\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 (a+b x^2)^3} \, dx\) [44]

Optimal result

Integrand size = 32, antiderivative size = 694 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=-\frac {d^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{\left (b c^2+a d^2\right )^3 (c+d x)}-\frac {a \left (b^2 c (B c-2 A d)+a^2 d^2 D+a b \left (2 c C d-B d^2-c^2 D\right )\right )-b \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right ) x}{4 a b \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^2}-\frac {4 a^2 \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (2 c^2 C d-3 B c d^2+4 A d^3-c^3 D\right )\right )-\left (A b \left (3 b^2 c^4+12 a b c^2 d^2-7 a^2 d^4\right )+a \left (b^2 c^3 (c C-2 B d)+3 a^2 d^3 (C d-2 c D)-2 a b c d \left (6 c C d-7 B d^2-5 c^2 D\right )\right )\right ) x}{8 a^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}+\frac {\left (3 A b \left (b^3 c^6+5 a b^2 c^4 d^2+15 a^2 b c^2 d^4-5 a^3 d^6\right )+a \left (b^3 c^5 (c C-2 B d)+3 a^3 d^5 (C d-2 c D)-3 a^2 b c d^3 \left (11 c C d-10 B d^2-12 c^2 D\right )+a b^2 c^3 d \left (13 c C d-20 B d^2-6 c^2 D\right )\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \left (b c^2+a d^2\right )^4}-\frac {d^2 \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (4 c^2 C d-5 B c d^2+6 A d^3-3 c^3 D\right )\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^4}+\frac {d^2 \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (4 c^2 C d-5 B c d^2+6 A d^3-3 c^3 D\right )\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^4} \] Output:

-d^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d^2+b*c^2)^3/(d*x+c)-1/4*(a*(b^2*c*( 
-2*A*d+B*c)+a^2*d^2*D+a*b*(-B*d^2+2*C*c*d-D*c^2))-b*(A*b*(-a*d^2+b*c^2)-a* 
(b*c*(-2*B*d+C*c)-a*d*(C*d-2*D*c)))*x)/a/b/(a*d^2+b*c^2)^2/(b*x^2+a)^2-1/8 
*(4*a^2*(a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(4*A*d^3-3*B*c*d^2+2*C*c^2*d-D 
*c^3))-(A*b*(-7*a^2*d^4+12*a*b*c^2*d^2+3*b^2*c^4)+a*(b^2*c^3*(-2*B*d+C*c)+ 
3*a^2*d^3*(C*d-2*D*c)-2*a*b*c*d*(-7*B*d^2+6*C*c*d-5*D*c^2)))*x)/a^2/(a*d^2 
+b*c^2)^3/(b*x^2+a)+1/8*(3*A*b*(-5*a^3*d^6+15*a^2*b*c^2*d^4+5*a*b^2*c^4*d^ 
2+b^3*c^6)+a*(b^3*c^5*(-2*B*d+C*c)+3*a^3*d^5*(C*d-2*D*c)-3*a^2*b*c*d^3*(-1 
0*B*d^2+11*C*c*d-12*D*c^2)+a*b^2*c^3*d*(-20*B*d^2+13*C*c*d-6*D*c^2)))*arct 
an(b^(1/2)*x/a^(1/2))/a^(5/2)/b^(1/2)/(a*d^2+b*c^2)^4-d^2*(a*d^2*(-B*d^2+2 
*C*c*d-3*D*c^2)-b*c*(6*A*d^3-5*B*c*d^2+4*C*c^2*d-3*D*c^3))*ln(d*x+c)/(a*d^ 
2+b*c^2)^4+1/2*d^2*(a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(6*A*d^3-5*B*c*d^2+ 
4*C*c^2*d-3*D*c^3))*ln(b*x^2+a)/(a*d^2+b*c^2)^4
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {8 d^2 \left (b c^2+a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{c+d x}+\frac {2 \left (b c^2+a d^2\right )^2 \left (-a^3 d^2 D+A b^3 c^2 x-a b^2 \left (c^2 C x+B c (c-2 d x)+A d (-2 c+d x)\right )+a^2 b \left (c^2 D+d^2 (B+C x)-2 c d (C+D x)\right )\right )}{a b \left (a+b x^2\right )^2}+\frac {\left (b c^2+a d^2\right ) \left (3 A b^3 c^4 x+a b^2 c^2 \left (c^2 C-2 B c d+12 A d^2\right ) x+a^3 d^2 \left (12 c^2 D+d^2 (4 B+3 C x)-2 c d (4 C+3 D x)\right )+a^2 b \left (-4 c^4 D-7 A d^4 x+2 c d^3 (8 A+7 B x)-12 c^2 d^2 (B+C x)+2 c^3 d (4 C+5 D x)\right )\right )}{a^2 \left (a+b x^2\right )}+\frac {\left (3 A b \left (b^3 c^6+5 a b^2 c^4 d^2+15 a^2 b c^2 d^4-5 a^3 d^6\right )+a \left (b^3 c^5 (c C-2 B d)+3 a^3 d^5 (C d-2 c D)+a b^2 c^3 d \left (13 c C d-20 B d^2-6 c^2 D\right )+3 a^2 b c d^3 \left (-11 c C d+10 B d^2+12 c^2 D\right )\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+8 \left (a d^4 \left (-2 c C d+B d^2+3 c^2 D\right )+b c d^2 \left (4 c^2 C d-5 B c d^2+6 A d^3-3 c^3 D\right )\right ) \log (c+d x)-4 d^2 \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (4 c^2 C d-5 B c d^2+6 A d^3-3 c^3 D\right )\right ) \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^4} \] Input:

