\(\int \frac {(a+b x^2)^3 (A+B x+C x^2+D x^3)}{(c+d x)^2} \, dx\) [20]

Optimal result

Integrand size = 32, antiderivative size = 683 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {\left (a^3 d^6 (C d-2 c D)+b^3 c^4 \left (7 c^2 C d-6 B c d^2+5 A d^3-8 c^3 D\right )+3 a b^2 c^2 d^2 \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )+3 a^2 b d^4 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x}{d^9}+\frac {\left (a^3 d^6 D-3 a^2 b d^4 \left (2 c C d-B d^2-3 c^2 D\right )-b^3 c^3 \left (6 c^2 C d-5 B c d^2+4 A d^3-7 c^3 D\right )-3 a b^2 c d^2 \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) x^2}{2 d^8}+\frac {b \left (3 a^2 d^4 (C d-2 c D)+b^2 c^2 \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )+3 a b d^2 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x^3}{3 d^7}+\frac {b \left (3 a^2 d^4 D-3 a b d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b^2 c \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) x^4}{4 d^6}+\frac {b^2 \left (3 a d^2 (C d-2 c D)+b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x^5}{5 d^5}+\frac {b^2 \left (3 a d^2 D-b \left (2 c C d-B d^2-3 c^2 D\right )\right ) x^6}{6 d^4}+\frac {b^3 (C d-2 c D) x^7}{7 d^3}+\frac {b^3 D x^8}{8 d^2}-\frac {\left (b c^2+a d^2\right )^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^{10} (c+d x)}-\frac {\left (b c^2+a d^2\right )^2 \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (8 c^2 C d-7 B c d^2+6 A d^3-9 c^3 D\right )\right ) \log (c+d x)}{d^{10}} \] Output:

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {420 a^3 d^6 \left (2 c^3 D-2 c^2 d (C+2 D x)+c d^2 (2 B+x (2 C-3 D x))+d^3 \left (-2 A+x^2 (2 C+D x)\right )\right )+210 a^2 b d^4 \left (12 c^5 D-12 c^4 d (C+4 D x)+6 c^3 d^2 (2 B+x (6 C-5 D x))-2 c^2 d^3 \left (6 A+x \left (12 B-12 C x-5 D x^2\right )\right )+d^5 x^2 \left (12 A+x \left (6 B+4 C x+3 D x^2\right )\right )-c d^4 x \left (-12 A+x \left (18 B+8 C x+5 D x^2\right )\right )\right )+42 a b^2 d^2 \left (60 c^7 D-60 c^6 d (C+6 D x)+30 c^5 d^2 (2 B+x (10 C-7 D x))+d^7 x^4 (20 A+x (15 B+2 x (6 C+5 D x)))-c d^6 x^3 (40 A+x (25 B+2 x (9 C+7 D x)))+c^2 d^5 x^2 (120 A+x (50 B+3 x (10 C+7 D x)))-5 c^3 d^4 x (-36 A+x (30 B+x (12 C+7 D x)))-10 c^4 d^3 (6 A+x (24 B-x (18 C+7 D x)))\right )+b^3 \left (840 c^9 D-840 c^8 d (C+8 D x)+420 c^7 d^2 (2 B+x (14 C-9 D x))-14 c^3 d^6 x^3 (60 A+x (35 B+6 x (4 C+3 D x)))-420 c^6 d^3 (2 A+x (12 B-x (8 C+3 D x)))+d^9 x^6 (168 A+5 x (28 B+3 x (8 C+7 D x)))-70 c^5 d^4 x (-60 A+x (42 B+x (16 C+9 D x)))-c d^8 x^5 (252 A+x (196 B+5 x (32 C+27 D x)))+14 c^4 d^5 x^2 (180 A+x (70 B+x (40 C+27 D x)))+2 c^2 d^7 x^4 (210 A+x (147 B+2 x (56 C+45 D x)))\right )+840 \left (b c^2+a d^2\right )^2 \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-8 c^2 C d+7 B c d^2-6 A d^3+9 c^3 D\right )\right ) (c+d x) \log (c+d x)}{840 d^{10} (c+d x)} \] Input:

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

Output:

(420*a^3*d^6*(2*c^3*D - 2*c^2*d*(C + 2*D*x) + c*d^2*(2*B + x*(2*C - 3*D*x) 
) + d^3*(-2*A + x^2*(2*C + D*x))) + 210*a^2*b*d^4*(12*c^5*D - 12*c^4*d*(C 
+ 4*D*x) + 6*c^3*d^2*(2*B + x*(6*C - 5*D*x)) - 2*c^2*d^3*(6*A + x*(12*B - 
12*C*x - 5*D*x^2)) + d^5*x^2*(12*A + x*(6*B + 4*C*x + 3*D*x^2)) - c*d^4*x* 
(-12*A + x*(18*B + 8*C*x + 5*D*x^2))) + 42*a*b^2*d^2*(60*c^7*D - 60*c^6*d* 
(C + 6*D*x) + 30*c^5*d^2*(2*B + x*(10*C - 7*D*x)) + d^7*x^4*(20*A + x*(15* 
B + 2*x*(6*C + 5*D*x))) - c*d^6*x^3*(40*A + x*(25*B + 2*x*(9*C + 7*D*x))) 
+ c^2*d^5*x^2*(120*A + x*(50*B + 3*x*(10*C + 7*D*x))) - 5*c^3*d^4*x*(-36*A 
 + x*(30*B + x*(12*C + 7*D*x))) - 10*c^4*d^3*(6*A + x*(24*B - x*(18*C + 7* 
D*x)))) + b^3*(840*c^9*D - 840*c^8*d*(C + 8*D*x) + 420*c^7*d^2*(2*B + x*(1 
4*C - 9*D*x)) - 14*c^3*d^6*x^3*(60*A + x*(35*B + 6*x*(4*C + 3*D*x))) - 420 
*c^6*d^3*(2*A + x*(12*B - x*(8*C + 3*D*x))) + d^9*x^6*(168*A + 5*x*(28*B + 
 3*x*(8*C + 7*D*x))) - 70*c^5*d^4*x*(-60*A + x*(42*B + x*(16*C + 9*D*x))) 
- c*d^8*x^5*(252*A + x*(196*B + 5*x*(32*C + 27*D*x))) + 14*c^4*d^5*x^2*(18 
0*A + x*(70*B + x*(40*C + 27*D*x))) + 2*c^2*d^7*x^4*(210*A + x*(147*B + 2* 
x*(56*C + 45*D*x)))) + 840*(b*c^2 + a*d^2)^2*(a*d^2*(-2*c*C*d + B*d^2 + 3* 
c^2*D) + b*c*(-8*c^2*C*d + 7*B*c*d^2 - 6*A*d^3 + 9*c^3*D))*(c + d*x)*Log[c 
 + d*x])/(840*d^10*(c + d*x))
 

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.

