Integrand size = 32, antiderivative size = 423 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\left (a^2 d^4 D-2 a b d^2 \left (3 c C d-B d^2-6 c^2 D\right )-b^2 c \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) x}{d^7}+\frac {b \left (2 a d^2 (C d-3 c D)+b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) x^2}{2 d^6}+\frac {b \left (2 a d^2 D-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) x^3}{3 d^5}+\frac {b^2 (C d-3 c D) x^4}{4 d^4}+\frac {b^2 D x^5}{5 d^3}-\frac {\left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{2 d^8 (c+d x)^2}+\frac {\left (b c^2+a d^2\right ) \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (6 c^2 C d-5 B c d^2+4 A d^3-7 c^3 D\right )\right )}{d^8 (c+d x)}+\frac {\left (a^2 d^4 (C d-3 c D)+b^2 c^2 \left (15 c^2 C d-10 B c d^2+6 A d^3-21 c^3 D\right )+2 a b d^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \log (c+d x)}{d^8} \] Output:
(a^2*d^4*D-2*a*b*d^2*(-B*d^2+3*C*c*d-6*D*c^2)-b^2*c*(3*A*d^3-6*B*c*d^2+10* C*c^2*d-15*D*c^3))*x/d^7+1/2*b*(2*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))*x^2/d^6+1/3*b*(2*a*d^2*D-b*(-B*d^2+3*C*c*d-6*D*c^2))*x^3 /d^5+1/4*b^2*(C*d-3*D*c)*x^4/d^4+1/5*b^2*D*x^5/d^3-1/2*(a*d^2+b*c^2)^2*(A* d^3-B*c*d^2+C*c^2*d-D*c^3)/d^8/(d*x+c)^2+(a*d^2+b*c^2)*(a*d^2*(-B*d^2+2*C* c*d-3*D*c^2)+b*c*(4*A*d^3-5*B*c*d^2+6*C*c^2*d-7*D*c^3))/d^8/(d*x+c)+(a^2*d ^4*(C*d-3*D*c)+b^2*c^2*(6*A*d^3-10*B*c*d^2+15*C*c^2*d-21*D*c^3)+2*a*b*d^2* (A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*ln(d*x+c)/d^8
Time = 0.17 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {60 d \left (a^2 d^4 D+2 a b d^2 \left (-3 c C d+B d^2+6 c^2 D\right )+b^2 c \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )\right ) x+30 b d^2 \left (2 a d^2 (C d-3 c D)+b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) x^2+20 b d^3 \left (2 a d^2 D+b \left (-3 c C d+B d^2+6 c^2 D\right )\right ) x^3+15 b^2 d^4 (C d-3 c D) x^4+12 b^2 d^5 D x^5+\frac {30 \left (b c^2+a d^2\right )^2 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{(c+d x)^2}-\frac {60 \left (b c^2+a d^2\right ) \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-6 c^2 C d+5 B c d^2-4 A d^3+7 c^3 D\right )\right )}{c+d x}+60 \left (a^2 d^4 (C d-3 c D)+b^2 c^2 \left (15 c^2 C d-10 B c d^2+6 A d^3-21 c^3 D\right )+2 a b d^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \log (c+d x)}{60 d^8} \] Input:
Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
(60*d*(a^2*d^4*D + 2*a*b*d^2*(-3*c*C*d + B*d^2 + 6*c^2*D) + b^2*c*(-10*c^2 *C*d + 6*B*c*d^2 - 3*A*d^3 + 15*c^3*D))*x + 30*b*d^2*(2*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))*x^2 + 20*b*d^3*(2*a*d^2* D + b*(-3*c*C*d + B*d^2 + 6*c^2*D))*x^3 + 15*b^2*d^4*(C*d - 3*c*D)*x^4 + 1 2*b^2*d^5*D*x^5 + (30*(b*c^2 + a*d^2)^2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^ 3*D))/(c + d*x)^2 - (60*(b*c^2 + a*d^2)*(a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D ) + b*c*(-6*c^2*C*d + 5*B*c*d^2 - 4*A*d^3 + 7*c^3*D)))/(c + d*x) + 60*(a^2 *d^4*(C*d - 3*c*D) + b^2*c^2*(15*c^2*C*d - 10*B*c*d^2 + 6*A*d^3 - 21*c^3*D ) + 2*a*b*d^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Log[c + d*x])/(6 0*d^8)
