\(\int \frac {x^2 (a+b x^2)^2}{(c+d x)^{3/2}} \, dx\) [547]

Optimal result

Integrand size = 22, antiderivative size = 211 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=-\frac {2 c^2 \left (b c^2+a d^2\right )^2}{d^7 \sqrt {c+d x}}-\frac {4 c \left (b c^2+a d^2\right ) \left (3 b c^2+a d^2\right ) \sqrt {c+d x}}{d^7}+\frac {2 \left (15 b^2 c^4+12 a b c^2 d^2+a^2 d^4\right ) (c+d x)^{3/2}}{3 d^7}-\frac {8 b c \left (5 b c^2+2 a d^2\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b \left (15 b c^2+2 a d^2\right ) (c+d x)^{7/2}}{7 d^7}-\frac {4 b^2 c (c+d x)^{9/2}}{3 d^7}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^7} \] Output:

-2*c^2*(a*d^2+b*c^2)^2/d^7/(d*x+c)^(1/2)-4*c*(a*d^2+b*c^2)*(a*d^2+3*b*c^2) 
*(d*x+c)^(1/2)/d^7+2/3*(a^2*d^4+12*a*b*c^2*d^2+15*b^2*c^4)*(d*x+c)^(3/2)/d 
^7-8/5*b*c*(2*a*d^2+5*b*c^2)*(d*x+c)^(5/2)/d^7+2/7*b*(2*a*d^2+15*b*c^2)*(d 
*x+c)^(7/2)/d^7-4/3*b^2*c*(d*x+c)^(9/2)/d^7+2/11*b^2*(d*x+c)^(11/2)/d^7
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=-\frac {2 \left (385 a^2 d^4 \left (8 c^2+4 c d x-d^2 x^2\right )+66 a b d^2 \left (128 c^4+64 c^3 d x-16 c^2 d^2 x^2+8 c d^3 x^3-5 d^4 x^4\right )+5 b^2 \left (1024 c^6+512 c^5 d x-128 c^4 d^2 x^2+64 c^3 d^3 x^3-40 c^2 d^4 x^4+28 c d^5 x^5-21 d^6 x^6\right )\right )}{1155 d^7 \sqrt {c+d x}} \] Input:

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

Output:

(-2*(385*a^2*d^4*(8*c^2 + 4*c*d*x - d^2*x^2) + 66*a*b*d^2*(128*c^4 + 64*c^ 
3*d*x - 16*c^2*d^2*x^2 + 8*c*d^3*x^3 - 5*d^4*x^4) + 5*b^2*(1024*c^6 + 512* 
c^5*d*x - 128*c^4*d^2*x^2 + 64*c^3*d^3*x^3 - 40*c^2*d^4*x^4 + 28*c*d^5*x^5 
 - 21*d^6*x^6)))/(1155*d^7*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {\sqrt {c+d x} \left (a^2 d^4+12 a b c^2 d^2+15 b^2 c^4\right )}{d^6}+\frac {\left (a c d^2+b c^3\right )^2}{d^6 (c+d x)^{3/2}}+\frac {b (c+d x)^{5/2} \left (2 a d^2+15 b c^2\right )}{d^6}-\frac {4 b c (c+d x)^{3/2} \left (2 a d^2+5 b c^2\right )}{d^6}+\frac {2 c \left (-a d^2-3 b c^2\right ) \left (a d^2+b c^2\right )}{d^6 \sqrt {c+d x}}+\frac {b^2 (c+d x)^{9/2}}{d^6}-\frac {6 b^2 c (c+d x)^{7/2}}{d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (c+d x)^{3/2} \left (a^2 d^4+12 a b c^2 d^2+15 b^2 c^4\right )}{3 d^7}+\frac {2 b (c+d x)^{7/2} \left (2 a d^2+15 b c^2\right )}{7 d^7}-\frac {8 b c (c+d x)^{5/2} \left (2 a d^2+5 b c^2\right )}{5 d^7}-\frac {4 c \sqrt {c+d x} \left (a d^2+b c^2\right ) \left (a d^2+3 b c^2\right )}{d^7}-\frac {2 c^2 \left (a d^2+b c^2\right )^2}{d^7 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^7}-\frac {4 b^2 c (c+d x)^{9/2}}{3 d^7}\)

Input:

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

Output:

(-2*c^2*(b*c^2 + a*d^2)^2)/(d^7*Sqrt[c + d*x]) - (4*c*(b*c^2 + a*d^2)*(3*b 
*c^2 + a*d^2)*Sqrt[c + d*x])/d^7 + (2*(15*b^2*c^4 + 12*a*b*c^2*d^2 + a^2*d 
^4)*(c + d*x)^(3/2))/(3*d^7) - (8*b*c*(5*b*c^2 + 2*a*d^2)*(c + d*x)^(5/2)) 
/(5*d^7) + (2*b*(15*b*c^2 + 2*a*d^2)*(c + d*x)^(7/2))/(7*d^7) - (4*b^2*c*( 
c + d*x)^(9/2))/(3*d^7) + (2*b^2*(c + d*x)^(11/2))/(11*d^7)
 

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 = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {\left (210 b^{2} x^{6}+660 a b \,x^{4}+770 a^{2} x^{2}\right ) d^{6}-3080 x \left (\frac {1}{11} b^{2} x^{4}+\frac {12}{35} a b \,x^{2}+a^{2}\right ) c \,d^{5}-6160 c^{2} \left (-\frac {5}{77} b^{2} x^{4}-\frac {12}{35} a b \,x^{2}+a^{2}\right ) d^{4}-8448 x b \,c^{3} \left (\frac {5 b \,x^{2}}{66}+a \right ) d^{3}-16896 \left (-\frac {5 b \,x^{2}}{66}+a \right ) b \,c^{4} d^{2}-5120 b^{2} c^{5} d x -10240 c^{6} b^{2}}{1155 \sqrt {d x +c}\, d^{7}}\)	\(148\)
gosper	\(-\frac {2 \left (-105 b^{2} d^{6} x^{6}+140 b^{2} c \,d^{5} x^{5}-330 x^{4} a b \,d^{6}-200 b^{2} c^{2} d^{4} x^{4}+528 x^{3} a b c \,d^{5}+320 b^{2} c^{3} d^{3} x^{3}-385 x^{2} a^{2} d^{6}-1056 x^{2} a b \,c^{2} d^{4}-640 b^{2} c^{4} d^{2} x^{2}+1540 x \,a^{2} c \,d^{5}+4224 x a b \,c^{3} d^{3}+2560 b^{2} c^{5} d x +3080 a^{2} c^{2} d^{4}+8448 a b \,c^{4} d^{2}+5120 c^{6} b^{2}\right )}{1155 \sqrt {d x +c}\, d^{7}}\)	\(184\)
trager	\(-\frac {2 \left (-105 b^{2} d^{6} x^{6}+140 b^{2} c \,d^{5} x^{5}-330 x^{4} a b \,d^{6}-200 b^{2} c^{2} d^{4} x^{4}+528 x^{3} a b c \,d^{5}+320 b^{2} c^{3} d^{3} x^{3}-385 x^{2} a^{2} d^{6}-1056 x^{2} a b \,c^{2} d^{4}-640 b^{2} c^{4} d^{2} x^{2}+1540 x \,a^{2} c \,d^{5}+4224 x a b \,c^{3} d^{3}+2560 b^{2} c^{5} d x +3080 a^{2} c^{2} d^{4}+8448 a b \,c^{4} d^{2}+5120 c^{6} b^{2}\right )}{1155 \sqrt {d x +c}\, d^{7}}\)	\(184\)
risch	\(-\frac {2 \left (-105 b^{2} x^{5} d^{5}+245 b^{2} c \,x^{4} d^{4}-330 a b \,d^{5} x^{3}-445 c^{2} d^{3} x^{3} b^{2}+858 a b c \,d^{4} x^{2}+765 b^{2} c^{3} d^{2} x^{2}-385 a^{2} x \,d^{5}-1914 a b \,c^{2} d^{3} x -1405 b^{2} c^{4} d x +1925 a^{2} c \,d^{4}+6138 a \,c^{3} d^{2} b +3965 c^{5} b^{2}\right ) \sqrt {d x +c}}{1155 d^{7}}-\frac {2 c^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{d^{7} \sqrt {d x +c}}\)	\(184\)
orering	\(-\frac {2 \left (-105 b^{2} d^{6} x^{6}+140 b^{2} c \,d^{5} x^{5}-330 x^{4} a b \,d^{6}-200 b^{2} c^{2} d^{4} x^{4}+528 x^{3} a b c \,d^{5}+320 b^{2} c^{3} d^{3} x^{3}-385 x^{2} a^{2} d^{6}-1056 x^{2} a b \,c^{2} d^{4}-640 b^{2} c^{4} d^{2} x^{2}+1540 x \,a^{2} c \,d^{5}+4224 x a b \,c^{3} d^{3}+2560 b^{2} c^{5} d x +3080 a^{2} c^{2} d^{4}+8448 a b \,c^{4} d^{2}+5120 c^{6} b^{2}\right )}{1155 \sqrt {d x +c}\, d^{7}}\)	\(184\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {4 b^{2} c \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {4 a b \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {30 b^{2} c^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {16 a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-8 b^{2} c^{3} \left (d x +c \right )^{\frac {5}{2}}+\frac {2 a^{2} d^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}+8 a b \,c^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}+10 b^{2} c^{4} \left (d x +c \right )^{\frac {3}{2}}-4 a^{2} c \,d^{4} \sqrt {d x +c}-16 a b \,c^{3} d^{2} \sqrt {d x +c}-12 b^{2} c^{5} \sqrt {d x +c}-\frac {2 c^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{\sqrt {d x +c}}}{d^{7}}\)	\(223\)
default	\(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {4 b^{2} c \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {4 a b \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {30 b^{2} c^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {16 a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-8 b^{2} c^{3} \left (d x +c \right )^{\frac {5}{2}}+\frac {2 a^{2} d^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}+8 a b \,c^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}+10 b^{2} c^{4} \left (d x +c \right )^{\frac {3}{2}}-4 a^{2} c \,d^{4} \sqrt {d x +c}-16 a b \,c^{3} d^{2} \sqrt {d x +c}-12 b^{2} c^{5} \sqrt {d x +c}-\frac {2 c^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{\sqrt {d x +c}}}{d^{7}}\)	\(223\)

