivative size = 103 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {6}{11} a \left (b^2+a c\right ) x^{11/2}+\frac {2}{15} b \left (b^2+6 a c\right ) x^{15/2}+\frac {6}{19} c \left (b^2+a c\right ) x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]
2/3*a^3*x^(3/2)+6/7*a^2*b*x^(7/2)+6/11*a*(a*c+b^2)*x^(11/2)+2/15*b*(6*a*c+ b^2)*x^(15/2)+6/19*c*(a*c+b^2)*x^(19/2)+6/23*b*c^2*x^(23/2)+2/27*c^3*x^(27 /2)
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{3} a^3 x^{3/2}+\frac {6}{77} a^2 x^{7/2} \left (11 b+7 c x^2\right )+\frac {2 a x^{11/2} \left (285 b^2+418 b c x^2+165 c^2 x^4\right )}{1045}+\frac {2 x^{15/2} \left (3933 b^3+9315 b^2 c x^2+7695 b c^2 x^4+2185 c^3 x^6\right )}{58995} \]
(2*a^3*x^(3/2))/3 + (6*a^2*x^(7/2)*(11*b + 7*c*x^2))/77 + (2*a*x^(11/2)*(2 85*b^2 + 418*b*c*x^2 + 165*c^2*x^4))/1045 + (2*x^(15/2)*(3933*b^3 + 9315*b ^2*c*x^2 + 7695*b*c^2*x^4 + 2185*c^3*x^6))/58995
Time = 0.23 (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.
\(\displaystyle \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \int \left (a^3 \sqrt {x}+3 a^2 b x^{5/2}+3 c x^{17/2} \left (a c+b^2\right )+b x^{13/2} \left (6 a c+b^2\right )+3 a x^{9/2} \left (a c+b^2\right )+3 b c^2 x^{21/2}+c^3 x^{25/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {6}{19} c x^{19/2} \left (a c+b^2\right )+\frac {2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac {6}{11} a x^{11/2} \left (a c+b^2\right )+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2}\) |
(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(b^2 + 6*a*c)*x^(15/2))/15 + (6*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2 *x^(23/2))/23 + (2*c^3*x^(27/2))/27
3.11.57.3.1 Defintions of rubi rules used
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 = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (168245 c^{3} x^{12}+592515 b \,c^{2} x^{10}+717255 a \,c^{2} x^{8}+717255 b^{2} c \,x^{8}+1817046 a b c \,x^{6}+302841 b^{3} x^{6}+1238895 a^{2} c \,x^{4}+1238895 b^{2} x^{4} a +1946835 a^{2} b \,x^{2}+1514205 a^{3}\right )}{4542615}\) | \(90\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (168245 c^{3} x^{12}+592515 b \,c^{2} x^{10}+717255 a \,c^{2} x^{8}+717255 b^{2} c \,x^{8}+1817046 a b c \,x^{6}+302841 b^{3} x^{6}+1238895 a^{2} c \,x^{4}+1238895 b^{2} x^{4} a +1946835 a^{2} b \,x^{2}+1514205 a^{3}\right )}{4542615}\) | \(90\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (168245 c^{3} x^{12}+592515 b \,c^{2} x^{10}+717255 a \,c^{2} x^{8}+717255 b^{2} c \,x^{8}+1817046 a b c \,x^{6}+302841 b^{3} x^{6}+1238895 a^{2} c \,x^{4}+1238895 b^{2} x^{4} a +1946835 a^{2} b \,x^{2}+1514205 a^{3}\right )}{4542615}\) | \(90\) |
derivativedivides | \(\frac {2 c^{3} x^{\frac {27}{2}}}{27}+\frac {6 b \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {2 a^{3} x^{\frac {3}{2}}}{3}\) | \(111\) |
default | \(\frac {2 c^{3} x^{\frac {27}{2}}}{27}+\frac {6 b \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {2 a^{3} x^{\frac {3}{2}}}{3}\) | \(111\) |
2/4542615*x^(3/2)*(168245*c^3*x^12+592515*b*c^2*x^10+717255*a*c^2*x^8+7172 55*b^2*c*x^8+1817046*a*b*c*x^6+302841*b^3*x^6+1238895*a^2*c*x^4+1238895*a* b^2*x^4+1946835*a^2*b*x^2+1514205*a^3)
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{4542615} \, {\left (168245 \, c^{3} x^{13} + 592515 \, b c^{2} x^{11} + 717255 \, {\left (b^{2} c + a c^{2}\right )} x^{9} + 302841 \, {\left (b^{3} + 6 \, a b c\right )} x^{7} + 1946835 \, a^{2} b x^{3} + 1238895 \, {\left (a b^{2} + a^{2} c\right )} x^{5} + 1514205 \, a^{3} x\right )} \sqrt {x} \]
2/4542615*(168245*c^3*x^13 + 592515*b*c^2*x^11 + 717255*(b^2*c + a*c^2)*x^ 9 + 302841*(b^3 + 6*a*b*c)*x^7 + 1946835*a^2*b*x^3 + 1238895*(a*b^2 + a^2* c)*x^5 + 1514205*a^3*x)*sqrt(x)
Time = 1.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 a^{3} x^{\frac {3}{2}}}{3} + \frac {6 a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 b c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} + \frac {2 x^{\frac {19}{2}} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{19} + \frac {2 x^{\frac {15}{2}} \cdot \left (6 a b c + b^{3}\right )}{15} + \frac {2 x^{\frac {11}{2}} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{11} \]
2*a**3*x**(3/2)/3 + 6*a**2*b*x**(7/2)/7 + 6*b*c**2*x**(23/2)/23 + 2*c**3*x **(27/2)/27 + 2*x**(19/2)*(3*a*c**2 + 3*b**2*c)/19 + 2*x**(15/2)*(6*a*b*c + b**3)/15 + 2*x**(11/2)*(3*a**2*c + 3*a*b**2)/11
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {15}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {6}{11} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]
2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*(b^2*c + a*c^2)*x^(19/2) + 2/15*(b^3 + 6*a*b*c)*x^(15/2) + 6/7*a^2*b*x^(7/2) + 6/11*(a*b^2 + a^2*c)*x ^(11/2) + 2/3*a^3*x^(3/2)
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} + \frac {4}{5} \, a b c x^{\frac {15}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]
2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*b^2*c*x^(19/2) + 6/19*a*c^2 *x^(19/2) + 2/15*b^3*x^(15/2) + 4/5*a*b*c*x^(15/2) + 6/11*a*b^2*x^(11/2) + 6/11*a^2*c*x^(11/2) + 6/7*a^2*b*x^(7/2) + 2/3*a^3*x^(3/2)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx=x^{15/2}\,\left (\frac {2\,b^3}{15}+\frac {4\,a\,c\,b}{5}\right )+\frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,a^2\,b\,x^{7/2}}{7}+\frac {6\,b\,c^2\,x^{23/2}}{23}+\frac {6\,a\,x^{11/2}\,\left (b^2+a\,c\right )}{11}+\frac {6\,c\,x^{19/2}\,\left (b^2+a\,c\right )}{19} \]