Time = 0.90 (sec) , antiderivative size = 423, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \int \left (\frac {a^2 d^4 (C d-3 c D)+2 a b d^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )+b^2 c^2 \left (6 A d^3-10 B c d^2-21 c^3 D+15 c^2 C d\right )}{d^7 (c+d x)}+\frac {a^2 d^4 D-2 a b d^2 \left (-B d^2-6 c^2 D+3 c C d\right )-b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )}{d^7}+\frac {\left (a d^2+b c^2\right ) \left (-a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-5 B c d^2-7 c^3 D+6 c^2 C d\right )\right )}{d^7 (c+d x)^2}+\frac {\left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 (c+d x)^3}+\frac {b x \left (2 a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac {b x^2 \left (2 a d^2 D-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac {b^2 x^3 (C d-3 c D)}{d^4}+\frac {b^2 D x^4}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (c+d x) \left (a^2 d^4 (C d-3 c D)+2 a b d^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )+b^2 c^2 \left (6 A d^3-10 B c d^2-21 c^3 D+15 c^2 C d\right )\right )}{d^8}+\frac {x \left (a^2 d^4 D-2 a b d^2 \left (-B d^2-6 c^2 D+3 c C d\right )-b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7}+\frac {\left (a d^2+b c^2\right ) \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (4 A d^3-5 B c d^2-7 c^3 D+6 c^2 C d\right )\right )}{d^8 (c+d x)}-\frac {\left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^8 (c+d x)^2}+\frac {b x^2 \left (2 a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{2 d^6}+\frac {b x^3 \left (2 a d^2 D-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac {b^2 x^4 (C d-3 c D)}{4 d^4}+\frac {b^2 D x^5}{5 d^3}\) |
Input:
Int[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
((a^2*d^4*D - 2*a*b*d^2*(3*c*C*d - B*d^2 - 6*c^2*D) - b^2*c*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*x)/d^7 + (b*(2*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))*x^2)/(2*d^6) + (b*(2*a*d^2*D - b *(3*c*C*d - B*d^2 - 6*c^2*D))*x^3)/(3*d^5) + (b^2*(C*d - 3*c*D)*x^4)/(4*d^ 4) + (b^2*D*x^5)/(5*d^3) - ((b*c^2 + a*d^2)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(2*d^8*(c + d*x)^2) + ((b*c^2 + a*d^2)*(a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(6*c^2*C*d - 5*B*c*d^2 + 4*A*d^3 - 7*c^3*D)))/(d^8*(c + d*x )) + ((a^2*d^4*(C*d - 3*c*D) + b^2*c^2*(15*c^2*C*d - 10*B*c*d^2 + 6*A*d^3 - 21*c^3*D) + 2*a*b*d^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Log[c + d*x])/d^8
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]
Time = 0.98 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.39
| method | result | size |
| norman | \(\frac {\frac {\left (4 A a b c \,d^{5}+12 A \,b^{2} c^{3} d^{3}-B \,a^{2} d^{6}-12 B a b \,c^{2} d^{4}-20 B \,b^{2} c^{4} d^{2}+2 C \,a^{2} c \,d^{5}+24 C a b \,c^{3} d^{3}+30 C \,b^{2} c^{5} d -6 D a^{2} c^{2} d^{4}-40 D a b \,c^{4} d^{2}-42 D b^{2} c^{6}\right ) x}{d^{7}}-\frac {A \,a^{2} d^{7}-6 A a b \,c^{2} d^{5}-18 A \,b^{2} c^{4} d^{3}+B \,a^{2} c \,d^{6}+18 B a b \,c^{3} d^{4}+30 B \,b^{2} c^{5} d^{2}-3 C \,a^{2} c^{2} d^{5}-36 C a b \,c^{4} d^{3}-45 C \,b^{2} c^{6} d +9 D a^{2} c^{3} d^{4}+60 