Input:

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

Output:

1/1155*((210*b^2*x^6+660*a*b*x^4+770*a^2*x^2)*d^6-3080*x*(1/11*b^2*x^4+12/ 
35*a*b*x^2+a^2)*c*d^5-6160*c^2*(-5/77*b^2*x^4-12/35*a*b*x^2+a^2)*d^4-8448* 
x*b*c^3*(5/66*b*x^2+a)*d^3-16896*(-5/66*b*x^2+a)*b*c^4*d^2-5120*b^2*c^5*d* 
x-10240*c^6*b^2)/(d*x+c)^(1/2)/d^7
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (105 \, b^{2} d^{6} x^{6} - 140 \, b^{2} c d^{5} x^{5} - 5120 \, b^{2} c^{6} - 8448 \, a b c^{4} d^{2} - 3080 \, a^{2} c^{2} d^{4} + 10 \, {\left (20 \, b^{2} c^{2} d^{4} + 33 \, a b d^{6}\right )} x^{4} - 16 \, {\left (20 \, b^{2} c^{3} d^{3} + 33 \, a b c d^{5}\right )} x^{3} + {\left (640 \, b^{2} c^{4} d^{2} + 1056 \, a b c^{2} d^{4} + 385 \, a^{2} d^{6}\right )} x^{2} - 4 \, {\left (640 \, b^{2} c^{5} d + 1056 \, a b c^{3} d^{3} + 385 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c}}{1155 \, {\left (d^{8} x + c d^{7}\right )}} \] Input:

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

Output:

2/1155*(105*b^2*d^6*x^6 - 140*b^2*c*d^5*x^5 - 5120*b^2*c^6 - 8448*a*b*c^4* 
d^2 - 3080*a^2*c^2*d^4 + 10*(20*b^2*c^2*d^4 + 33*a*b*d^6)*x^4 - 16*(20*b^2 
*c^3*d^3 + 33*a*b*c*d^5)*x^3 + (640*b^2*c^4*d^2 + 1056*a*b*c^2*d^4 + 385*a 
^2*d^6)*x^2 - 4*(640*b^2*c^5*d + 1056*a*b*c^3*d^3 + 385*a^2*c*d^5)*x)*sqrt 
(d*x + c)/(d^8*x + c*d^7)
 

Sympy [A] (verification not implemented)

