Integrand size = 20, antiderivative size = 99 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=-\frac {2 a^3}{\sqrt {x}}+2 a^2 b x^{3/2}+\frac {6}{7} a \left (b^2+a c\right ) x^{7/2}+\frac {2}{11} b \left (b^2+6 a c\right ) x^{11/2}+\frac {2}{5} c \left (b^2+a c\right ) x^{15/2}+\frac {6}{19} b c^2 x^{19/2}+\frac {2}{23} c^3 x^{23/2} \] Output:
-2*a^3/x^(1/2)+2*a^2*b*x^(3/2)+6/7*a*(a*c+b^2)*x^(7/2)+2/11*b*(6*a*c+b^2)* x^(11/2)+2/5*c*(a*c+b^2)*x^(15/2)+6/19*b*c^2*x^(19/2)+2/23*c^3*x^(23/2)
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=-\frac {2 \left (168245 a^3-168245 a^2 b x^2-72105 a b^2 x^4-72105 a^2 c x^4-15295 b^3 x^6-91770 a b c x^6-33649 b^2 c x^8-33649 a c^2 x^8-26565 b c^2 x^{10}-7315 c^3 x^{12}\right )}{168245 \sqrt {x}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^3/x^(3/2),x]
Output:
(-2*(168245*a^3 - 168245*a^2*b*x^2 - 72105*a*b^2*x^4 - 72105*a^2*c*x^4 - 1 5295*b^3*x^6 - 91770*a*b*c*x^6 - 33649*b^2*c*x^8 - 33649*a*c^2*x^8 - 26565 *b*c^2*x^10 - 7315*c^3*x^12))/(168245*Sqrt[x])
Time = 0.38 (sec) , antiderivative size = 99, 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.
\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \int \left (\frac {a^3}{x^{3/2}}+3 a^2 b \sqrt {x}+3 c x^{13/2} \left (a c+b^2\right )+b x^{9/2} \left (6 a c+b^2\right )+3 a x^{5/2} \left (a c+b^2\right )+3 b c^2 x^{17/2}+c^3 x^{21/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^3}{\sqrt {x}}+2 a^2 b x^{3/2}+\frac {2}{5} c x^{15/2} \left (a c+b^2\right )+\frac {2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac {6}{7} a x^{7/2} \left (a c+b^2\right )+\frac {6}{19} b c^2 x^{19/2}+\frac {2}{23} c^3 x^{23/2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^3/x^(3/2),x]
Output:
(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*(b^2 + a*c)*x^(7/2))/7 + (2*b*(b ^2 + 6*a*c)*x^(11/2))/11 + (2*c*(b^2 + a*c)*x^(15/2))/5 + (6*b*c^2*x^(19/2 ))/19 + (2*c^3*x^(23/2))/23
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 = 88, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 c^{3} x^{\frac {23}{2}}}{23}+\frac {6 b \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 a \,c^{2} x^{\frac {15}{2}}}{5}+\frac {2 b^{2} c \,x^{\frac {15}{2}}}{5}+\frac {12 a b c \,x^{\frac {11}{2}}}{11}+\frac {2 b^{3} x^{\frac {11}{2}}}{11}+\frac {6 a^{2} c \,x^{\frac {7}{2}}}{7}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+2 a^{2} b \,x^{\frac {3}{2}}-\frac {2 a^{3}}{\sqrt {x}}\) | \(88\) |
default | \(\frac {2 c^{3} x^{\frac {23}{2}}}{23}+\frac {6 b \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 a \,c^{2} x^{\frac {15}{2}}}{5}+\frac {2 b^{2} c \,x^{\frac {15}{2}}}{5}+\frac {12 a b c \,x^{\frac {11}{2}}}{11}+\frac {2 b^{3} x^{\frac {11}{2}}}{11}+\frac {6 a^{2} c \,x^{\frac {7}{2}}}{7}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+2 a^{2} b \,x^{\frac {3}{2}}-\frac {2 a^{3}}{\sqrt {x}}\) | \(88\) |