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

\(\Big \downarrow \) 2160

\(\displaystyle \int \left (\frac {b x^2 \left (3 a^2 d^4 (C d-2 c D)+3 a b d^2 \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )+b^2 c^2 \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac {b x^3 \left (3 a^2 d^4 D-3 a b d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b^2 c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6}+\frac {x \left (a^3 d^6 D-3 a^2 b d^4 \left (-B d^2-3 c^2 D+2 c C d\right )-3 a b^2 c d^2 \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )-b^3 c^3 \left (4 A d^3-5 B c d^2-7 c^3 D+6 c^2 C d\right )\right )}{d^8}+\frac {a^3 d^6 (C d-2 c D)+3 a^2 b d^4 \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )+3 a b^2 c^2 d^2 \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )+b^3 c^4 \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )}{d^9}+\frac {b^2 x^4 \left (3 a d^2 (C d-2 c D)+b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5}+\frac {\left (a d^2+b c^2\right )^2 \left (-a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (6 A d^3-7 B c d^2-9 c^3 D+8 c^2 C d\right )\right )}{d^9 (c+d x)}+\frac {\left (a d^2+b c^2\right )^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^9 (c+d x)^2}+\frac {b^2 x^5 \left (3 a d^2 D-b \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^4}+\frac {b^3 x^6 (C d-2 c D)}{d^3}+\frac {b^3 D x^7}{d^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b x^3 \left (3 a^2 d^4 (C d-2 c D)+3 a b d^2 \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )+b^2 c^2 \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{3 d^7}+\frac {b x^4 \left (3 a^2 d^4 D-3 a b d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b^2 c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{4 d^6}+\frac {x^2 \left (a^3 d^6 D-3 a^2 b d^4 \left (-B d^2-3 c^2 D+2 c C d\right )-3 a b^2 c d^2 \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )-b^3 c^3 \left (4 A d^3-5 B c d^2-7 c^3 D+6 c^2 C d\right )\right )}{2 d^8}+\frac {x \left (a^3 d^6 (C d-2 c D)+3 a^2 b d^4 \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )+3 a b^2 c^2 d^2 \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )+b^3 c^4 \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )}{d^9}+\frac {b^2 x^5 \left (3 a d^2 (C d-2 c D)+b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{5 d^5}-\frac {\left (a d^2+b c^2\right )^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^{10} (c+d x)}-\frac {\left (a d^2+b c^2\right )^2 \log (c+d x) \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (6 A d^3-7 B c d^2-9 c^3 D+8 c^2 C d\right )\right )}{d^{10}}+\frac {b^2 x^6 \left (3 a d^2 D-b \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{6 d^4}+\frac {b^3 x^7 (C d-2 c D)}{7 d^3}+\frac {b^3 D x^8}{8 d^2}\)

Input:

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

Output:

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

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 = 0.96 (sec) , antiderivative size = 1039, normalized size of antiderivative = 1.52

method	result	size
norman	\(\text {Expression too large to display}\)	\(1039\)
default	\(\text {Expression too large to display}\)	\(1170\)
parallelrisch	\(\text {Expression too large to display}\)	\(1576\)

Input:

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

Output:

((A*a^3*d^9+6*A*a^2*b*c^2*d^7+12*A*a*b^2*c^4*d^5+6*A*b^3*c^6*d^3-B*a^3*c*d 
^8-9*B*a^2*b*c^3*d^6-15*B*a*b^2*c^5*d^4-7*B*b^3*c^7*d^2+2*C*a^3*c^2*d^7+12 
*C*a^2*b*c^4*d^5+18*C*a*b^2*c^6*d^3+8*C*b^3*c^8*d-3*D*a^3*c^3*d^6-15*D*a^2 
*b*c^5*d^4-21*D*a*b^2*c^7*d^2-9*D*b^3*c^9)/d^9/c*x-1/6*(12*A*a*b^2*c*d^5+6 
*A*b^3*c^3*d^3-9*B*a^2*b*d^6-15*B*a*b^2*c^2*d^4-7*B*b^3*c^4*d^2+12*C*a^2*b 
*c*d^5+18*C*a*b^2*c^3*d^3+8*C*b^3*c^5*d-3*D*a^3*d^6-15*D*a^2*b*c^2*d^4-21* 
D*a*b^2*c^4*d^2-9*D*b^3*c^6)/d^7*x^3+1/2*(6*A*a^2*b*d^7+12*A*a*b^2*c^2*d^5 
+6*A*b^3*c^4*d^3-9*B*a^2*b*c*d^6-15*B*a*b^2*c^3*d^4-7*B*b^3*c^5*d^2+2*C*a^ 
3*d^7+12*C*a^2*b*c^2*d^5+18*C*a*b^2*c^4*d^3+8*C*b^3*c^6*d-3*D*a^3*c*d^6-15 
*D*a^2*b*c^3*d^4-21*D*a*b^2*c^5*d^2-9*D*b^3*c^7)/d^8*x^2+1/8*D*b^3/d*x^9-1 
/20*b*(6*A*b^2*c*d^3-15*B*a*b*d^4-7*B*b^2*c^2*d^2+18*C*a*b*c*d^3+8*C*b^2*c 
^3*d-15*D*a^2*d^4-21*D*a*b*c^2*d^2-9*D*b^2*c^4)/d^5*x^5+1/12*b*(12*A*a*b*d 
^5+6*A*b^2*c^2*d^3-15*B*a*b*c*d^4-7*B*b^2*c^3*d^2+12*C*a^2*d^5+18*C*a*b*c^ 
2*d^3+8*C*b^2*c^4*d-15*D*a^2*c*d^4-21*D*a*b*c^3*d^2-9*D*b^2*c^5)/d^6*x^4+1 
/42*b^2*(7*B*b*d^2-8*C*b*c*d+21*D*a*d^2+9*D*b*c^2)/d^3*x^7+1/30*b^2*(6*A*b 
*d^3-7*B*b*c*d^2+18*C*a*d^3+8*C*b*c^2*d-21*D*a*c*d^2-9*D*b*c^3)/d^4*x^6+1/ 
56*b^3*(8*C*d-9*D*c)/d^2*x^8)/(d*x+c)-(6*A*a^2*b*c*d^7+12*A*a*b^2*c^3*d^5+ 
6*A*b^3*c^5*d^3-B*a^3*d^8-9*B*a^2*b*c^2*d^6-15*B*a*b^2*c^4*d^4-7*B*b^3*c^6 
*d^2+2*C*a^3*c*d^7+12*C*a^2*b*c^3*d^5+18*C*a*b^2*c^5*d^3+8*C*b^3*c^7*d-3*D 
*a^3*c^2*d^6-15*D*a^2*b*c^4*d^4-21*D*a*b^2*c^6*d^2-9*D*b^3*c^8)/d^10*ln...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 1278, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/840*(105*D*b^3*d^9*x^9 + 840*D*b^3*c^9 - 840*C*b^3*c^8*d + 840*B*a^3*c*d 