Time = 2.70 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {2 b^{2} c \left (c + d x\right )^{\frac {9}{2}}}{3 d^{4}} + \frac {b^{2} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{4}} - \frac {c^{2} \left (a d^{2} + b c^{2}\right )^{2}}{d^{4} \sqrt {c + d x}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 a b d^{2} + 15 b^{2} c^{2}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (- 8 a b c d^{2} - 20 b^{2} c^{3}\right )}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{2} d^{4} + 12 a b c^{2} d^{2} + 15 b^{2} c^{4}\right )}{3 d^{4}} + \frac {\sqrt {c + d x} \left (- 2 a^{2} c d^{4} - 8 a b c^{3} d^{2} - 6 b^{2} c^{5}\right )}{d^{4}}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(-2*b**2*c*(c + d*x)**(9/2)/(3*d**4) + b**2*(c + d*x)**(11/2) 
/(11*d**4) - c**2*(a*d**2 + b*c**2)**2/(d**4*sqrt(c + d*x)) + (c + d*x)**( 
7/2)*(2*a*b*d**2 + 15*b**2*c**2)/(7*d**4) + (c + d*x)**(5/2)*(-8*a*b*c*d** 
2 - 20*b**2*c**3)/(5*d**4) + (c + d*x)**(3/2)*(a**2*d**4 + 12*a*b*c**2*d** 
2 + 15*b**2*c**4)/(3*d**4) + sqrt(c + d*x)*(-2*a**2*c*d**4 - 8*a*b*c**3*d* 
*2 - 6*b**2*c**5)/d**4)/d**3, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b* 
*2*x**7/7)/c**(3/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{2} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{2} c + 165 \, {\left (15 \, b^{2} c^{2} + 2 \, a b d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 924 \, {\left (5 \, b^{2} c^{3} + 2 \, a b c d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 385 \, {\left (15 \, b^{2} c^{4} + 12 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 2310 \, {\left (3 \, b^{2} c^{5} + 4 \, a b c^{3} d^{2} + a^{2} c d^{4}\right )} \sqrt {d x + c}}{d^{4}} - \frac {1155 \, {\left (b^{2} c^{6} + 2 \, a b c^{4} d^{2} + a^{2} c^{2} d^{4}\right )}}{\sqrt {d x + c} d^{4}}\right )}}{1155 \, d^{3}} \] Input:

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

Output:

2/1155*((105*(d*x + c)^(11/2)*b^2 - 770*(d*x + c)^(9/2)*b^2*c + 165*(15*b^ 
2*c^2 + 2*a*b*d^2)*(d*x + c)^(7/2) - 924*(5*b^2*c^3 + 2*a*b*c*d^2)*(d*x + 
c)^(5/2) + 385*(15*b^2*c^4 + 12*a*b*c^2*d^2 + a^2*d^4)*(d*x + c)^(3/2) - 2 
310*(3*b^2*c^5 + 4*a*b*c^3*d^2 + a^2*c*d^4)*sqrt(d*x + c))/d^4 - 1155*(b^2 
*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4)/(sqrt(d*x + c)*d^4))/d^3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{2} c^{6} + 2 \, a b c^{4} d^{2} + a^{2} c^{2} d^{4}\right )}}{\sqrt {d x + c} d^{7}} + \frac {2 \, {\left (105 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{2} d^{70} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{2} c d^{70} + 2475 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c^{2} d^{70} - 4620 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{3} d^{70} + 5775 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{4} d^{70} - 6930 \, \sqrt {d x + c} b^{2} c^{5} d^{70} + 330 \, {\left (d x + c\right )}^{\frac {7}{2}} a b d^{72} - 1848 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{72} + 4620 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{72} - 9240 \, \sqrt {d x + c} a b c^{3} d^{72} + 385 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{74} - 2310 \, \sqrt {d x + c} a^{2} c d^{74}\right )}}{1155 \, d^{77}} \] Input:

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

Output:

-2*(b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4)/(sqrt(d*x + c)*d^7) + 2/1155*(1 
05*(d*x + c)^(11/2)*b^2*d^70 - 770*(d*x + c)^(9/2)*b^2*c*d^70 + 2475*(d*x 
+ c)^(7/2)*b^2*c^2*d^70 - 4620*(d*x + c)^(5/2)*b^2*c^3*d^70 + 5775*(d*x + 
c)^(3/2)*b^2*c^4*d^70 - 6930*sqrt(d*x + c)*b^2*c^5*d^70 + 330*(d*x + c)^(7 
/2)*a*b*d^72 - 1848*(d*x + c)^(5/2)*a*b*c*d^72 + 4620*(d*x + c)^(3/2)*a*b* 
c^2*d^72 - 9240*sqrt(d*x + c)*a*b*c^3*d^72 + 385*(d*x + c)^(3/2)*a^2*d^74 
- 2310*sqrt(d*x + c)*a^2*c*d^74)/d^77
 

Mupad [B] (verification not implemented)

Time = 8.50 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {2\,b^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^7}-\frac {2\,a^2\,c^2\,d^4+4\,a\,b\,c^4\,d^2+2\,b^2\,c^6}{d^7\,\sqrt {c+d\,x}}-\frac {\left (40\,b^2\,c^3+16\,a\,b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^7}+\frac {{\left (c+d\,x\right )}^{3/2}\,\left (2\,a^2\,d^4+24\,a\,b\,c^2\,d^2+30\,b^2\,c^4\right )}{3\,d^7}+\frac {\left (30\,b^2\,c^2+4\,a\,b\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^7}-\frac {\sqrt {c+d\,x}\,\left (4\,a^2\,c\,d^4+16\,a\,b\,c^3\,d^2+12\,b^2\,c^5\right )}{d^7}-\frac {4\,b^2\,c\,{\left (c+d\,x\right )}^{9/2}}{3\,d^7} \] Input:

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

Output:

(2*b^2*(c + d*x)^(11/2))/(11*d^7) - (2*b^2*c^6 + 2*a^2*c^2*d^4 + 4*a*b*c^4 
*d^2)/(d^7*(c + d*x)^(1/2)) - ((40*b^2*c^3 + 16*a*b*c*d^2)*(c + d*x)^(5/2) 
)/(5*d^7) + ((c + d*x)^(3/2)*(2*a^2*d^4 + 30*b^2*c^4 + 24*a*b*c^2*d^2))/(3 
*d^7) + ((30*b^2*c^2 + 4*a*b*d^2)*(c + d*x)^(7/2))/(7*d^7) - ((c + d*x)^(1 
/2)*(12*b^2*c^5 + 4*a^2*c*d^4 + 16*a*b*c^3*d^2))/d^7 - (4*b^2*c*(c + d*x)^ 
(9/2))/(3*d^7)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {\frac {2}{11} b^{2} d^{6} x^{6}-\frac {8}{33} b^{2} c \,d^{5} x^{5}+\frac {4}{7} a b \,d^{6} x^{4}+\frac {80}{231} b^{2} c^{2} d^{4} x^{4}-\frac {32}{35} a b c \,d^{5} x^{3}-\frac {128}{231} b^{2} c^{3} d^{3} x^{3}+\frac {2}{3} a^{2} d^{6} x^{2}+\frac {64}{35} a b \,c^{2} d^{4} x^{2}+\frac {256}{231} b^{2} c^{4} d^{2} x^{2}-\frac {8}{3} a^{2} c \,d^{5} x -\frac {256}{35} a b \,c^{3} d^{3} x -\frac {1024}{231} b^{2} c^{5} d x -\frac {16}{3} a^{2} c^{2} d^{4}-\frac {512}{35} a b \,c^{4} d^{2}-\frac {2048}{231} b^{2} c^{6}}{\sqrt {d x +c}\, d^{7}} \] Input:

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

Output:

(2*( - 3080*a**2*c**2*d**4 - 1540*a**2*c*d**5*x + 385*a**2*d**6*x**2 - 844 
8*a*b*c**4*d**2 - 4224*a*b*c**3*d**3*x + 1056*a*b*c**2*d**4*x**2 - 528*a*b 
*c*d**5*x**3 + 330*a*b*d**6*x**4 - 5120*b**2*c**6 - 2560*b**2*c**5*d*x + 6 
40*b**2*c**4*d**2*x**2 - 320*b**2*c**3*d**3*x**3 + 200*b**2*c**2*d**4*x**4 
 - 140*b**2*c*d**5*x**5 + 105*b**2*d**6*x**6))/(1155*sqrt(c + d*x)*d**7)