\(\int (e x)^{3/2} (c+d x)^2 (a+b x^2)^3 \, dx\) [367]

Optimal result

Integrand size = 24, antiderivative size = 204 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2 a^3 c^2 (e x)^{5/2}}{5 e}+\frac {4 a^3 c d (e x)^{7/2}}{7 e^2}+\frac {2 a^2 \left (3 b c^2+a d^2\right ) (e x)^{9/2}}{9 e^3}+\frac {12 a^2 b c d (e x)^{11/2}}{11 e^4}+\frac {6 a b \left (b c^2+a d^2\right ) (e x)^{13/2}}{13 e^5}+\frac {4 a b^2 c d (e x)^{15/2}}{5 e^6}+\frac {2 b^2 \left (b c^2+3 a d^2\right ) (e x)^{17/2}}{17 e^7}+\frac {4 b^3 c d (e x)^{19/2}}{19 e^8}+\frac {2 b^3 d^2 (e x)^{21/2}}{21 e^9} \] Output:

2/5*a^3*c^2*(e*x)^(5/2)/e+4/7*a^3*c*d*(e*x)^(7/2)/e^2+2/9*a^2*(a*d^2+3*b*c 
^2)*(e*x)^(9/2)/e^3+12/11*a^2*b*c*d*(e*x)^(11/2)/e^4+6/13*a*b*(a*d^2+b*c^2 
)*(e*x)^(13/2)/e^5+4/5*a*b^2*c*d*(e*x)^(15/2)/e^6+2/17*b^2*(3*a*d^2+b*c^2) 
*(e*x)^(17/2)/e^7+4/19*b^3*c*d*(e*x)^(19/2)/e^8+2/21*b^3*d^2*(e*x)^(21/2)/ 
e^9
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.59 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2 x (e x)^{3/2} \left (46189 a^3 \left (63 c^2+90 c d x+35 d^2 x^2\right )+33915 a^2 b x^2 \left (143 c^2+234 c d x+99 d^2 x^2\right )+13167 a b^2 x^4 \left (255 c^2+442 c d x+195 d^2 x^2\right )+2145 b^3 x^6 \left (399 c^2+714 c d x+323 d^2 x^2\right )\right )}{14549535} \] Input:

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

Output:

(2*x*(e*x)^(3/2)*(46189*a^3*(63*c^2 + 90*c*d*x + 35*d^2*x^2) + 33915*a^2*b 
*x^2*(143*c^2 + 234*c*d*x + 99*d^2*x^2) + 13167*a*b^2*x^4*(255*c^2 + 442*c 
*d*x + 195*d^2*x^2) + 2145*b^3*x^6*(399*c^2 + 714*c*d*x + 323*d^2*x^2)))/1 
4549535
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 204, 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 = {522, 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.

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

\(\Big \downarrow \) 522

\(\displaystyle \int \left (a^3 c^2 (e x)^{3/2}+\frac {2 a^3 c d (e x)^{5/2}}{e}+\frac {a^2 (e x)^{7/2} \left (a d^2+3 b c^2\right )}{e^2}+\frac {6 a^2 b c d (e x)^{9/2}}{e^3}+\frac {b^2 (e x)^{15/2} \left (3 a d^2+b c^2\right )}{e^6}+\frac {6 a b^2 c d (e x)^{13/2}}{e^5}+\frac {3 a b (e x)^{11/2} \left (a d^2+b c^2\right )}{e^4}+\frac {2 b^3 c d (e x)^{17/2}}{e^7}+\frac {b^3 d^2 (e x)^{19/2}}{e^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^3 c^2 (e x)^{5/2}}{5 e}+\frac {4 a^3 c d (e x)^{7/2}}{7 e^2}+\frac {2 a^2 (e x)^{9/2} \left (a d^2+3 b c^2\right )}{9 e^3}+\frac {12 a^2 b c d (e x)^{11/2}}{11 e^4}+\frac {2 b^2 (e x)^{17/2} \left (3 a d^2+b c^2\right )}{17 e^7}+\frac {4 a b^2 c d (e x)^{15/2}}{5 e^6}+\frac {6 a b (e x)^{13/2} \left (a d^2+b c^2\right )}{13 e^5}+\frac {4 b^3 c d (e x)^{19/2}}{19 e^8}+\frac {2 b^3 d^2 (e x)^{21/2}}{21 e^9}\)

Input:

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

Output:

(2*a^3*c^2*(e*x)^(5/2))/(5*e) + (4*a^3*c*d*(e*x)^(7/2))/(7*e^2) + (2*a^2*( 
3*b*c^2 + a*d^2)*(e*x)^(9/2))/(9*e^3) + (12*a^2*b*c*d*(e*x)^(11/2))/(11*e^ 
4) + (6*a*b*(b*c^2 + a*d^2)*(e*x)^(13/2))/(13*e^5) + (4*a*b^2*c*d*(e*x)^(1 
5/2))/(5*e^6) + (2*b^2*(b*c^2 + 3*a*d^2)*(e*x)^(17/2))/(17*e^7) + (4*b^3*c 
*d*(e*x)^(19/2))/(19*e^8) + (2*b^3*d^2*(e*x)^(21/2))/(21*e^9)
 

Defintions of rubi rules used

rule 522

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

rule 2009

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {2 x^{2} e \sqrt {e x}\, \left (\frac {5 b^{3} d^{2} x^{8}}{21}+\frac {10 d \,x^{7} c \,b^{3}}{19}+\frac {5 \left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) x^{6}}{17}+2 a \,b^{2} c d \,x^{5}+\frac {15 b a \left (a \,d^{2}+b \,c^{2}\right ) x^{4}}{13}+\frac {30 a^{2} b c d \,x^{3}}{11}+\frac {5 a^{2} \left (a \,d^{2}+3 b \,c^{2}\right ) x^{2}}{9}+\frac {10 a^{3} c d x}{7}+c^{2} a^{3}\right )}{5}\)	\(131\)
gosper	\(\frac {2 x \left (692835 b^{3} d^{2} x^{8}+1531530 d \,x^{7} c \,b^{3}+2567565 a \,b^{2} d^{2} x^{6}+855855 b^{3} c^{2} x^{6}+5819814 a \,b^{2} c d \,x^{5}+3357585 a^{2} b \,d^{2} x^{4}+3357585 a \,b^{2} c^{2} x^{4}+7936110 a^{2} b c d \,x^{3}+1616615 a^{3} d^{2} x^{2}+4849845 a^{2} b \,c^{2} x^{2}+4157010 a^{3} c d x +2909907 c^{2} a^{3}\right ) \left (e x \right )^{\frac {3}{2}}}{14549535}\)	\(139\)
orering	\(\frac {2 x \left (692835 b^{3} d^{2} x^{8}+1531530 d \,x^{7} c \,b^{3}+2567565 a \,b^{2} d^{2} x^{6}+855855 b^{3} c^{2} x^{6}+5819814 a \,b^{2} c d \,x^{5}+3357585 a^{2} b \,d^{2} x^{4}+3357585 a \,b^{2} c^{2} x^{4}+7936110 a^{2} b c d \,x^{3}+1616615 a^{3} d^{2} x^{2}+4849845 a^{2} b \,c^{2} x^{2}+4157010 a^{3} c d x +2909907 c^{2} a^{3}\right ) \left (e x \right )^{\frac {3}{2}}}{14549535}\)	\(139\)
trager	\(\frac {2 e \,x^{2} \left (692835 b^{3} d^{2} x^{8}+1531530 d \,x^{7} c \,b^{3}+2567565 a \,b^{2} d^{2} x^{6}+855855 b^{3} c^{2} x^{6}+5819814 a \,b^{2} c d \,x^{5}+3357585 a^{2} b \,d^{2} x^{4}+3357585 a \,b^{2} c^{2} x^{4}+7936110 a^{2} b c d \,x^{3}+1616615 a^{3} d^{2} x^{2}+4849845 a^{2} b \,c^{2} x^{2}+4157010 a^{3} c d x +2909907 c^{2} a^{3}\right ) \sqrt {e x}}{14549535}\)	\(142\)
risch	\(\frac {2 e^{2} x^{3} \left (692835 b^{3} d^{2} x^{8}+1531530 d \,x^{7} c \,b^{3}+2567565 a \,b^{2} d^{2} x^{6}+855855 b^{3} c^{2} x^{6}+5819814 a \,b^{2} c d \,x^{5}+3357585 a^{2} b \,d^{2} x^{4}+3357585 a \,b^{2} c^{2} x^{4}+7936110 a^{2} b c d \,x^{3}+1616615 a^{3} d^{2} x^{2}+4849845 a^{2} b \,c^{2} x^{2}+4157010 a^{3} c d x +2909907 c^{2} a^{3}\right )}{14549535 \sqrt {e x}}\)	\(144\)
derivativedivides	\(\frac {\frac {2 d^{2} b^{3} \left (e x \right )^{\frac {21}{2}}}{21}+\frac {4 c e d \,b^{3} \left (e x \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} d^{2} e^{2}+b^{3} c^{2} e^{2}\right ) \left (e x \right )^{\frac {17}{2}}}{17}+\frac {4 c \,e^{3} d a \,b^{2} \left (e x \right )^{\frac {15}{2}}}{5}+\frac {2 \left (3 a^{2} b \,d^{2} e^{4}+3 a \,b^{2} c^{2} e^{4}\right ) \left (e x \right )^{\frac {13}{2}}}{13}+\frac {12 c \,e^{5} d \,a^{2} b \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{3} d^{2} e^{6}+3 a^{2} b \,c^{2} e^{6}\right ) \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 c \,e^{7} d \,a^{3} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 c^{2} e^{8} a^{3} \left (e x \right )^{\frac {5}{2}}}{5}}{e^{9}}\)	\(188\)
default	\(\frac {\frac {2 d^{2} b^{3} \left (e x \right )^{\frac {21}{2}}}{21}+\frac {4 c e d \,b^{3} \left (e x \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} d^{2} e^{2}+b^{3} c^{2} e^{2}\right ) \left (e x \right )^{\frac {17}{2}}}{17}+\frac {4 c \,e^{3} d a \,b^{2} \left (e x \right )^{\frac {15}{2}}}{5}+\frac {2 \left (3 a^{2} b \,d^{2} e^{4}+3 a \,b^{2} c^{2} e^{4}\right ) \left (e x \right )^{\frac {13}{2}}}{13}+\frac {12 c \,e^{5} d \,a^{2} b \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{3} d^{2} e^{6}+3 a^{2} b \,c^{2} e^{6}\right ) \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 c \,e^{7} d \,a^{3} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 c^{2} e^{8} a^{3} \left (e x \right )^{\frac {5}{2}}}{5}}{e^{9}}\)	\(188\)

