\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{x^{5/2}} \, dx\) [735]

Optimal result

Integrand size = 24, antiderivative size = 137 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 (2 b c+3 a d) \sqrt {x}+\frac {2}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5/2}+\frac {2}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{9/2}+\frac {2}{13} b d^2 (3 b c+2 a d) x^{13/2}+\frac {2}{17} b^2 d^3 x^{17/2} \] Output:

-2/3*a^2*c^3/x^(3/2)+2*a*c^2*(3*a*d+2*b*c)*x^(1/2)+2/5*c*(3*a^2*d^2+6*a*b* 
c*d+b^2*c^2)*x^(5/2)+2/9*d*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)*x^(9/2)+2/13*b*d^ 
2*(2*a*d+3*b*c)*x^(13/2)+2/17*b^2*d^3*x^(17/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=\frac {-442 a^2 \left (15 c^3-135 c^2 d x^2-27 c d^2 x^4-5 d^3 x^6\right )+204 a b x^2 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )+6 b^2 x^4 \left (663 c^3+1105 c^2 d x^2+765 c d^2 x^4+195 d^3 x^6\right )}{9945 x^{3/2}} \] Input:

Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]
 

Output:

(-442*a^2*(15*c^3 - 135*c^2*d*x^2 - 27*c*d^2*x^4 - 5*d^3*x^6) + 204*a*b*x^ 
2*(195*c^3 + 117*c^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6) + 6*b^2*x^4*(663*c 
^3 + 1105*c^2*d*x^2 + 765*c*d^2*x^4 + 195*d^3*x^6))/(9945*x^(3/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {355, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx\)

\(\Big \downarrow \) 355

\(\displaystyle \int \left (d x^{7/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x^{3/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {a^2 c^3}{x^{5/2}}+\frac {a c^2 (3 a d+2 b c)}{\sqrt {x}}+b d^2 x^{11/2} (2 a d+3 b c)+b^2 d^3 x^{15/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt {x} (3 a d+2 b c)+\frac {2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac {2}{17} b^2 d^3 x^{17/2}\)

Input:

Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]
 

Output:

(-2*a^2*c^3)/(3*x^(3/2)) + 2*a*c^2*(2*b*c + 3*a*d)*Sqrt[x] + (2*c*(b^2*c^2 
 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2))/5 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d 
^2)*x^(9/2))/9 + (2*b*d^2*(3*b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d^3*x^(17/ 
2))/17
 

