\(\int \frac {(a+b x^2) (A+B x+C x^2+D x^3)}{(c+d x)^3} \, dx\) [7]

Optimal result

Integrand size = 30, antiderivative size = 232 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\left (a d^2 D-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) x}{d^5}+\frac {b (C d-3 c D) x^2}{2 d^4}+\frac {b D x^3}{3 d^3}-\frac {\left (b c^2+a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{2 d^6 (c+d x)^2}+\frac {a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )}{d^6 (c+d x)}+\frac {\left (a d^2 (C d-3 c D)+b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \log (c+d x)}{d^6} \] Output:

(a*d^2*D-b*(-B*d^2+3*C*c*d-6*D*c^2))*x/d^5+1/2*b*(C*d-3*D*c)*x^2/d^4+1/3*b 
*D*x^3/d^3-1/2*(a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^6/(d*x+c)^2+( 
a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))/ 
d^6/(d*x+c)+(a*d^2*(C*d-3*D*c)+b*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*ln( 
d*x+c)/d^6
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {6 d \left (a d^2 D+b \left (-3 c C d+B d^2+6 c^2 D\right )\right ) x+3 b d^2 (C d-3 c D) x^2+2 b d^3 D x^3+\frac {3 \left (b c^2+a d^2\right ) \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{(c+d x)^2}-\frac {6 \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-4 c^2 C d+3 B c d^2-2 A d^3+5 c^3 D\right )\right )}{c+d x}+6 \left (a d^2 (C d-3 c D)+b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \log (c+d x)}{6 d^6} \] Input:

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

Output:

(6*d*(a*d^2*D + b*(-3*c*C*d + B*d^2 + 6*c^2*D))*x + 3*b*d^2*(C*d - 3*c*D)* 
x^2 + 2*b*d^3*D*x^3 + (3*(b*c^2 + a*d^2)*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c 
^3*D))/(c + d*x)^2 - (6*(a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(-4*c^2* 
C*d + 3*B*c*d^2 - 2*A*d^3 + 5*c^3*D)))/(c + d*x) + 6*(a*d^2*(C*d - 3*c*D) 
+ b*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Log[c + d*x])/(6*d^6)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2160, 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.

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

\(\Big \downarrow \) 2160

\(\displaystyle \int \left (\frac {a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )}{d^5 (c+d x)}+\frac {-a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )}{d^5 (c+d x)^2}+\frac {\left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 (c+d x)^3}+\frac {a d^2 D-b \left (-B d^2-6 c^2 D+3 c C d\right )}{d^5}+\frac {b x (C d-3 c D)}{d^4}+\frac {b D x^2}{d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )}{d^6 (c+d x)}-\frac {\left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^6 (c+d x)^2}+\frac {\log (c+d x) \left (a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac {x \left (a d^2 D-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac {b x^2 (C d-3 c D)}{2 d^4}+\frac {b D x^3}{3 d^3}\)

Input:

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

Output:

((a*d^2*D - b*(3*c*C*d - B*d^2 - 6*c^2*D))*x)/d^5 + (b*(C*d - 3*c*D)*x^2)/ 
(2*d^4) + (b*D*x^3)/(3*d^3) - ((b*c^2 + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 
- c^3*D))/(2*d^6*(c + d*x)^2) + (a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*( 
4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))/(d^6*(c + d*x)) + ((a*d^2*(C*d 
 - 3*c*D) + b*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Log[c + d*x])/d^ 
6
 