Input:

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

Output:

2/5*x^2*e*(e*x)^(1/2)*(5/21*b^3*d^2*x^8+10/19*d*x^7*c*b^3+5/17*(3*a*b^2*d^ 
2+b^3*c^2)*x^6+2*a*b^2*c*d*x^5+15/13*b*a*(a*d^2+b*c^2)*x^4+30/11*a^2*b*c*d 
*x^3+5/9*a^2*(a*d^2+3*b*c^2)*x^2+10/7*a^3*c*d*x+c^2*a^3)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.72 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2}{14549535} \, {\left (692835 \, b^{3} d^{2} e x^{10} + 1531530 \, b^{3} c d e x^{9} + 5819814 \, a b^{2} c d e x^{7} + 7936110 \, a^{2} b c d e x^{5} + 855855 \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} e x^{8} + 4157010 \, a^{3} c d e x^{3} + 2909907 \, a^{3} c^{2} e x^{2} + 3357585 \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} e x^{6} + 1616615 \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} e x^{4}\right )} \sqrt {e x} \] Input:

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

Output:

2/14549535*(692835*b^3*d^2*e*x^10 + 1531530*b^3*c*d*e*x^9 + 5819814*a*b^2* 
c*d*e*x^7 + 7936110*a^2*b*c*d*e*x^5 + 855855*(b^3*c^2 + 3*a*b^2*d^2)*e*x^8 
 + 4157010*a^3*c*d*e*x^3 + 2909907*a^3*c^2*e*x^2 + 3357585*(a*b^2*c^2 + a^ 
2*b*d^2)*e*x^6 + 1616615*(3*a^2*b*c^2 + a^3*d^2)*e*x^4)*sqrt(e*x)
 