Defintions of rubi rules used

rule 355

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, 
x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & 
& IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{3} x^{\frac {17}{2}}}{17}+\frac {4 a b \,d^{3} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,d^{2} x^{\frac {13}{2}}}{13}+\frac {2 a^{2} d^{3} x^{\frac {9}{2}}}{9}+\frac {4 a b c \,d^{2} x^{\frac {9}{2}}}{3}+\frac {2 b^{2} c^{2} d \,x^{\frac {9}{2}}}{3}+\frac {6 a^{2} c \,d^{2} x^{\frac {5}{2}}}{5}+\frac {12 a b \,c^{2} d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{3} x^{\frac {5}{2}}}{5}+6 a^{2} c^{2} d \sqrt {x}+4 a b \,c^{3} \sqrt {x}-\frac {2 a^{2} c^{3}}{3 x^{\frac {3}{2}}}\)	\(136\)
default	\(\frac {2 b^{2} d^{3} x^{\frac {17}{2}}}{17}+\frac {4 a b \,d^{3} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,d^{2} x^{\frac {13}{2}}}{13}+\frac {2 a^{2} d^{3} x^{\frac {9}{2}}}{9}+\frac {4 a b c \,d^{2} x^{\frac {9}{2}}}{3}+\frac {2 b^{2} c^{2} d \,x^{\frac {9}{2}}}{3}+\frac {6 a^{2} c \,d^{2} x^{\frac {5}{2}}}{5}+\frac {12 a b \,c^{2} d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{3} x^{\frac {5}{2}}}{5}+6 a^{2} c^{2} d \sqrt {x}+4 a b \,c^{3} \sqrt {x}-\frac {2 a^{2} c^{3}}{3 x^{\frac {3}{2}}}\)	\(136\)
gosper	\(-\frac {2 \left (-585 b^{2} d^{3} x^{10}-1530 x^{8} a \,d^{3} b -2295 x^{8} b^{2} c \,d^{2}-1105 a^{2} d^{3} x^{6}-6630 a b c \,d^{2} x^{6}-3315 b^{2} c^{2} d \,x^{6}-5967 a^{2} c \,d^{2} x^{4}-11934 a b \,c^{2} d \,x^{4}-1989 b^{2} c^{3} x^{4}-29835 a^{2} c^{2} d \,x^{2}-19890 a b \,c^{3} x^{2}+3315 a^{2} c^{3}\right )}{9945 x^{\frac {3}{2}}}\)	\(138\)
trager	\(-\frac {2 \left (-585 b^{2} d^{3} x^{10}-1530 x^{8} a \,d^{3} b -2295 x^{8} b^{2} c \,d^{2}-1105 a^{2} d^{3} x^{6}-6630 a b c \,d^{2} x^{6}-3315 b^{2} c^{2} d \,x^{6}-5967 a^{2} c \,d^{2} x^{4}-11934 a b \,c^{2} d \,x^{4}-1989 b^{2} c^{3} x^{4}-29835 a^{2} c^{2} d \,x^{2}-19890 a b \,c^{3} x^{2}+3315 a^{2} c^{3}\right )}{9945 x^{\frac {3}{2}}}\)	\(138\)
risch	\(-\frac {2 \left (-585 b^{2} d^{3} x^{10}-1530 x^{8} a \,d^{3} b -2295 x^{8} b^{2} c \,d^{2}-1105 a^{2} d^{3} x^{6}-6630 a b c \,d^{2} x^{6}-3315 b^{2} c^{2} d \,x^{6}-5967 a^{2} c \,d^{2} x^{4}-11934 a b \,c^{2} d \,x^{4}-1989 b^{2} c^{3} x^{4}-29835 a^{2} c^{2} d \,x^{2}-19890 a b \,c^{3} x^{2}+3315 a^{2} c^{3}\right )}{9945 x^{\frac {3}{2}}}\)	\(138\)
orering	\(-\frac {2 \left (-585 b^{2} d^{3} x^{10}-1530 x^{8} a \,d^{3} b -2295 x^{8} b^{2} c \,d^{2}-1105 a^{2} d^{3} x^{6}-6630 a b c \,d^{2} x^{6}-3315 b^{2} c^{2} d \,x^{6}-5967 a^{2} c \,d^{2} x^{4}-11934 a b \,c^{2} d \,x^{4}-1989 b^{2} c^{3} x^{4}-29835 a^{2} c^{2} d \,x^{2}-19890 a b \,c^{3} x^{2}+3315 a^{2} c^{3}\right )}{9945 x^{\frac {3}{2}}}\)	\(138\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/17*b^2*d^3*x^(17/2)+4/13*a*b*d^3*x^(13/2)+6/13*b^2*c*d^2*x^(13/2)+2/9*a^ 
2*d^3*x^(9/2)+4/3*a*b*c*d^2*x^(9/2)+2/3*b^2*c^2*d*x^(9/2)+6/5*a^2*c*d^2*x^ 
(5/2)+12/5*a*b*c^2*d*x^(5/2)+2/5*b^2*c^3*x^(5/2)+6*a^2*c^2*d*x^(1/2)+4*a*b 
*c^3*x^(1/2)-2/3*a^2*c^3/x^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=\frac {2 \, {\left (585 \, b^{2} d^{3} x^{10} + 765 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 1105 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 3315 \, a^{2} c^{3} + 1989 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 9945 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{9945 \, x^{\frac {3}{2}}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="fricas")
 

Output:

2/9945*(585*b^2*d^3*x^10 + 765*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 1105*(3*b^2 
*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6 - 3315*a^2*c^3 + 1989*(b^2*c^3 + 6*a*b 
*c^2*d + 3*a^2*c*d^2)*x^4 + 9945*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x^(3/2)
 

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=- \frac {2 a^{2} c^{3}}{3 x^{\frac {3}{2}}} + 6 a^{2} c^{2} d \sqrt {x} + \frac {6 a^{2} c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 a^{2} d^{3} x^{\frac {9}{2}}}{9} + 4 a b c^{3} \sqrt {x} + \frac {12 a b c^{2} d x^{\frac {5}{2}}}{5} + \frac {4 a b c d^{2} x^{\frac {9}{2}}}{3} + \frac {4 a b d^{3} x^{\frac {13}{2}}}{13} + \frac {2 b^{2} c^{3} x^{\frac {5}{2}}}{5} + \frac {2 b^{2} c^{2} d x^{\frac {9}{2}}}{3} + \frac {6 b^{2} c d^{2} x^{\frac {13}{2}}}{13} + \frac {2 b^{2} d^{3} x^{\frac {17}{2}}}{17} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**3/x**(5/2),x)
 

Output:

-2*a**2*c**3/(3*x**(3/2)) + 6*a**2*c**2*d*sqrt(x) + 6*a**2*c*d**2*x**(5/2) 
/5 + 2*a**2*d**3*x**(9/2)/9 + 4*a*b*c**3*sqrt(x) + 12*a*b*c**2*d*x**(5/2)/ 
5 + 4*a*b*c*d**2*x**(9/2)/3 + 4*a*b*d**3*x**(13/2)/13 + 2*b**2*c**3*x**(5/ 
2)/5 + 2*b**2*c**2*d*x**(9/2)/3 + 6*b**2*c*d**2*x**(13/2)/13 + 2*b**2*d**3 
*x**(17/2)/17
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{17} \, b^{2} d^{3} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {9}{2}} - \frac {2 \, a^{2} c^{3}}{3 \, x^{\frac {3}{2}}} + \frac {2}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {5}{2}} + 2 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \sqrt {x} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="maxima")
 