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

Output:

((-8*d^2*(b*c^2 + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(c + d*x) + 
(2*(b*c^2 + a*d^2)^2*(-(a^3*d^2*D) + A*b^3*c^2*x - a*b^2*(c^2*C*x + B*c*(c 
 - 2*d*x) + A*d*(-2*c + d*x)) + a^2*b*(c^2*D + d^2*(B + C*x) - 2*c*d*(C + 
D*x))))/(a*b*(a + b*x^2)^2) + ((b*c^2 + a*d^2)*(3*A*b^3*c^4*x + a*b^2*c^2* 
(c^2*C - 2*B*c*d + 12*A*d^2)*x + a^3*d^2*(12*c^2*D + d^2*(4*B + 3*C*x) - 2 
*c*d*(4*C + 3*D*x)) + a^2*b*(-4*c^4*D - 7*A*d^4*x + 2*c*d^3*(8*A + 7*B*x) 
- 12*c^2*d^2*(B + C*x) + 2*c^3*d*(4*C + 5*D*x))))/(a^2*(a + b*x^2)) + ((3* 
A*b*(b^3*c^6 + 5*a*b^2*c^4*d^2 + 15*a^2*b*c^2*d^4 - 5*a^3*d^6) + a*(b^3*c^ 
5*(c*C - 2*B*d) + 3*a^3*d^5*(C*d - 2*c*D) + a*b^2*c^3*d*(13*c*C*d - 20*B*d 
^2 - 6*c^2*D) + 3*a^2*b*c*d^3*(-11*c*C*d + 10*B*d^2 + 12*c^2*D)))*ArcTan[( 
Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]) + 8*(a*d^4*(-2*c*C*d + B*d^2 + 3*c^ 
2*D) + b*c*d^2*(4*c^2*C*d - 5*B*c*d^2 + 6*A*d^3 - 3*c^3*D))*Log[c + d*x] - 
 4*d^2*(a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(4*c^2*C*d - 5*B*c*d^2 + 
6*A*d^3 - 3*c^3*D))*Log[a + b*x^2])/(8*(b*c^2 + a*d^2)^4)
 

Rubi [A] (verified)

Time = 2.84 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2178, 25, 2178, 25, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2178