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.24 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2 a^{3} c^{2} x \left (e x\right )^{\frac {3}{2}}}{5} + \frac {4 a^{3} c d x^{2} \left (e x\right )^{\frac {3}{2}}}{7} + \frac {2 a^{3} d^{2} x^{3} \left (e x\right )^{\frac {3}{2}}}{9} + \frac {2 a^{2} b c^{2} x^{3} \left (e x\right )^{\frac {3}{2}}}{3} + \frac {12 a^{2} b c d x^{4} \left (e x\right )^{\frac {3}{2}}}{11} + \frac {6 a^{2} b d^{2} x^{5} \left (e x\right )^{\frac {3}{2}}}{13} + \frac {6 a b^{2} c^{2} x^{5} \left (e x\right )^{\frac {3}{2}}}{13} + \frac {4 a b^{2} c d x^{6} \left (e x\right )^{\frac {3}{2}}}{5} + \frac {6 a b^{2} d^{2} x^{7} \left (e x\right )^{\frac {3}{2}}}{17} + \frac {2 b^{3} c^{2} x^{7} \left (e x\right )^{\frac {3}{2}}}{17} + \frac {4 b^{3} c d x^{8} \left (e x\right )^{\frac {3}{2}}}{19} + \frac {2 b^{3} d^{2} x^{9} \left (e x\right )^{\frac {3}{2}}}{21} \] Input:

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

Output:

2*a**3*c**2*x*(e*x)**(3/2)/5 + 4*a**3*c*d*x**2*(e*x)**(3/2)/7 + 2*a**3*d** 
2*x**3*(e*x)**(3/2)/9 + 2*a**2*b*c**2*x**3*(e*x)**(3/2)/3 + 12*a**2*b*c*d* 
x**4*(e*x)**(3/2)/11 + 6*a**2*b*d**2*x**5*(e*x)**(3/2)/13 + 6*a*b**2*c**2* 
x**5*(e*x)**(3/2)/13 + 4*a*b**2*c*d*x**6*(e*x)**(3/2)/5 + 6*a*b**2*d**2*x* 
*7*(e*x)**(3/2)/17 + 2*b**3*c**2*x**7*(e*x)**(3/2)/17 + 4*b**3*c*d*x**8*(e 
*x)**(3/2)/19 + 2*b**3*d**2*x**9*(e*x)**(3/2)/21
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.86 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2 \, {\left (692835 \, \left (e x\right )^{\frac {21}{2}} b^{3} d^{2} + 1531530 \, \left (e x\right )^{\frac {19}{2}} b^{3} c d e + 5819814 \, \left (e x\right )^{\frac {15}{2}} a b^{2} c d e^{3} + 7936110 \, \left (e x\right )^{\frac {11}{2}} a^{2} b c d e^{5} + 4157010 \, \left (e x\right )^{\frac {7}{2}} a^{3} c d e^{7} + 2909907 \, \left (e x\right )^{\frac {5}{2}} a^{3} c^{2} e^{8} + 855855 \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} \left (e x\right )^{\frac {17}{2}} e^{2} + 3357585 \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} \left (e x\right )^{\frac {13}{2}} e^{4} + 1616615 \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} \left (e x\right )^{\frac {9}{2}} e^{6}\right )}}{14549535 \, e^{9}} \] Input:

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

Output:

2/14549535*(692835*(e*x)^(21/2)*b^3*d^2 + 1531530*(e*x)^(19/2)*b^3*c*d*e + 
 5819814*(e*x)^(15/2)*a*b^2*c*d*e^3 + 7936110*(e*x)^(11/2)*a^2*b*c*d*e^5 + 
 4157010*(e*x)^(7/2)*a^3*c*d*e^7 + 2909907*(e*x)^(5/2)*a^3*c^2*e^8 + 85585 
5*(b^3*c^2 + 3*a*b^2*d^2)*(e*x)^(17/2)*e^2 + 3357585*(a*b^2*c^2 + a^2*b*d^ 
2)*(e*x)^(13/2)*e^4 + 1616615*(3*a^2*b*c^2 + a^3*d^2)*(e*x)^(9/2)*e^6)/e^9
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2}{14549535} \, {\left (692835 \, \sqrt {e x} b^{3} d^{2} x^{10} + 1531530 \, \sqrt {e x} b^{3} c d x^{9} + 855855 \, \sqrt {e x} b^{3} c^{2} x^{8} + 2567565 \, \sqrt {e x} a b^{2} d^{2} x^{8} + 5819814 \, \sqrt {e x} a b^{2} c d x^{7} + 3357585 \, \sqrt {e x} a b^{2} c^{2} x^{6} + 3357585 \, \sqrt {e x} a^{2} b d^{2} x^{6} + 7936110 \, \sqrt {e x} a^{2} b c d x^{5} + 4849845 \, \sqrt {e x} a^{2} b c^{2} x^{4} + 1616615 \, \sqrt {e x} a^{3} d^{2} x^{4} + 4157010 \, \sqrt {e x} a^{3} c d x^{3} + 2909907 \, \sqrt {e x} a^{3} c^{2} x^{2}\right )} e \] Input:

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

Output:

2/14549535*(692835*sqrt(e*x)*b^3*d^2*x^10 + 1531530*sqrt(e*x)*b^3*c*d*x^9 
+ 855855*sqrt(e*x)*b^3*c^2*x^8 + 2567565*sqrt(e*x)*a*b^2*d^2*x^8 + 5819814 
*sqrt(e*x)*a*b^2*c*d*x^7 + 3357585*sqrt(e*x)*a*b^2*c^2*x^6 + 3357585*sqrt( 
e*x)*a^2*b*d^2*x^6 + 7936110*sqrt(e*x)*a^2*b*c*d*x^5 + 4849845*sqrt(e*x)*a 
^2*b*c^2*x^4 + 1616615*sqrt(e*x)*a^3*d^2*x^4 + 4157010*sqrt(e*x)*a^3*c*d*x 
^3 + 2909907*sqrt(e*x)*a^3*c^2*x^2)*e
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.82 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2\,a^3\,c^2\,{\left (e\,x\right )}^{5/2}}{5\,e}+\frac {2\,b^3\,d^2\,{\left (e\,x\right )}^{21/2}}{21\,e^9}+\frac {2\,a^2\,{\left (e\,x\right )}^{9/2}\,\left (3\,b\,c^2+a\,d^2\right )}{9\,e^3}+\frac {2\,b^2\,{\left (e\,x\right )}^{17/2}\,\left (b\,c^2+3\,a\,d^2\right )}{17\,e^7}+\frac {4\,a^3\,c\,d\,{\left (e\,x\right )}^{7/2}}{7\,e^2}+\frac {4\,b^3\,c\,d\,{\left (e\,x\right )}^{19/2}}{19\,e^8}+\frac {6\,a\,b\,{\left (e\,x\right )}^{13/2}\,\left (b\,c^2+a\,d^2\right )}{13\,e^5}+\frac {12\,a^2\,b\,c\,d\,{\left (e\,x\right )}^{11/2}}{11\,e^4}+\frac {4\,a\,b^2\,c\,d\,{\left (e\,x\right )}^{15/2}}{5\,e^6} \] Input:

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

Output:

(2*a^3*c^2*(e*x)^(5/2))/(5*e) + (2*b^3*d^2*(e*x)^(21/2))/(21*e^9) + (2*a^2 
*(e*x)^(9/2)*(a*d^2 + 3*b*c^2))/(9*e^3) + (2*b^2*(e*x)^(17/2)*(3*a*d^2 + b 
*c^2))/(17*e^7) + (4*a^3*c*d*(e*x)^(7/2))/(7*e^2) + (4*b^3*c*d*(e*x)^(19/2 
))/(19*e^8) + (6*a*b*(e*x)^(13/2)*(a*d^2 + b*c^2))/(13*e^5) + (12*a^2*b*c* 
d*(e*x)^(11/2))/(11*e^4) + (4*a*b^2*c*d*(e*x)^(15/2))/(5*e^6)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.69 \[ \int (e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2 \sqrt {x}\, \sqrt {e}\, e \,x^{2} \left (692835 b^{3} d^{2} x^{8}+1531530 b^{3} c d \,x^{7}+2567565 a \,b^{2} d^{2} x^{6}+855855 b^{3} c^{2} x^{6}+5819814 a \,b^{2} c d \,x^{5}+3357585 a^{2} b \,d^{2} x^{4}+3357585 a \,b^{2} c^{2} x^{4}+7936110 a^{2} b c d \,x^{3}+1616615 a^{3} d^{2} x^{2}+4849845 a^{2} b \,c^{2} x^{2}+4157010 a^{3} c d x +2909907 a^{3} c^{2}\right )}{14549535} \] Input:

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

Output:

(2*sqrt(x)*sqrt(e)*e*x**2*(2909907*a**3*c**2 + 4157010*a**3*c*d*x + 161661 
5*a**3*d**2*x**2 + 4849845*a**2*b*c**2*x**2 + 7936110*a**2*b*c*d*x**3 + 33 
57585*a**2*b*d**2*x**4 + 3357585*a*b**2*c**2*x**4 + 5819814*a*b**2*c*d*x** 
5 + 2567565*a*b**2*d**2*x**6 + 855855*b**3*c**2*x**6 + 1531530*b**3*c*d*x* 
*7 + 692835*b**3*d**2*x**8))/14549535