gosper | \(-\frac {2 \left (-7315 c^{3} x^{12}-26565 b \,c^{2} x^{10}-33649 a \,c^{2} x^{8}-33649 b^{2} c \,x^{8}-91770 a b c \,x^{6}-15295 b^{3} x^{6}-72105 a^{2} c \,x^{4}-72105 b^{2} x^{4} a -168245 a^{2} b \,x^{2}+168245 a^{3}\right )}{168245 \sqrt {x}}\) | \(90\) |
trager | \(-\frac {2 \left (-7315 c^{3} x^{12}-26565 b \,c^{2} x^{10}-33649 a \,c^{2} x^{8}-33649 b^{2} c \,x^{8}-91770 a b c \,x^{6}-15295 b^{3} x^{6}-72105 a^{2} c \,x^{4}-72105 b^{2} x^{4} a -168245 a^{2} b \,x^{2}+168245 a^{3}\right )}{168245 \sqrt {x}}\) | \(90\) |
risch | \(-\frac {2 \left (-7315 c^{3} x^{12}-26565 b \,c^{2} x^{10}-33649 a \,c^{2} x^{8}-33649 b^{2} c \,x^{8}-91770 a b c \,x^{6}-15295 b^{3} x^{6}-72105 a^{2} c \,x^{4}-72105 b^{2} x^{4} a -168245 a^{2} b \,x^{2}+168245 a^{3}\right )}{168245 \sqrt {x}}\) | \(90\) |
orering | \(-\frac {2 \left (-7315 c^{3} x^{12}-26565 b \,c^{2} x^{10}-33649 a \,c^{2} x^{8}-33649 b^{2} c \,x^{8}-91770 a b c \,x^{6}-15295 b^{3} x^{6}-72105 a^{2} c \,x^{4}-72105 b^{2} x^{4} a -168245 a^{2} b \,x^{2}+168245 a^{3}\right )}{168245 \sqrt {x}}\) | \(90\) |
Input:
int((c*x^4+b*x^2+a)^3/x^(3/2),x,method=_RETURNVERBOSE)
Output:
2/23*c^3*x^(23/2)+6/19*b*c^2*x^(19/2)+2/5*a*c^2*x^(15/2)+2/5*b^2*c*x^(15/2 )+12/11*a*b*c*x^(11/2)+2/11*b^3*x^(11/2)+6/7*a^2*c*x^(7/2)+6/7*a*b^2*x^(7/ 2)+2*a^2*b*x^(3/2)-2*a^3/x^(1/2)
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=\frac {2 \, {\left (7315 \, c^{3} x^{12} + 26565 \, b c^{2} x^{10} + 33649 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 15295 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 168245 \, a^{2} b x^{2} + 72105 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 168245 \, a^{3}\right )}}{168245 \, \sqrt {x}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(3/2),x, algorithm="fricas")
Output:
2/168245*(7315*c^3*x^12 + 26565*b*c^2*x^10 + 33649*(b^2*c + a*c^2)*x^8 + 1 5295*(b^3 + 6*a*b*c)*x^6 + 168245*a^2*b*x^2 + 72105*(a*b^2 + a^2*c)*x^4 - 168245*a^3)/sqrt(x)
Time = 0.92 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=- \frac {2 a^{3}}{\sqrt {x}} + 2 a^{2} b x^{\frac {3}{2}} + \frac {6 a^{2} c x^{\frac {7}{2}}}{7} + \frac {6 a b^{2} x^{\frac {7}{2}}}{7} + \frac {12 a b c x^{\frac {11}{2}}}{11} + \frac {2 a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 b^{3} x^{\frac {11}{2}}}{11} + \frac {2 b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 c^{3} x^{\frac {23}{2}}}{23} \] Input:
integrate((c*x**4+b*x**2+a)**3/x**(3/2),x)
Output:
-2*a**3/sqrt(x) + 2*a**2*b*x**(3/2) + 6*a**2*c*x**(7/2)/7 + 6*a*b**2*x**(7 /2)/7 + 12*a*b*c*x**(11/2)/11 + 2*a*c**2*x**(15/2)/5 + 2*b**3*x**(11/2)/11 + 2*b**2*c*x**(15/2)/5 + 6*b*c**2*x**(19/2)/19 + 2*c**3*x**(23/2)/23
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=\frac {2}{23} \, c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b c^{2} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {11}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} + \frac {6}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {7}{2}} - \frac {2 \, a^{3}}{\sqrt {x}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(3/2),x, algorithm="maxima")
Output:
2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*(b^2*c + a*c^2)*x^(15/2) + 2 /11*(b^3 + 6*a*b*c)*x^(11/2) + 2*a^2*b*x^(3/2) + 6/7*(a*b^2 + a^2*c)*x^(7/ 2) - 2*a^3/sqrt(x)
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=\frac {2}{23} \, c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b c^{2} x^{\frac {19}{2}} + \frac {2}{5} \, b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, a c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {12}{11} \, a b c x^{\frac {11}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{7} \, a^{2} c x^{\frac {7}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} - \frac {2 \, a^{3}}{\sqrt {x}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(3/2),x, algorithm="giac")
Output:
2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*b^2*c*x^(15/2) + 2/5*a*c^2*x ^(15/2) + 2/11*b^3*x^(11/2) + 12/11*a*b*c*x^(11/2) + 6/7*a*b^2*x^(7/2) + 6 /7*a^2*c*x^(7/2) + 2*a^2*b*x^(3/2) - 2*a^3/sqrt(x)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=x^{11/2}\,\left (\frac {2\,b^3}{11}+\frac {12\,a\,c\,b}{11}\right )-\frac {2\,a^3}{\sqrt {x}}+\frac {2\,c^3\,x^{23/2}}{23}+2\,a^2\,b\,x^{3/2}+\frac {6\,b\,c^2\,x^{19/2}}{19}+\frac {6\,a\,x^{7/2}\,\left (b^2+a\,c\right )}{7}+\frac {2\,c\,x^{15/2}\,\left (b^2+a\,c\right )}{5} \] Input:
int((a + b*x^2 + c*x^4)^3/x^(3/2),x)
Output:
x^(11/2)*((2*b^3)/11 + (12*a*b*c)/11) - (2*a^3)/x^(1/2) + (2*c^3*x^(23/2)) /23 + 2*a^2*b*x^(3/2) + (6*b*c^2*x^(19/2))/19 + (6*a*x^(7/2)*(a*c + b^2))/ 7 + (2*c*x^(15/2)*(a*c + b^2))/5
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx=\frac {\frac {2}{23} c^{3} x^{12}+\frac {6}{19} b \,c^{2} x^{10}+\frac {2}{5} a \,c^{2} x^{8}+\frac {2}{5} b^{2} c \,x^{8}+\frac {12}{11} a b c \,x^{6}+\frac {2}{11} b^{3} x^{6}+\frac {6}{7} a^{2} c \,x^{4}+\frac {6}{7} a \,b^{2} x^{4}+2 a^{2} b \,x^{2}-2 a^{3}}{\sqrt {x}} \] Input:
int((c*x^4+b*x^2+a)^3/x^(3/2),x)
Output:
(2*( - 168245*a**3 + 168245*a**2*b*x**2 + 72105*a**2*c*x**4 + 72105*a*b**2 *x**4 + 91770*a*b*c*x**6 + 33649*a*c**2*x**8 + 15295*b**3*x**6 + 33649*b** 2*c*x**8 + 26565*b*c**2*x**10 + 7315*c**3*x**12))/(168245*sqrt(x))