^8 - 840*A*a^3*d^9 + 840*(3*D*a*b^2 + B*b^3)*c^7*d^2 - 840*(3*C*a*b^2 + A* 
b^3)*c^6*d^3 + 2520*(D*a^2*b + B*a*b^2)*c^5*d^4 - 2520*(C*a^2*b + A*a*b^2) 
*c^4*d^5 + 840*(D*a^3 + 3*B*a^2*b)*c^3*d^6 - 840*(C*a^3 + 3*A*a^2*b)*c^2*d 
^7 - 15*(9*D*b^3*c*d^8 - 8*C*b^3*d^9)*x^8 + 20*(9*D*b^3*c^2*d^7 - 8*C*b^3* 
c*d^8 + 7*(3*D*a*b^2 + B*b^3)*d^9)*x^7 - 28*(9*D*b^3*c^3*d^6 - 8*C*b^3*c^2 
*d^7 + 7*(3*D*a*b^2 + B*b^3)*c*d^8 - 6*(3*C*a*b^2 + A*b^3)*d^9)*x^6 + 42*( 
9*D*b^3*c^4*d^5 - 8*C*b^3*c^3*d^6 + 7*(3*D*a*b^2 + B*b^3)*c^2*d^7 - 6*(3*C 
*a*b^2 + A*b^3)*c*d^8 + 15*(D*a^2*b + B*a*b^2)*d^9)*x^5 - 70*(9*D*b^3*c^5* 
d^4 - 8*C*b^3*c^4*d^5 + 7*(3*D*a*b^2 + B*b^3)*c^3*d^6 - 6*(3*C*a*b^2 + A*b 
^3)*c^2*d^7 + 15*(D*a^2*b + B*a*b^2)*c*d^8 - 12*(C*a^2*b + A*a*b^2)*d^9)*x 
^4 + 140*(9*D*b^3*c^6*d^3 - 8*C*b^3*c^5*d^4 + 7*(3*D*a*b^2 + B*b^3)*c^4*d^ 
5 - 6*(3*C*a*b^2 + A*b^3)*c^3*d^6 + 15*(D*a^2*b + B*a*b^2)*c^2*d^7 - 12*(C 
*a^2*b + A*a*b^2)*c*d^8 + 3*(D*a^3 + 3*B*a^2*b)*d^9)*x^3 - 420*(9*D*b^3*c^ 
7*d^2 - 8*C*b^3*c^6*d^3 + 7*(3*D*a*b^2 + B*b^3)*c^5*d^4 - 6*(3*C*a*b^2 + A 
*b^3)*c^4*d^5 + 15*(D*a^2*b + B*a*b^2)*c^3*d^6 - 12*(C*a^2*b + A*a*b^2)*c^ 
2*d^7 + 3*(D*a^3 + 3*B*a^2*b)*c*d^8 - 2*(C*a^3 + 3*A*a^2*b)*d^9)*x^2 - 840 
*(8*D*b^3*c^8*d - 7*C*b^3*c^7*d^2 + 6*(3*D*a*b^2 + B*b^3)*c^6*d^3 - 5*(3*C 
*a*b^2 + A*b^3)*c^5*d^4 + 12*(D*a^2*b + B*a*b^2)*c^4*d^5 - 9*(C*a^2*b + A* 
a*b^2)*c^3*d^6 + 2*(D*a^3 + 3*B*a^2*b)*c^2*d^7 - (C*a^3 + 3*A*a^2*b)*c*...
 

