Integrand size = 20, antiderivative size = 101 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=-\frac {2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt {x}+\frac {6}{5} a \left (b^2+a c\right ) x^{5/2}+\frac {2}{9} b \left (b^2+6 a c\right ) x^{9/2}+\frac {6}{13} c \left (b^2+a c\right ) x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \] Output:
-2/3*a^3/x^(3/2)+6*a^2*b*x^(1/2)+6/5*a*(a*c+b^2)*x^(5/2)+2/9*b*(6*a*c+b^2) *x^(9/2)+6/13*c*(a*c+b^2)*x^(13/2)+6/17*b*c^2*x^(17/2)+2/21*c^3*x^(21/2)
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=-\frac {2 \left (23205 a^3-208845 a^2 b x^2-41769 a b^2 x^4-41769 a^2 c x^4-7735 b^3 x^6-46410 a b c x^6-16065 b^2 c x^8-16065 a c^2 x^8-12285 b c^2 x^{10}-3315 c^3 x^{12}\right )}{69615 x^{3/2}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^3/x^(5/2),x]
Output:
(-2*(23205*a^3 - 208845*a^2*b*x^2 - 41769*a*b^2*x^4 - 41769*a^2*c*x^4 - 77 35*b^3*x^6 - 46410*a*b*c*x^6 - 16065*b^2*c*x^8 - 16065*a*c^2*x^8 - 12285*b *c^2*x^10 - 3315*c^3*x^12))/(69615*x^(3/2))
Time = 0.39 (sec) , antiderivative size = 101, 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.
\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \int \left (\frac {a^3}{x^{5/2}}+\frac {3 a^2 b}{\sqrt {x}}+3 c x^{11/2} \left (a c+b^2\right )+b x^{7/2} \left (6 a c+b^2\right )+3 a x^{3/2} \left (a c+b^2\right )+3 b c^2 x^{15/2}+c^3 x^{19/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt {x}+\frac {6}{13} c x^{13/2} \left (a c+b^2\right )+\frac {2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac {6}{5} a x^{5/2} \left (a c+b^2\right )+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^3/x^(5/2),x]
Output:
(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2* b*(b^2 + 6*a*c)*x^(9/2))/9 + (6*c*(b^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(1 7/2))/17 + (2*c^3*x^(21/2))/21
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 = 0.87
method | result | size |
derivativedivides | \(\frac {2 c^{3} x^{\frac {21}{2}}}{21}+\frac {6 b \,c^{2} x^{\frac {17}{2}}}{17}+\frac {6 a \,c^{2} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,x^{\frac {13}{2}}}{13}+\frac {4 a b c \,x^{\frac {9}{2}}}{3}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}+\frac {6 a^{2} c \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {5}{2}}}{5}+6 a^{2} b \sqrt {x}-\frac {2 a^{3}}{3 x^{\frac {3}{2}}}\) | \(88\) |
default | \(\frac {2 c^{3} x^{\frac {21}{2}}}{21}+\frac {6 b \,c^{2} x^{\frac {17}{2}}}{17}+\frac {6 a \,c^{2} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,x^{\frac {13}{2}}}{13}+\frac {4 a b c \,x^{\frac {9}{2}}}{3}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}+\frac {6 a^{2} c \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {5}{2}}}{5}+6 a^{2} b \sqrt {x}-\frac {2 a^{3}}{3 x^{\frac {3}{2}}}\) | \(88\) |
gosper | \(-\frac {2 \left (-3315 c^{3} x^{12}-12285 b \,c^{2} x^{10}-16065 a \,c^{2} x^{8}-16065 b^{2} c \,x^{8}-46410 a b c \,x^{6}-7735 b^{3} x^{6}-41769 a^{2} c \,x^{4}-41769 b^{2} x^{4} a -208845 a^{2} b \,x^{2}+23205 a^{3}\right )}{69615 x^{\frac {3}{2}}}\) | \(90\) |
trager | \(-\frac {2 \left (-3315 c^{3} x^{12}-12285 b \,c^{2} x^{10}-16065 a \,c^{2} x^{8}-16065 b^{2} c \,x^{8}-46410 a b c \,x^{6}-7735 b^{3} x^{6}-41769 a^{2} c \,x^{4}-41769 b^{2} x^{4} a -208845 a^{2} b \,x^{2}+23205 a^{3}\right )}{69615 x^{\frac {3}{2}}}\) | \(90\) |
risch | \(-\frac {2 \left (-3315 c^{3} x^{12}-12285 b \,c^{2} x^{10}-16065 a \,c^{2} x^{8}-16065 b^{2} c \,x^{8}-46410 a b c \,x^{6}-7735 b^{3} x^{6}-41769 a^{2} c \,x^{4}-41769 b^{2} x^{4} a -208845 a^{2} b \,x^{2}+23205 a^{3}\right )}{69615 x^{\frac {3}{2}}}\) | \(90\) |
orering | \(-\frac {2 \left (-3315 c^{3} x^{12}-12285 b \,c^{2} x^{10}-16065 a \,c^{2} x^{8}-16065 b^{2} c \,x^{8}-46410 a b c \,x^{6}-7735 b^{3} x^{6}-41769 a^{2} c \,x^{4}-41769 b^{2} x^{4} a -208845 a^{2} b \,x^{2}+23205 a^{3}\right )}{69615 x^{\frac {3}{2}}}\) | \(90\) |