Defintions of rubi rules used

rule 2009

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

rule 2160

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

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.15

method	result	size
norman	\(\frac {\frac {\left (2 A \,d^{3} b c -B a \,d^{4}-6 B b \,c^{2} d^{2}+2 C a c \,d^{3}+12 C b \,c^{3} d -6 D a \,c^{2} d^{2}-20 D b \,c^{4}\right ) x}{d^{5}}-\frac {A a \,d^{5}-3 A \,d^{3} b \,c^{2}+B a c \,d^{4}+9 B b \,c^{3} d^{2}-3 C a \,c^{2} d^{3}-18 C b \,c^{4} d +9 D a \,c^{3} d^{2}+30 D b \,c^{5}}{2 d^{6}}+\frac {\left (3 B b \,d^{2}-6 C b c d +3 a \,d^{2} D+10 D b \,c^{2}\right ) x^{3}}{3 d^{3}}+\frac {b D x^{5}}{3 d}+\frac {b \left (3 C d -5 D c \right ) x^{4}}{6 d^{2}}}{\left (d x +c \right )^{2}}+\frac {\left (A b \,d^{3}-3 B b c \,d^{2}+C a \,d^{3}+6 C b \,c^{2} d -3 D a c \,d^{2}-10 D b \,c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}\)	\(267\)
default	\(\frac {\frac {1}{3} D b \,x^{3} d^{2}+\frac {1}{2} C b \,x^{2} d^{2}-\frac {3}{2} D b c d \,x^{2}+B b \,d^{2} x -3 C b c d x +a \,d^{2} D x +6 D b \,c^{2} x}{d^{5}}-\frac {-2 A \,d^{3} b c +B a \,d^{4}+3 B b \,c^{2} d^{2}-2 C a c \,d^{3}-4 C b \,c^{3} d +3 D a \,c^{2} d^{2}+5 D b \,c^{4}}{d^{6} \left (d x +c \right )}-\frac {A a \,d^{5}+A \,d^{3} b \,c^{2}-B a c \,d^{4}-B b \,c^{3} d^{2}+C a \,c^{2} d^{3}+C b \,c^{4} d -D a \,c^{3} d^{2}-D b \,c^{5}}{2 d^{6} \left (d x +c \right )^{2}}+\frac {\left (A b \,d^{3}-3 B b c \,d^{2}+C a \,d^{3}+6 C b \,c^{2} d -3 D a c \,d^{2}-10 D b \,c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}\)	\(268\)
parallelrisch	\(\frac {-18 B \ln \left (d x +c \right ) x^{2} b c \,d^{4}+36 C \ln \left (d x +c \right ) x^{2} b \,c^{2} d^{3}-18 D \ln \left (d x +c \right ) x^{2} a c \,d^{4}+6 C \ln \left (d x +c \right ) a \,c^{2} d^{3}+12 A x b c \,d^{4}-36 B x b \,c^{2} d^{3}-5 D x^{4} b c \,d^{4}+6 A \ln \left (d x +c \right ) x^{2} b \,d^{5}+12 C \ln \left (d x +c \right ) x a c \,d^{4}+9 A \,d^{3} b \,c^{2}+36 C \ln \left (d x +c \right ) b \,c^{4} d +9 C a \,c^{2} d^{3}-3 B a c \,d^{4}-27 B b \,c^{3} d^{2}+72 C x b \,c^{3} d^{2}-36 D x a \,c^{2} d^{3}-120 D x b \,c^{4} d -18 D \ln \left (d x +c \right ) a \,c^{3} d^{2}+12 C x a c \,d^{4}+6 C \ln \left (d x +c \right ) x^{2} a \,d^{5}-27 D a \,c^{3} d^{2}-12 C \,x^{3} b c \,d^{4}+20 D x^{3} b \,c^{2} d^{3}+6 A \ln \left (d x +c \right ) b \,c^{2} d^{3}-18 B \ln \left (d x +c \right ) b \,c^{3} d^{2}+54 C b \,c^{4} d +12 A \ln \left (d x +c \right ) x b c \,d^{4}-36 B \ln \left (d x +c \right ) x b \,c^{2} d^{3}+2 D x^{5} b \,d^{5}+3 C \,x^{4} b \,d^{5}+6 D x^{3} a \,d^{5}-6 B x a \,d^{5}-60 D \ln \left (d x +c \right ) b \,c^{5}+6 B \,x^{3} b \,d^{5}+72 C \ln \left (d x +c \right ) x b \,c^{3} d^{2}-36 D \ln \left (d x +c \right ) x a \,c^{2} d^{3}-120 D \ln \left (d x +c \right ) x b \,c^{4} d -60 D \ln \left (d x +c \right ) x^{2} b \,c^{3} d^{2}-90 D b \,c^{5}-3 A a \,d^{5}}{6 d^{6} \left (d x +c \right )^{2}}\)	\(518\)

Input:

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

Output:

((2*A*b*c*d^3-B*a*d^4-6*B*b*c^2*d^2+2*C*a*c*d^3+12*C*b*c^3*d-6*D*a*c^2*d^2 
-20*D*b*c^4)/d^5*x-1/2*(A*a*d^5-3*A*b*c^2*d^3+B*a*c*d^4+9*B*b*c^3*d^2-3*C* 
a*c^2*d^3-18*C*b*c^4*d+9*D*a*c^3*d^2+30*D*b*c^5)/d^6+1/3*(3*B*b*d^2-6*C*b* 
c*d+3*D*a*d^2+10*D*b*c^2)/d^3*x^3+1/3*b*D*x^5/d+1/6*b*(3*C*d-5*D*c)/d^2*x^ 
4)/(d*x+c)^2+1/d^6*(A*b*d^3-3*B*b*c*d^2+C*a*d^3+6*C*b*c^2*d-3*D*a*c*d^2-10 
*D*b*c^3)*ln(d*x+c)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {2 \, D b d^{5} x^{5} - 27 \, D b c^{5} + 21 \, C b c^{4} d - 3 \, B a c d^{4} - 3 \, A a d^{5} - 15 \, {\left (D a + B b\right )} c^{3} d^{2} + 9 \, {\left (C a + A b\right )} c^{2} d^{3} - {\left (5 \, D b c d^{4} - 3 \, C b d^{5}\right )} x^{4} + 2 \, {\left (10 \, D b c^{2} d^{3} - 6 \, C b c d^{4} + 3 \, {\left (D a + B b\right )} d^{5}\right )} x^{3} + 3 \, {\left (21 \, D b c^{3} d^{2} - 11 \, C b c^{2} d^{3} + 4 \, {\left (D a + B b\right )} c d^{4}\right )} x^{2} + 6 \, {\left (D b c^{4} d + C b c^{3} d^{2} - B a d^{5} - 2 \, {\left (D a + B b\right )} c^{2} d^{3} + 2 \, {\left (C a + A b\right )} c d^{4}\right )} x - 6 \, {\left (10 \, D b c^{5} - 6 \, C b c^{4} d + 3 \, {\left (D a + B b\right )} c^{3} d^{2} - {\left (C a + A b\right )} c^{2} d^{3} + {\left (10 \, D b c^{3} d^{2} - 6 \, C b c^{2} d^{3} + 3 \, {\left (D a + B b\right )} c d^{4} - {\left (C a + A b\right )} d^{5}\right )} x^{2} + 2 \, {\left (10 \, D b c^{4} d - 6 \, C b c^{3} d^{2} + 3 \, {\left (D a + B b\right )} c^{2} d^{3} - {\left (C a + A b\right )} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \] Input:

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

Output:

1/6*(2*D*b*d^5*x^5 - 27*D*b*c^5 + 21*C*b*c^4*d - 3*B*a*c*d^4 - 3*A*a*d^5 - 
 15*(D*a + B*b)*c^3*d^2 + 9*(C*a + A*b)*c^2*d^3 - (5*D*b*c*d^4 - 3*C*b*d^5 
)*x^4 + 2*(10*D*b*c^2*d^3 - 6*C*b*c*d^4 + 3*(D*a + B*b)*d^5)*x^3 + 3*(21*D 
*b*c^3*d^2 - 11*C*b*c^2*d^3 + 4*(D*a + B*b)*c*d^4)*x^2 + 6*(D*b*c^4*d + C* 
b*c^3*d^2 - B*a*d^5 - 2*(D*a + B*b)*c^2*d^3 + 2*(C*a + A*b)*c*d^4)*x - 6*( 
10*D*b*c^5 - 6*C*b*c^4*d + 3*(D*a + B*b)*c^3*d^2 - (C*a + A*b)*c^2*d^3 + ( 
10*D*b*c^3*d^2 - 6*C*b*c^2*d^3 + 3*(D*a + B*b)*c*d^4 - (C*a + A*b)*d^5)*x^ 
2 + 2*(10*D*b*c^4*d - 6*C*b*c^3*d^2 + 3*(D*a + B*b)*c^2*d^3 - (C*a + A*b)* 
c*d^4)*x)*log(d*x + c))/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)
 

Sympy [A] (verification not implemented)