Sympy [A] (verification not implemented)

Time = 3.73 (sec) , antiderivative size = 1114, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

D*b**3*x**8/(8*d**2) + x**7*(C*b**3/(7*d**2) - 2*D*b**3*c/(7*d**3)) + x**6 
*(B*b**3/(6*d**2) - C*b**3*c/(3*d**3) + D*a*b**2/(2*d**2) + D*b**3*c**2/(2 
*d**4)) + x**5*(A*b**3/(5*d**2) - 2*B*b**3*c/(5*d**3) + 3*C*a*b**2/(5*d**2 
) + 3*C*b**3*c**2/(5*d**4) - 6*D*a*b**2*c/(5*d**3) - 4*D*b**3*c**3/(5*d**5 
)) + x**4*(-A*b**3*c/(2*d**3) + 3*B*a*b**2/(4*d**2) + 3*B*b**3*c**2/(4*d** 
4) - 3*C*a*b**2*c/(2*d**3) - C*b**3*c**3/d**5 + 3*D*a**2*b/(4*d**2) + 9*D* 
a*b**2*c**2/(4*d**4) + 5*D*b**3*c**4/(4*d**6)) + x**3*(A*a*b**2/d**2 + A*b 
**3*c**2/d**4 - 2*B*a*b**2*c/d**3 - 4*B*b**3*c**3/(3*d**5) + C*a**2*b/d**2 
 + 3*C*a*b**2*c**2/d**4 + 5*C*b**3*c**4/(3*d**6) - 2*D*a**2*b*c/d**3 - 4*D 
*a*b**2*c**3/d**5 - 2*D*b**3*c**5/d**7) + x**2*(-3*A*a*b**2*c/d**3 - 2*A*b 
**3*c**3/d**5 + 3*B*a**2*b/(2*d**2) + 9*B*a*b**2*c**2/(2*d**4) + 5*B*b**3* 
c**4/(2*d**6) - 3*C*a**2*b*c/d**3 - 6*C*a*b**2*c**3/d**5 - 3*C*b**3*c**5/d 
**7 + D*a**3/(2*d**2) + 9*D*a**2*b*c**2/(2*d**4) + 15*D*a*b**2*c**4/(2*d** 
6) + 7*D*b**3*c**6/(2*d**8)) + x*(3*A*a**2*b/d**2 + 9*A*a*b**2*c**2/d**4 + 
 5*A*b**3*c**4/d**6 - 6*B*a**2*b*c/d**3 - 12*B*a*b**2*c**3/d**5 - 6*B*b**3 
*c**5/d**7 + C*a**3/d**2 + 9*C*a**2*b*c**2/d**4 + 15*C*a*b**2*c**4/d**6 + 
7*C*b**3*c**6/d**8 - 2*D*a**3*c/d**3 - 12*D*a**2*b*c**3/d**5 - 18*D*a*b**2 
*c**5/d**7 - 8*D*b**3*c**7/d**9) + (-A*a**3*d**9 - 3*A*a**2*b*c**2*d**7 - 
3*A*a*b**2*c**4*d**5 - A*b**3*c**6*d**3 + B*a**3*c*d**8 + 3*B*a**2*b*c**3* 
d**6 + 3*B*a*b**2*c**5*d**4 + B*b**3*c**7*d**2 - C*a**3*c**2*d**7 - 3*C...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 961, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(D*b^3*c^9 - C*b^3*c^8*d + B*a^3*c*d^8 - A*a^3*d^9 + (3*D*a*b^2 + B*b^3)*c 
^7*d^2 - (3*C*a*b^2 + A*b^3)*c^6*d^3 + 3*(D*a^2*b + B*a*b^2)*c^5*d^4 - 3*( 
C*a^2*b + A*a*b^2)*c^4*d^5 + (D*a^3 + 3*B*a^2*b)*c^3*d^6 - (C*a^3 + 3*A*a^ 
2*b)*c^2*d^7)/(d^11*x + c*d^10) + 1/840*(105*D*b^3*d^7*x^8 - 120*(2*D*b^3* 
c*d^6 - C*b^3*d^7)*x^7 + 140*(3*D*b^3*c^2*d^5 - 2*C*b^3*c*d^6 + (3*D*a*b^2 
 + B*b^3)*d^7)*x^6 - 168*(4*D*b^3*c^3*d^4 - 3*C*b^3*c^2*d^5 + 2*(3*D*a*b^2 
 + B*b^3)*c*d^6 - (3*C*a*b^2 + A*b^3)*d^7)*x^5 + 210*(5*D*b^3*c^4*d^3 - 4* 
C*b^3*c^3*d^4 + 3*(3*D*a*b^2 + B*b^3)*c^2*d^5 - 2*(3*C*a*b^2 + A*b^3)*c*d^ 
6 + 3*(D*a^2*b + B*a*b^2)*d^7)*x^4 - 280*(6*D*b^3*c^5*d^2 - 5*C*b^3*c^4*d^ 
3 + 4*(3*D*a*b^2 + B*b^3)*c^3*d^4 - 3*(3*C*a*b^2 + A*b^3)*c^2*d^5 + 6*(D*a 