Output:

2/17*b^2*d^3*x^(17/2) + 2/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(13/2) + 2/9*(3*b 
^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(9/2) - 2/3*a^2*c^3/x^(3/2) + 2/5*(b^2 
*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(5/2) + 2*(2*a*b*c^3 + 3*a^2*c^2*d)*sq 
rt(x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{17} \, b^{2} d^{3} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c d^{2} x^{\frac {13}{2}} + \frac {4}{13} \, a b d^{3} x^{\frac {13}{2}} + \frac {2}{3} \, b^{2} c^{2} d x^{\frac {9}{2}} + \frac {4}{3} \, a b c d^{2} x^{\frac {9}{2}} + \frac {2}{9} \, a^{2} d^{3} x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c^{3} x^{\frac {5}{2}} + \frac {12}{5} \, a b c^{2} d x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c d^{2} x^{\frac {5}{2}} + 4 \, a b c^{3} \sqrt {x} + 6 \, a^{2} c^{2} d \sqrt {x} - \frac {2 \, a^{2} c^{3}}{3 \, x^{\frac {3}{2}}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/17*b^2*d^3*x^(17/2) + 6/13*b^2*c*d^2*x^(13/2) + 4/13*a*b*d^3*x^(13/2) + 
2/3*b^2*c^2*d*x^(9/2) + 4/3*a*b*c*d^2*x^(9/2) + 2/9*a^2*d^3*x^(9/2) + 2/5* 
b^2*c^3*x^(5/2) + 12/5*a*b*c^2*d*x^(5/2) + 6/5*a^2*c*d^2*x^(5/2) + 4*a*b*c 
^3*sqrt(x) + 6*a^2*c^2*d*sqrt(x) - 2/3*a^2*c^3/x^(3/2)
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=x^{5/2}\,\left (\frac {6\,a^2\,c\,d^2}{5}+\frac {12\,a\,b\,c^2\,d}{5}+\frac {2\,b^2\,c^3}{5}\right )+x^{9/2}\,\left (\frac {2\,a^2\,d^3}{9}+\frac {4\,a\,b\,c\,d^2}{3}+\frac {2\,b^2\,c^2\,d}{3}\right )-\frac {2\,a^2\,c^3}{3\,x^{3/2}}+\frac {2\,b^2\,d^3\,x^{17/2}}{17}+2\,a\,c^2\,\sqrt {x}\,\left (3\,a\,d+2\,b\,c\right )+\frac {2\,b\,d^2\,x^{13/2}\,\left (2\,a\,d+3\,b\,c\right )}{13} \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x)
 

Output:

x^(5/2)*((2*b^2*c^3)/5 + (6*a^2*c*d^2)/5 + (12*a*b*c^2*d)/5) + x^(9/2)*((2 
*a^2*d^3)/9 + (2*b^2*c^2*d)/3 + (4*a*b*c*d^2)/3) - (2*a^2*c^3)/(3*x^(3/2)) 
 + (2*b^2*d^3*x^(17/2))/17 + 2*a*c^2*x^(1/2)*(3*a*d + 2*b*c) + (2*b*d^2*x^ 
(13/2)*(2*a*d + 3*b*c))/13
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx=\frac {\frac {2}{17} b^{2} d^{3} x^{10}+\frac {4}{13} a b \,d^{3} x^{8}+\frac {6}{13} b^{2} c \,d^{2} x^{8}+\frac {2}{9} a^{2} d^{3} x^{6}+\frac {4}{3} a b c \,d^{2} x^{6}+\frac {2}{3} b^{2} c^{2} d \,x^{6}+\frac {6}{5} a^{2} c \,d^{2} x^{4}+\frac {12}{5} a b \,c^{2} d \,x^{4}+\frac {2}{5} b^{2} c^{3} x^{4}+6 a^{2} c^{2} d \,x^{2}+4 a b \,c^{3} x^{2}-\frac {2}{3} a^{2} c^{3}}{\sqrt {x}\, x} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x)
 

Output:

(2*( - 3315*a**2*c**3 + 29835*a**2*c**2*d*x**2 + 5967*a**2*c*d**2*x**4 + 1 
105*a**2*d**3*x**6 + 19890*a*b*c**3*x**2 + 11934*a*b*c**2*d*x**4 + 6630*a* 
b*c*d**2*x**6 + 1530*a*b*d**3*x**8 + 1989*b**2*c**3*x**4 + 3315*b**2*c**2* 
d*x**6 + 2295*b**2*c*d**2*x**8 + 585*b**2*d**3*x**10))/(9945*sqrt(x)*x)