\(\displaystyle \frac {-\frac {\int -\frac {\frac {d^2 \left (A b \left (3 b^2 c^4+12 a b d^2 c^2-7 a^2 d^4\right )+a \left (b^2 (c C-2 B d) c^3-2 a b d \left (-5 D c^2+6 C d c-7 B d^2\right ) c+3 a^2 d^3 (C d-2 c D)\right )\right ) x^2 b^2}{\left (b c^2+a d^2\right )^3}+\frac {\left (a \left (b^2 (c C-2 B d) c^3+6 a b d \left (-D c^2+2 C d c-3 B d^2\right ) c-5 a^2 d^3 (C d-2 c D)\right ) c^2+A \left (3 b^3 c^6+12 a b^2 d^2 c^4+33 a^2 b d^4 c^2+8 a^3 d^6\right )\right ) b^2}{\left (b c^2+a d^2\right )^3}+\frac {2 d \left (3 A b c \left (b c^2+3 a d^2\right )+a \left (b c^2 (c C-2 B d)-a d \left (-6 D c^2+5 C d c-4 B d^2\right )\right )\right ) x b^2}{\left (b c^2+a d^2\right )^2}}{(c+d x)^2 \left (b x^2+a\right )}dx}{2 a b}-\frac {b \left (4 a^2 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-3 B c d^2+c^3 (-D)+2 c^2 C d\right )\right )-x \left (A b \left (-7 a^2 d^4+12 a b c^2 d^2+3 b^2 c^4\right )+a \left (3 a^2 d^3 (C d-2 c D)-2 a b c d \left (-7 B d^2-5 c^2 D+6 c C d\right )+b^2 c^3 (c C-2 B d)\right )\right )\right )}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{4 a b}-\frac {a \left (a^2 d^2 D+a b \left (-B d^2+c^2 (-D)+2 c C d\right )+b^2 c (B c-2 A d)\right )-b x \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right )}{4 a b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {d^2 \left (A b \left (3 b^2 c^4+12 a b d^2 c^2-7 a^2 d^4\right )+a \left (b^2 (c C-2 B d) c^3-2 a b d \left (-5 D c^2+6 C d c-7 B d^2\right ) c+3 a^2 d^3 (C d-2 c D)\right )\right ) x^2 b^2}{\left (b c^2+a d^2\right )^3}+\frac {\left (a \left (b^2 (c C-2 B d) c^3+6 a b d \left (-D c^2+2 C d c-3 B d^2\right ) c-5 a^2 d^3 (C d-2 c D)\right ) c^2+A \left (3 b^3 c^6+12 a b^2 d^2 c^4+33 a^2 b d^4 c^2+8 a^3 d^6\right )\right ) b^2}{\left (b c^2+a d^2\right )^3}+\frac {2 d \left (3 A b c \left (b c^2+3 a d^2\right )+a \left (b c^2 (c C-2 B d)-a d \left (-6 D c^2+5 C d c-4 B d^2\right )\right )\right ) x b^2}{\left (b c^2+a d^2\right )^2}}{(c+d x)^2 \left (b x^2+a\right )}dx}{2 a b}-\frac {b \left (4 a^2 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-3 B c d^2+c^3 (-D)+2 c^2 C d\right )\right )-x \left (A b \left (-7 a^2 d^4+12 a b c^2 d^2+3 b^2 c^4\right )+a \left (3 a^2 d^3 (C d-2 c D)-2 a b c d \left (-7 B d^2-5 c^2 D+6 c C d\right )+b^2 c^3 (c C-2 B d)\right )\right )\right )}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{4 a b}-\frac {a \left (a^2 d^2 D+a b \left (-B d^2+c^2 (-D)+2 c C d\right )+b^2 c (B c-2 A d)\right )-b x \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right )}{4 a b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\frac {\int \left (\frac {8 a^2 b^2 \left (b c \left (-3 D c^3+4 C d c^2-5 B d^2 c+6 A d^3\right )-a d^2 \left (-3 D c^2+2 C d c-B d^2\right )\right ) d^3}{\left (b c^2+a d^2\right )^4 (c+d x)}+\frac {8 a^2 b^2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) d^3}{\left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {b^2 \left (8 a^2 b \left (a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-3 D c^3+4 C d c^2-5 B d^2 c+6 A d^3\right )\right ) x d^2+3 A b \left (b^3 c^6+5 a b^2 d^2 c^4+15 a^2 b d^4 c^2-5 a^3 d^6\right )+a \left (b^3 (c C-2 B d) c^5+a b^2 d \left (-6 D c^2+13 C d c-20 B d^2\right ) c^3-3 a^2 b d^3 \left (-12 D c^2+11 C d c-10 B d^2\right ) c+3 a^3 d^5 (C d-2 c D)\right )\right )}{\left (b c^2+a d^2\right )^4 \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {b \left (4 a^2 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-3 B c d^2+c^3 (-D)+2 c^2 C d\right )\right )-x \left (A b \left (-7 a^2 d^4+12 a b c^2 d^2+3 b^2 c^4\right )+a \left (3 a^2 d^3 (C d-2 c D)-2 a b c d \left (-7 B d^2-5 c^2 D+6 c C d\right )+b^2 c^3 (c C-2 B d)\right )\right )\right )}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{4 a b}-\frac {a \left (a^2 d^2 D+a b \left (-B d^2+c^2 (-D)+2 c C d\right )+b^2 c (B c-2 A d)\right )-b x \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right )}{4 a b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\frac {4 a^2 b^2 d^2 \log \left (a+b x^2\right ) \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (6 A d^3-5 B c d^2-3 c^3 D+4 c^2 C d\right )\right )}{\left (a d^2+b c^2\right )^4}-\frac {8 a^2 b^2 d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{(c+d x) \left (a d^2+b c^2\right )^3}-\frac {8 a^2 b^2 d^2 \log (c+d x) \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (6 A d^3-5 B c d^2-3 c^3 D+4 c^2 C d\right )\right )}{\left (a d^2+b c^2\right )^4}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 A b \left (-5 a^3 d^6+15 a^2 b c^2 d^4+5 a b^2 c^4 d^2+b^3 c^6\right )+a \left (3 a^3 d^5 (C d-2 c D)-3 a^2 b c d^3 \left (-10 B d^2-12 c^2 D+11 c C d\right )+a b^2 c^3 d \left (-20 B d^2-6 c^2 D+13 c C d\right )+b^3 c^5 (c C-2 B d)\right )\right )}{\sqrt {a} \left (a d^2+b c^2\right )^4}}{2 a b}-\frac {b \left (4 a^2 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-3 B c d^2+c^3 (-D)+2 c^2 C d\right )\right )-x \left (A b \left (-7 a^2 d^4+12 a b c^2 d^2+3 b^2 c^4\right )+a \left (3 a^2 d^3 (C d-2 c D)-2 a b c d \left (-7 B d^2-5 c^2 D+6 c C d\right )+b^2 c^3 (c C-2 B d)\right )\right )\right )}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{4 a b}-\frac {a \left (a^2 d^2 D+a b \left (-B d^2+c^2 (-D)+2 c C d\right )+b^2 c (B c-2 A d)\right )-b x \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right )}{4 a b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