Input:
int((c*x^4+b*x^2+a)^3/x^(5/2),x,method=_RETURNVERBOSE)
Output:
2/21*c^3*x^(21/2)+6/17*b*c^2*x^(17/2)+6/13*a*c^2*x^(13/2)+6/13*b^2*c*x^(13 /2)+4/3*a*b*c*x^(9/2)+2/9*b^3*x^(9/2)+6/5*a^2*c*x^(5/2)+6/5*a*b^2*x^(5/2)+ 6*a^2*b*x^(1/2)-2/3*a^3/x^(3/2)
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=\frac {2 \, {\left (3315 \, c^{3} x^{12} + 12285 \, b c^{2} x^{10} + 16065 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 7735 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 208845 \, a^{2} b x^{2} + 41769 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 23205 \, a^{3}\right )}}{69615 \, x^{\frac {3}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="fricas")
Output:
2/69615*(3315*c^3*x^12 + 12285*b*c^2*x^10 + 16065*(b^2*c + a*c^2)*x^8 + 77 35*(b^3 + 6*a*b*c)*x^6 + 208845*a^2*b*x^2 + 41769*(a*b^2 + a^2*c)*x^4 - 23 205*a^3)/x^(3/2)
Time = 1.02 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=- \frac {2 a^{3}}{3 x^{\frac {3}{2}}} + 6 a^{2} b \sqrt {x} + \frac {6 a^{2} c x^{\frac {5}{2}}}{5} + \frac {6 a b^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b c x^{\frac {9}{2}}}{3} + \frac {6 a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 b^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {21}{2}}}{21} \] Input:
integrate((c*x**4+b*x**2+a)**3/x**(5/2),x)
Output:
-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a**2*c*x**(5/2)/5 + 6*a*b**2*x **(5/2)/5 + 4*a*b*c*x**(9/2)/3 + 6*a*c**2*x**(13/2)/13 + 2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=\frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {9}{2}} + 6 \, a^{2} b \sqrt {x} + \frac {6}{5} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {5}{2}} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="maxima")
Output:
2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*(b^2*c + a*c^2)*x^(13/2) + 2/9*(b^3 + 6*a*b*c)*x^(9/2) + 6*a^2*b*sqrt(x) + 6/5*(a*b^2 + a^2*c)*x^(5/2 ) - 2/3*a^3/x^(3/2)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=\frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, a c^{2} x^{\frac {13}{2}} + \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {4}{3} \, a b c x^{\frac {9}{2}} + \frac {6}{5} \, a b^{2} x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c x^{\frac {5}{2}} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="giac")
Output:
2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 6/13*a*c^2 *x^(13/2) + 2/9*b^3*x^(9/2) + 4/3*a*b*c*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6/5* a^2*c*x^(5/2) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=x^{9/2}\,\left (\frac {2\,b^3}{9}+\frac {4\,a\,c\,b}{3}\right )-\frac {2\,a^3}{3\,x^{3/2}}+\frac {2\,c^3\,x^{21/2}}{21}+6\,a^2\,b\,\sqrt {x}+\frac {6\,b\,c^2\,x^{17/2}}{17}+\frac {6\,a\,x^{5/2}\,\left (b^2+a\,c\right )}{5}+\frac {6\,c\,x^{13/2}\,\left (b^2+a\,c\right )}{13} \] Input:
int((a + b*x^2 + c*x^4)^3/x^(5/2),x)
Output:
x^(9/2)*((2*b^3)/9 + (4*a*b*c)/3) - (2*a^3)/(3*x^(3/2)) + (2*c^3*x^(21/2)) /21 + 6*a^2*b*x^(1/2) + (6*b*c^2*x^(17/2))/17 + (6*a*x^(5/2)*(a*c + b^2))/ 5 + (6*c*x^(13/2)*(a*c + b^2))/13
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx=\frac {\frac {2}{21} c^{3} x^{12}+\frac {6}{17} b \,c^{2} x^{10}+\frac {6}{13} a \,c^{2} x^{8}+\frac {6}{13} b^{2} c \,x^{8}+\frac {4}{3} a b c \,x^{6}+\frac {2}{9} b^{3} x^{6}+\frac {6}{5} a^{2} c \,x^{4}+\frac {6}{5} a \,b^{2} x^{4}+6 a^{2} b \,x^{2}-\frac {2}{3} a^{3}}{\sqrt {x}\, x} \] Input:
int((c*x^4+b*x^2+a)^3/x^(5/2),x)
Output:
(2*( - 23205*a**3 + 208845*a**2*b*x**2 + 41769*a**2*c*x**4 + 41769*a*b**2* x**4 + 46410*a*b*c*x**6 + 16065*a*c**2*x**8 + 7735*b**3*x**6 + 16065*b**2* c*x**8 + 12285*b*c**2*x**10 + 3315*c**3*x**12))/(69615*sqrt(x)*x)