^2*b + B*a*b^2)*c*d^6 - 3*(C*a^2*b + A*a*b^2)*d^7)*x^3 + 420*(7*D*b^3*c^6* 
d - 6*C*b^3*c^5*d^2 + 5*(3*D*a*b^2 + B*b^3)*c^4*d^3 - 4*(3*C*a*b^2 + A*b^3 
)*c^3*d^4 + 9*(D*a^2*b + B*a*b^2)*c^2*d^5 - 6*(C*a^2*b + A*a*b^2)*c*d^6 + 
(D*a^3 + 3*B*a^2*b)*d^7)*x^2 - 840*(8*D*b^3*c^7 - 7*C*b^3*c^6*d + 6*(3*D*a 
*b^2 + B*b^3)*c^5*d^2 - 5*(3*C*a*b^2 + A*b^3)*c^4*d^3 + 12*(D*a^2*b + B*a* 
b^2)*c^3*d^4 - 9*(C*a^2*b + A*a*b^2)*c^2*d^5 + 2*(D*a^3 + 3*B*a^2*b)*c*d^6 
 - (C*a^3 + 3*A*a^2*b)*d^7)*x)/d^9 + (9*D*b^3*c^8 - 8*C*b^3*c^7*d + B*a^3* 
d^8 + 7*(3*D*a*b^2 + B*b^3)*c^6*d^2 - 6*(3*C*a*b^2 + A*b^3)*c^5*d^3 + 15*( 
D*a^2*b + B*a*b^2)*c^4*d^4 - 12*(C*a^2*b + A*a*b^2)*c^3*d^5 + 3*(D*a^3 + 3 
*B*a^2*b)*c^2*d^6 - 2*(C*a^3 + 3*A*a^2*b)*c*d^7)*log(d*x + c)/d^10
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1252, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/840*(105*D*b^3 - 120*(9*D*b^3*c*d - C*b^3*d^2)/((d*x + c)*d) + 140*(36*D 
*b^3*c^2*d^2 - 8*C*b^3*c*d^3 + 3*D*a*b^2*d^4 + B*b^3*d^4)/((d*x + c)^2*d^2 
) - 168*(84*D*b^3*c^3*d^3 - 28*C*b^3*c^2*d^4 + 21*D*a*b^2*c*d^5 + 7*B*b^3* 
c*d^5 - 3*C*a*b^2*d^6 - A*b^3*d^6)/((d*x + c)^3*d^3) + 210*(126*D*b^3*c^4* 
d^4 - 56*C*b^3*c^3*d^5 + 63*D*a*b^2*c^2*d^6 + 21*B*b^3*c^2*d^6 - 18*C*a*b^ 
2*c*d^7 - 6*A*b^3*c*d^7 + 3*D*a^2*b*d^8 + 3*B*a*b^2*d^8)/((d*x + c)^4*d^4) 
 - 280*(126*D*b^3*c^5*d^5 - 70*C*b^3*c^4*d^6 + 105*D*a*b^2*c^3*d^7 + 35*B* 
b^3*c^3*d^7 - 45*C*a*b^2*c^2*d^8 - 15*A*b^3*c^2*d^8 + 15*D*a^2*b*c*d^9 + 1 
5*B*a*b^2*c*d^9 - 3*C*a^2*b*d^10 - 3*A*a*b^2*d^10)/((d*x + c)^5*d^5) + 420 
*(84*D*b^3*c^6*d^6 - 56*C*b^3*c^5*d^7 + 105*D*a*b^2*c^4*d^8 + 35*B*b^3*c^4 
*d^8 - 60*C*a*b^2*c^3*d^9 - 20*A*b^3*c^3*d^9 + 30*D*a^2*b*c^2*d^10 + 30*B* 
a*b^2*c^2*d^10 - 12*C*a^2*b*c*d^11 - 12*A*a*b^2*c*d^11 + D*a^3*d^12 + 3*B* 
a^2*b*d^12)/((d*x + c)^6*d^6) - 840*(36*D*b^3*c^7*d^7 - 28*C*b^3*c^6*d^8 + 
 63*D*a*b^2*c^5*d^9 + 21*B*b^3*c^5*d^9 - 45*C*a*b^2*c^4*d^10 - 15*A*b^3*c^ 
4*d^10 + 30*D*a^2*b*c^3*d^11 + 30*B*a*b^2*c^3*d^11 - 18*C*a^2*b*c^2*d^12 - 
 18*A*a*b^2*c^2*d^12 + 3*D*a^3*c*d^13 + 9*B*a^2*b*c*d^13 - C*a^3*d^14 - 3* 
A*a^2*b*d^14)/((d*x + c)^7*d^7))*(d*x + c)^8/d^10 - (9*D*b^3*c^8 - 8*C*b^3 
*c^7*d + 21*D*a*b^2*c^6*d^2 + 7*B*b^3*c^6*d^2 - 18*C*a*b^2*c^5*d^3 - 6*A*b 
^3*c^5*d^3 + 15*D*a^2*b*c^4*d^4 + 15*B*a*b^2*c^4*d^4 - 12*C*a^2*b*c^3*d^5 
- 12*A*a*b^2*c^3*d^5 + 3*D*a^3*c^2*d^6 + 9*B*a^2*b*c^2*d^6 - 2*C*a^3*c*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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