Input:

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

Output:

-1/4*(a*(b^2*c*(B*c - 2*A*d) + a^2*d^2*D + a*b*(2*c*C*d - B*d^2 - c^2*D)) 
- b*(A*b*(b*c^2 - a*d^2) - a*(b*c*(c*C - 2*B*d) - a*d*(C*d - 2*c*D)))*x)/( 
a*b*(b*c^2 + a*d^2)^2*(a + b*x^2)^2) + (-1/2*(b*(4*a^2*(a*d^2*(2*c*C*d - B 
*d^2 - 3*c^2*D) - b*c*(2*c^2*C*d - 3*B*c*d^2 + 4*A*d^3 - c^3*D)) - (A*b*(3 
*b^2*c^4 + 12*a*b*c^2*d^2 - 7*a^2*d^4) + a*(b^2*c^3*(c*C - 2*B*d) + 3*a^2* 
d^3*(C*d - 2*c*D) - 2*a*b*c*d*(6*c*C*d - 7*B*d^2 - 5*c^2*D)))*x))/(a*(b*c^ 
2 + a*d^2)^3*(a + b*x^2)) + ((-8*a^2*b^2*d^2*(c^2*C*d - B*c*d^2 + A*d^3 - 
c^3*D))/((b*c^2 + a*d^2)^3*(c + d*x)) + (b^(3/2)*(3*A*b*(b^3*c^6 + 5*a*b^2 
*c^4*d^2 + 15*a^2*b*c^2*d^4 - 5*a^3*d^6) + a*(b^3*c^5*(c*C - 2*B*d) + 3*a^ 
3*d^5*(C*d - 2*c*D) - 3*a^2*b*c*d^3*(11*c*C*d - 10*B*d^2 - 12*c^2*D) + a*b 
^2*c^3*d*(13*c*C*d - 20*B*d^2 - 6*c^2*D)))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(S 
qrt[a]*(b*c^2 + a*d^2)^4) - (8*a^2*b^2*d^2*(a*d^2*(2*c*C*d - B*d^2 - 3*c^2 
*D) - b*c*(4*c^2*C*d - 5*B*c*d^2 + 6*A*d^3 - 3*c^3*D))*Log[c + d*x])/(b*c^ 
2 + a*d^2)^4 + (4*a^2*b^2*d^2*(a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*(4* 
c^2*C*d - 5*B*c*d^2 + 6*A*d^3 - 3*c^3*D))*Log[a + b*x^2])/(b*c^2 + a*d^2)^ 
4)/(2*a*b))/(4*a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 1095, normalized size of antiderivative = 1.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1095\)