Time = 6.06 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {D b x^{3}}{3 d^{3}} + x^{2} \left (\frac {C b}{2 d^{3}} - \frac {3 D b c}{2 d^{4}}\right ) + x \left (\frac {B b}{d^{3}} - \frac {3 C b c}{d^{4}} + \frac {D a}{d^{3}} + \frac {6 D b c^{2}}{d^{5}}\right ) + \frac {- A a d^{5} + 3 A b c^{2} d^{3} - B a c d^{4} - 5 B b c^{3} d^{2} + 3 C a c^{2} d^{3} + 7 C b c^{4} d - 5 D a c^{3} d^{2} - 9 D b c^{5} + x \left (4 A b c d^{4} - 2 B a d^{5} - 6 B b c^{2} d^{3} + 4 C a c d^{4} + 8 C b c^{3} d^{2} - 6 D a c^{2} d^{3} - 10 D b c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} - \frac {\left (- A b d^{3} + 3 B b c d^{2} - C a d^{3} - 6 C b c^{2} d + 3 D a c d^{2} + 10 D b c^{3}\right ) \log {\left (c + d x \right )}}{d^{6}} \] Input:

integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**3,x)
                                                                                    
                                                                                    
 

Output:

D*b*x**3/(3*d**3) + x**2*(C*b/(2*d**3) - 3*D*b*c/(2*d**4)) + x*(B*b/d**3 - 
 3*C*b*c/d**4 + D*a/d**3 + 6*D*b*c**2/d**5) + (-A*a*d**5 + 3*A*b*c**2*d**3 
 - B*a*c*d**4 - 5*B*b*c**3*d**2 + 3*C*a*c**2*d**3 + 7*C*b*c**4*d - 5*D*a*c 
**3*d**2 - 9*D*b*c**5 + x*(4*A*b*c*d**4 - 2*B*a*d**5 - 6*B*b*c**2*d**3 + 4 
*C*a*c*d**4 + 8*C*b*c**3*d**2 - 6*D*a*c**2*d**3 - 10*D*b*c**4*d))/(2*c**2* 
d**6 + 4*c*d**7*x + 2*d**8*x**2) - (-A*b*d**3 + 3*B*b*c*d**2 - C*a*d**3 - 
6*C*b*c**2*d + 3*D*a*c*d**2 + 10*D*b*c**3)*log(c + d*x)/d**6
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=-\frac {9 \, D b c^{5} - 7 \, C b c^{4} d + B a c d^{4} + A a d^{5} + 5 \, {\left (D a + B b\right )} c^{3} d^{2} - 3 \, {\left (C a + A b\right )} c^{2} d^{3} + 2 \, {\left (5 \, D b c^{4} d - 4 \, C b c^{3} d^{2} + B a d^{5} + 3 \, {\left (D a + B b\right )} c^{2} d^{3} - 2 \, {\left (C a + A b\right )} c d^{4}\right )} x}{2 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {2 \, D b d^{2} x^{3} - 3 \, {\left (3 \, D b c d - C b d^{2}\right )} x^{2} + 6 \, {\left (6 \, D b c^{2} - 3 \, C b c d + {\left (D a + B b\right )} d^{2}\right )} x}{6 \, d^{5}} - \frac {{\left (10 \, D b c^{3} - 6 \, C b c^{2} d + 3 \, {\left (D a + B b\right )} c d^{2} - {\left (C a + A b\right )} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \] Input:

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

Output:

-1/2*(9*D*b*c^5 - 7*C*b*c^4*d + B*a*c*d^4 + A*a*d^5 + 5*(D*a + B*b)*c^3*d^ 
2 - 3*(C*a + A*b)*c^2*d^3 + 2*(5*D*b*c^4*d - 4*C*b*c^3*d^2 + B*a*d^5 + 3*( 
D*a + B*b)*c^2*d^3 - 2*(C*a + A*b)*c*d^4)*x)/(d^8*x^2 + 2*c*d^7*x + c^2*d^ 
6) + 1/6*(2*D*b*d^2*x^3 - 3*(3*D*b*c*d - C*b*d^2)*x^2 + 6*(6*D*b*c^2 - 3*C 
*b*c*d + (D*a + B*b)*d^2)*x)/d^5 - (10*D*b*c^3 - 6*C*b*c^2*d + 3*(D*a + B* 
b)*c*d^2 - (C*a + A*b)*d^3)*log(d*x + c)/d^6
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=-\frac {{\left (10 \, D b c^{3} - 6 \, C b c^{2} d + 3 \, D a c d^{2} + 3 \, B b c d^{2} - C a d^{3} - A b d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} - \frac {9 \, D b c^{5} - 7 \, C b c^{4} d + 5 \, D a c^{3} d^{2} + 5 \, B b c^{3} d^{2} - 3 \, C a c^{2} d^{3} - 3 \, A b c^{2} d^{3} + B a c d^{4} + A a d^{5} + 2 \, {\left (5 \, D b c^{4} d - 4 \, C b c^{3} d^{2} + 3 \, D a c^{2} d^{3} + 3 \, B b c^{2} d^{3} - 2 \, C a c d^{4} - 2 \, A b c d^{4} + B a d^{5}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{6}} + \frac {2 \, D b d^{6} x^{3} - 9 \, D b c d^{5} x^{2} + 3 \, C b d^{6} x^{2} + 36 \, D b c^{2} d^{4} x - 18 \, C b c d^{5} x + 6 \, D a d^{6} x + 6 \, B b d^{6} x}{6 \, d^{9}} \] Input:

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

Output:

-(10*D*b*c^3 - 6*C*b*c^2*d + 3*D*a*c*d^2 + 3*B*b*c*d^2 - C*a*d^3 - A*b*d^3 
)*log(abs(d*x + c))/d^6 - 1/2*(9*D*b*c^5 - 7*C*b*c^4*d + 5*D*a*c^3*d^2 + 5 
*B*b*c^3*d^2 - 3*C*a*c^2*d^3 - 3*A*b*c^2*d^3 + B*a*c*d^4 + A*a*d^5 + 2*(5* 
D*b*c^4*d - 4*C*b*c^3*d^2 + 3*D*a*c^2*d^3 + 3*B*b*c^2*d^3 - 2*C*a*c*d^4 - 
2*A*b*c*d^4 + B*a*d^5)*x)/((d*x + c)^2*d^6) + 1/6*(2*D*b*d^6*x^3 - 9*D*b*c 
*d^5*x^2 + 3*C*b*d^6*x^2 + 36*D*b*c^2*d^4*x - 18*C*b*c*d^5*x + 6*D*a*d^6*x 
 + 6*B*b*d^6*x)/d^9
 

Mupad [B] (verification not implemented)

Time = 16.88 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\frac {3\,A\,b\,c^2}{2\,d^3}+\frac {2\,A\,b\,c\,x}{d^2}}{c^2+2\,c\,d\,x+d^2\,x^2}+\frac {\frac {3\,C\,a\,c^2}{2\,d^3}+\frac {2\,C\,a\,c\,x}{d^2}}{c^2+2\,c\,d\,x+d^2\,x^2}-\frac {\frac {B\,a\,c}{2\,d^2}+\frac {B\,a\,x}{d}}{c^2+2\,c\,d\,x+d^2\,x^2}+\frac {C\,b\,\left (\frac {{\left (c+d\,x\right )}^2}{2}+\frac {4\,c^3}{c+d\,x}-\frac {c^4}{2\,{\left (c+d\,x\right )}^2}+6\,c^2\,\ln \left (c+d\,x\right )-4\,c\,d\,x\right )}{d^5}+\frac {A\,b\,\ln \left (c+d\,x\right )}{d^3}+\frac {C\,a\,\ln \left (c+d\,x\right )}{d^3}-\frac {B\,b\,\left (3\,c\,\ln \left (c+d\,x\right )-d\,x+\frac {3\,c^2}{c+d\,x}-\frac {c^3}{2\,{\left (c+d\,x\right )}^2}\right )}{d^4}-\frac {A\,a}{2\,d\,\left (c^2+2\,c\,d\,x+d^2\,x^2\right )}-\frac {b\,D\,\left (\frac {5\,c\,{\left (c+d\,x\right )}^2}{2}-\frac {{\left (c+d\,x\right )}^3}{3}+\frac {5\,c^4}{c+d\,x}-\frac {c^5}{2\,{\left (c+d\,x\right )}^2}+10\,c^3\,\ln \left (c+d\,x\right )-10\,c^2\,d\,x\right )}{d^6}-\frac {a\,D\,\left (3\,c\,\ln \left (c+d\,x\right )-d\,x+\frac {3\,c^2}{c+d\,x}-\frac {c^3}{2\,{\left (c+d\,x\right )}^2}\right )}{d^4} \] Input:

int(((a + b*x^2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3,x)
 

Output:

((3*A*b*c^2)/(2*d^3) + (2*A*b*c*x)/d^2)/(c^2 + d^2*x^2 + 2*c*d*x) + ((3*C* 
a*c^2)/(2*d^3) + (2*C*a*c*x)/d^2)/(c^2 + d^2*x^2 + 2*c*d*x) - ((B*a*c)/(2* 
d^2) + (B*a*x)/d)/(c^2 + d^2*x^2 + 2*c*d*x) + (C*b*((c + d*x)^2/2 + (4*c^3 
)/(c + d*x) - c^4/(2*(c + d*x)^2) + 6*c^2*log(c + d*x) - 4*c*d*x))/d^5 + ( 
A*b*log(c + d*x))/d^3 + (C*a*log(c + d*x))/d^3 - (B*b*(3*c*log(c + d*x) - 
d*x + (3*c^2)/(c + d*x) - c^3/(2*(c + d*x)^2)))/d^4 - (A*a)/(2*d*(c^2 + d^ 
2*x^2 + 2*c*d*x)) - (b*D*((5*c*(c + d*x)^2)/2 - (c + d*x)^3/3 + (5*c^4)/(c 
 + d*x) - c^5/(2*(c + d*x)^2) + 10*c^3*log(c + d*x) - 10*c^2*d*x))/d^6 - ( 
a*D*(3*c*log(c + d*x) - d*x + (3*c^2)/(c + d*x) - c^3/(2*(c + d*x)^2)))/d^ 
4
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {-2 b \,c^{2} d^{4} x^{4}-12 \,\mathrm {log}\left (d x +c \right ) a \,c^{4} d^{2}-18 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{4} d +12 a \,c^{2} d^{4} x^{2}+6 a c \,d^{5} x^{3}+18 b^{2} c^{2} d^{3} x^{2}+6 b^{2} c \,d^{4} x^{3}+24 b \,c^{4} d^{2} x^{2}+8 b \,c^{3} d^{3} x^{3}+2 b c \,d^{5} x^{5}-12 \,\mathrm {log}\left (d x +c \right ) a \,c^{2} d^{4} x^{2}-18 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{2} d^{3} x^{2}-24 \,\mathrm {log}\left (d x +c \right ) b \,c^{4} d^{2} x^{2}+6 \,\mathrm {log}\left (d x +c \right ) a b \,c^{3} d^{2}-24 \,\mathrm {log}\left (d x +c \right ) a \,c^{3} d^{3} x -36 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{3} d^{2} x -48 \,\mathrm {log}\left (d x +c \right ) b \,c^{5} d x -6 a b c \,d^{4} x^{2}-24 \,\mathrm {log}\left (d x +c \right ) b \,c^{6}-12 b \,c^{6}+3 a b \,c^{3} d^{2}+3 a b \,d^{5} x^{2}+12 \,\mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{3} x +6 \,\mathrm {log}\left (d x +c \right ) a b c \,d^{4} x^{2}-3 a^{2} c \,d^{4}-6 a \,c^{4} d^{2}-9 b^{2} c^{4} d}{6 c \,d^{5} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((b*x^2+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x)
 

Output:

(6*log(c + d*x)*a*b*c**3*d**2 + 12*log(c + d*x)*a*b*c**2*d**3*x + 6*log(c 
+ d*x)*a*b*c*d**4*x**2 - 12*log(c + d*x)*a*c**4*d**2 - 24*log(c + d*x)*a*c 
**3*d**3*x - 12*log(c + d*x)*a*c**2*d**4*x**2 - 18*log(c + d*x)*b**2*c**4* 
d - 36*log(c + d*x)*b**2*c**3*d**2*x - 18*log(c + d*x)*b**2*c**2*d**3*x**2 
 - 24*log(c + d*x)*b*c**6 - 48*log(c + d*x)*b*c**5*d*x - 24*log(c + d*x)*b 
*c**4*d**2*x**2 - 3*a**2*c*d**4 + 3*a*b*c**3*d**2 - 6*a*b*c*d**4*x**2 + 3* 
a*b*d**5*x**2 - 6*a*c**4*d**2 + 12*a*c**2*d**4*x**2 + 6*a*c*d**5*x**3 - 9* 
b**2*c**4*d + 18*b**2*c**2*d**3*x**2 + 6*b**2*c*d**4*x**3 - 12*b*c**6 + 24 
*b*c**4*d**2*x**2 + 8*b*c**3*d**3*x**3 - 2*b*c**2*d**4*x**4 + 2*b*c*d**5*x 
**5)/(6*c*d**5*(c**2 + 2*c*d*x + d**2*x**2))