Input:

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

Output:

-1/(a*d^2+b*c^2)^4*((1/8*b*(7*A*a^3*b*d^6-5*A*a^2*b^2*c^2*d^4-15*A*a*b^3*c 
^4*d^2-3*A*b^4*c^6-14*B*a^3*b*c*d^5-12*B*a^2*b^2*c^3*d^3+2*B*a*b^3*c^5*d-3 
*C*a^4*d^6+9*C*a^3*b*c^2*d^4+11*C*a^2*b^2*c^4*d^2-C*a*b^3*c^6+6*D*a^4*c*d^ 
5-4*D*a^3*b*c^3*d^3-10*D*a^2*b^2*c^5*d)/a^2*x^3+(-2*A*a*b^2*c*d^5-2*A*b^3* 
c^3*d^3-1/2*B*a^2*b*d^6+B*a*b^2*c^2*d^4+3/2*B*b^3*c^4*d^2+C*a^2*b*c*d^5-C* 
b^3*c^5*d-3/2*D*a^2*b*c^2*d^4-D*a*b^2*c^4*d^2+1/2*b^3*c^6*D)*x^2+1/8*(9*A* 
a^3*b*d^6-3*A*a^2*b^2*c^2*d^4-17*A*a*b^3*c^4*d^2-5*A*b^4*c^6-18*B*a^3*b*c* 
d^5-20*B*a^2*b^2*c^3*d^3-2*B*a*b^3*c^5*d-5*C*a^4*d^6+7*C*a^3*b*c^2*d^4+13* 
C*a^2*b^2*c^4*d^2+C*a*b^3*c^6+10*D*a^4*c*d^5+4*D*a^3*b*c^3*d^3-6*D*a^2*b^2 
*c^5*d)/a*x-1/4*(10*A*a^2*b^2*c*d^5+12*A*a*b^3*c^3*d^3+2*A*b^4*c^5*d+3*B*a 
^3*b*d^6-3*B*a^2*b^2*c^2*d^4-7*B*a*b^3*c^4*d^2-B*b^4*c^6-6*C*a^3*b*c*d^5-4 
*C*a^2*b^2*c^3*d^3+2*C*a*b^3*c^5*d-D*a^4*d^6+5*D*a^3*b*c^2*d^4+5*D*a^2*b^2 
*c^4*d^2-D*a*b^3*c^6)/b)/(b*x^2+a)^2+1/8/a^2*(1/2*(48*A*a^2*b^2*c*d^5+8*B* 
a^3*b*d^6-40*B*a^2*b^2*c^2*d^4-16*C*a^3*b*c*d^5+32*C*a^2*b^2*c^3*d^3+24*D* 
a^3*b*c^2*d^4-24*D*a^2*b^2*c^4*d^2)/b*ln(b*x^2+a)+(15*A*a^3*b*d^6-45*A*a^2 
*b^2*c^2*d^4-15*A*a*b^3*c^4*d^2-3*A*b^4*c^6-30*B*a^3*b*c*d^5+20*B*a^2*b^2* 
c^3*d^3+2*B*a*b^3*c^5*d-3*C*a^4*d^6+33*C*a^3*b*c^2*d^4-13*C*a^2*b^2*c^4*d^ 
2-C*a*b^3*c^6+6*D*a^4*c*d^5-36*D*a^3*b*c^3*d^3+6*D*a^2*b^2*c^5*d)/(a*b)^(1 
/2)*arctan(b*x/(a*b)^(1/2))))+d^2*(6*A*b*c*d^3+B*a*d^4-5*B*b*c^2*d^2-2*C*a 
*c*d^3+4*C*b*c^3*d+3*D*a*c^2*d^2-3*D*b*c^4)/(a*d^2+b*c^2)^4*ln(d*x+c)-d...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1440 vs. \(2 (673) = 1346\).

Time = 0.16 (sec) , antiderivative size = 1440, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(3*D*b*c^4*d^2 - 4*C*b*c^3*d^3 - B*a*d^6 - (3*D*a - 5*B*b)*c^2*d^4 + 2 
*(C*a - 3*A*b)*c*d^5)*log(b*x^2 + a)/(b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^ 
2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8) - (3*D*b*c^4*d^2 - 4*C*b*c^3*d^3 - 
B*a*d^6 - (3*D*a - 5*B*b)*c^2*d^4 + 2*(C*a - 3*A*b)*c*d^5)*log(d*x + c)/(b 
^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8) 
+ 1/8*((C*a*b^3 + 3*A*b^4)*c^6 - 2*(3*D*a^2*b^2 + B*a*b^3)*c^5*d + (13*C*a 
^2*b^2 + 15*A*a*b^3)*c^4*d^2 + 4*(9*D*a^3*b - 5*B*a^2*b^2)*c^3*d^3 - 3*(11 
*C*a^3*b - 15*A*a^2*b^2)*c^2*d^4 - 6*(D*a^4 - 5*B*a^3*b)*c*d^5 + 3*(C*a^4 
- 5*A*a^3*b)*d^6)*arctan(b*x/sqrt(a*b))/((a^2*b^4*c^8 + 4*a^3*b^3*c^6*d^2 
+ 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + a^6*d^8)*sqrt(a*b)) - 1/8*(8*A*a^4 
*b*d^5 + 2*(D*a^3*b^2 + B*a^2*b^3)*c^5 - 4*(C*a^3*b^2 + A*a^2*b^3)*c^4*d - 
 4*(5*D*a^4*b - 3*B*a^3*b^2)*c^3*d^2 + 20*(C*a^4*b - A*a^3*b^2)*c^2*d^3 + 
2*(D*a^5 - 7*B*a^4*b)*c*d^4 - ((C*a*b^4 + 3*A*b^5)*c^4*d + 2*(9*D*a^2*b^3 
- B*a*b^4)*c^3*d^2 - 4*(5*C*a^2*b^3 - 3*A*a*b^4)*c^2*d^3 - 2*(3*D*a^3*b^2 
- 11*B*a^2*b^3)*c*d^4 + 3*(C*a^3*b^2 - 5*A*a^2*b^3)*d^5)*x^4 - (4*B*a^3*b^ 
2*d^5 + (C*a*b^4 + 3*A*b^5)*c^5 + 2*(3*D*a^2*b^3 - B*a*b^4)*c^4*d - 4*(C*a 
^2*b^3 - 3*A*a*b^4)*c^3*d^2 + 2*(3*D*a^3*b^2 + B*a^2*b^3)*c^2*d^3 - (5*C*a 
^3*b^2 - 9*A*a^2*b^3)*c*d^4)*x^3 + (4*D*a^2*b^3*c^5 - (7*C*a^2*b^3 + 5*A*a 
*b^4)*c^4*d - 2*(17*D*a^3*b^2 - 5*B*a^2*b^3)*c^3*d^2 + 4*(9*C*a^3*b^2 - 7* 
A*a^2*b^3)*c^2*d^3 + 2*(5*D*a^4*b - 19*B*a^3*b^2)*c*d^4 - 5*(C*a^4*b - ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1435 vs. \(2 (673) = 1346\).

Time = 0.38 (sec) , antiderivative size = 1435, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(3*D*b*c^4*d^2 - 4*C*b*c^3*d^3 - 3*D*a*c^2*d^4 + 5*B*b*c^2*d^4 + 2*C*a 
*c*d^5 - 6*A*b*c*d^5 - B*a*d^6)*log(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^ 
2 + a*d^2/(d*x + c)^2)/(b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4* 
a^3*b*c^2*d^6 + a^4*d^8) + (D*c^3*d^8/(d*x + c) - C*c^2*d^9/(d*x + c) + B* 
c*d^10/(d*x + c) - A*d^11/(d*x + c))/(b^3*c^6*d^6 + 3*a*b^2*c^4*d^8 + 3*a^ 
2*b*c^2*d^10 + a^3*d^12) + 1/8*(C*a*b^3*c^6*d^2 + 3*A*b^4*c^6*d^2 - 6*D*a^ 
2*b^2*c^5*d^3 - 2*B*a*b^3*c^5*d^3 + 13*C*a^2*b^2*c^4*d^4 + 15*A*a*b^3*c^4* 
d^4 + 36*D*a^3*b*c^3*d^5 - 20*B*a^2*b^2*c^3*d^5 - 33*C*a^3*b*c^2*d^6 + 45* 
A*a^2*b^2*c^2*d^6 - 6*D*a^4*c*d^7 + 30*B*a^3*b*c*d^7 + 3*C*a^4*d^8 - 15*A* 
a^3*b*d^8)*arctan((b*c - b*c^2/(d*x + c) - a*d^2/(d*x + c))/(sqrt(a*b)*d)) 
/((a^2*b^4*c^8 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + 
 a^6*d^8)*sqrt(a*b)*d^2) + 1/8*(C*a*b^4*c^5*d + 3*A*b^5*c^5*d + 14*D*a^2*b 
^3*c^4*d^2 - 2*B*a*b^4*c^4*d^2 - 22*C*a^2*b^3*c^3*d^3 + 14*A*a*b^4*c^3*d^3 
 - 24*D*a^3*b^2*c^2*d^4 + 32*B*a^2*b^3*c^2*d^4 + 17*C*a^3*b^2*c*d^5 - 29*A 
*a^2*b^3*c*d^5 + 2*D*a^4*b*d^6 - 6*B*a^3*b^2*d^6 - (3*C*a*b^4*c^6*d^2 + 9* 
A*b^5*c^6*d^2 + 46*D*a^2*b^3*c^5*d^3 - 6*B*a*b^4*c^5*d^3 - 77*C*a^2*b^3*c^ 
4*d^4 + 41*A*a*b^4*c^4*d^4 - 100*D*a^3*b^2*c^3*d^5 + 116*B*a^2*b^3*c^3*d^5 
 + 77*C*a^3*b^2*c^2*d^6 - 121*A*a^2*b^3*c^2*d^6 + 14*D*a^4*b*c*d^7 - 38*B* 
a^3*b^2*c*d^7 - 3*C*a^4*b*d^8 + 7*A*a^3*b^2*d^8)/((d*x + c)*d) + (3*C*a*b^ 
4*c^7*d^3 + 9*A*b^5*c^7*d^3 + 50*D*a^2*b^3*c^6*d^4 - 6*B*a*b^4*c^6*d^4 ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 37.23 (sec) , antiderivative size = 4707, normalized size of antiderivative = 6.78 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b*c**2*d**6 - 15 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b*c*d**7*x - 3*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**3*d**6 - 3*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**2*d**7*x + 45*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**4*b**2*c**4*d**4 + 45*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**4*b**2*c**3*d**5*x + 30*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**4*b**2*c**3*d**5 - 30*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**4*b**2*c**2*d**6*x**2 + 30*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**4*b**2*c**2*d**6*x - 30*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**4*b**2*c*d**7*x**3 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**4*b*c**5*d**4 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**4*b*c**4*d**5*x - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*a**4*b*c**3*d**6*x**2 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)) 
)*a**4*b*c**2*d**7*x**3 + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) 
*a**3*b**3*c**6*d**2 + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a* 
*3*b**3*c**5*d**3*x - 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a** 
3*b**3*c**5*d**3 + 90*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b 
**3*c**4*d**4*x**2 - 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3 
*b**3*c**4*d**4*x + 90*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3* 
b**3*c**3*d**5*x**3 + 60*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*...