D a b \,c^{5} d^{2}+63 D b^{2} c^{7}}{2 d^{8}}-\frac {\left (6 A \,d^{3} b^{2} c -6 B a b \,d^{4}-10 B \,b^{2} c^{2} d^{2}+12 C a b c \,d^{3}+15 C \,b^{2} c^{3} d -3 a^{2} d^{4} D-20 D a b \,c^{2} d^{2}-21 D b^{2} c^{4}\right ) x^{3}}{3 d^{5}}+\frac {b \left (10 B b \,d^{2}-15 C b c d +20 a \,d^{2} D+21 D b \,c^{2}\right ) x^{5}}{30 d^{3}}+\frac {b \left (6 A b \,d^{3}-10 B b c \,d^{2}+12 C a \,d^{3}+15 C b \,c^{2} d -20 D a c \,d^{2}-21 D b \,c^{3}\right ) x^{4}}{12 d^{4}}+\frac {b^{2} D x^{7}}{5 d}+\frac {b^{2} \left (5 C d -7 D c \right ) x^{6}}{20 d^{2}}}{\left (d x +c \right )^{2}}+\frac {\left (2 A a b \,d^{5}+6 A \,b^{2} c^{2} d^{3}-6 B a b c \,d^{4}-10 B \,b^{2} c^{3} d^{2}+C \,a^{2} d^{5}+12 C a b \,c^{2} d^{3}+15 C \,b^{2} c^{4} d -3 D a^{2} c \,d^{4}-20 D a b \,c^{3} d^{2}-21 b^{2} c^{5} D\right ) \ln \left (d x +c \right )}{d^{8}}\) | \(589\) |
| default | \(-\frac {-\frac {1}{5} b^{2} D x^{5} d^{4}-\frac {1}{4} C \,b^{2} d^{4} x^{4}+\frac {3}{4} D b^{2} c \,d^{3} x^{4}-\frac {1}{3} B \,b^{2} d^{4} x^{3}+C \,b^{2} c \,d^{3} x^{3}-\frac {2}{3} D a b \,d^{4} x^{3}-2 D b^{2} c^{2} d^{2} x^{3}-\frac {1}{2} A \,b^{2} d^{4} x^{2}+\frac {3}{2} B \,b^{2} c \,d^{3} x^{2}-C a b \,d^{4} x^{2}-3 C \,b^{2} c^{2} d^{2} x^{2}+3 D a b c \,d^{3} x^{2}+5 D b^{2} c^{3} d \,x^{2}+3 A \,b^{2} c \,d^{3} x -2 B a b \,d^{4} x -6 B \,b^{2} c^{2} d^{2} x +6 C a b c \,d^{3} x +10 C \,b^{2} c^{3} d x -a^{2} d^{4} D x -12 D a b \,c^{2} d^{2} x -15 D b^{2} c^{4} x}{d^{7}}-\frac {-4 A a b c \,d^{5}-4 A \,b^{2} c^{3} d^{3}+B \,a^{2} d^{6}+6 B a b \,c^{2} d^{4}+5 B \,b^{2} c^{4} d^{2}-2 C \,a^{2} c \,d^{5}-8 C a b \,c^{3} d^{3}-6 C \,b^{2} c^{5} d +3 D a^{2} c^{2} d^{4}+10 D a b \,c^{4} d^{2}+7 D b^{2} c^{6}}{d^{8} \left (d x +c \right )}-\frac {A \,a^{2} d^{7}+2 A a b \,c^{2} d^{5}+A \,b^{2} c^{4} d^{3}-B \,a^{2} c \,d^{6}-2 B a b \,c^{3} d^{4}-B \,b^{2} c^{5} d^{2}+C \,a^{2} c^{2} d^{5}+2 C a b \,c^{4} d^{3}+C \,b^{2} c^{6} d -D a^{2} c^{3} d^{4}-2 D a b \,c^{5} d^{2}-D b^{2} c^{7}}{2 d^{8} \left (d x +c \right )^{2}}+\frac {\left (2 A a b \,d^{5}+6 A \,b^{2} c^{2} d^{3}-6 B a b c \,d^{4}-10 B \,b^{2} c^{3} d^{2}+C \,a^{2} d^{5}+12 C a b \,c^{2} d^{3}+15 C \,b^{2} c^{4} d -3 D a^{2} c \,d^{4}-20 D a b \,c^{3} d^{2}-21 b^{2} c^{5} D\right ) \ln \left (d x +c \right )}{d^{8}}\) | \(635\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1094\) |
Input:
int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
((4*A*a*b*c*d^5+12*A*b^2*c^3*d^3-B*a^2*d^6-12*B*a*b*c^2*d^4-20*B*b^2*c^4*d ^2+2*C*a^2*c*d^5+24*C*a*b*c^3*d^3+30*C*b^2*c^5*d-6*D*a^2*c^2*d^4-40*D*a*b* c^4*d^2-42*D*b^2*c^6)/d^7*x-1/2*(A*a^2*d^7-6*A*a*b*c^2*d^5-18*A*b^2*c^4*d^ 3+B*a^2*c*d^6+18*B*a*b*c^3*d^4+30*B*b^2*c^5*d^2-3*C*a^2*c^2*d^5-36*C*a*b*c ^4*d^3-45*C*b^2*c^6*d+9*D*a^2*c^3*d^4+60*D*a*b*c^5*d^2+63*D*b^2*c^7)/d^8-1 /3*(6*A*b^2*c*d^3-6*B*a*b*d^4-10*B*b^2*c^2*d^2+12*C*a*b*c*d^3+15*C*b^2*c^3 *d-3*D*a^2*d^4-20*D*a*b*c^2*d^2-21*D*b^2*c^4)/d^5*x^3+1/30*b*(10*B*b*d^2-1 5*C*b*c*d+20*D*a*d^2+21*D*b*c^2)/d^3*x^5+1/12*b*(6*A*b*d^3-10*B*b*c*d^2+12 *C*a*d^3+15*C*b*c^2*d-20*D*a*c*d^2-21*D*b*c^3)/d^4*x^4+1/5*b^2*D*x^7/d+1/2 0*b^2*(5*C*d-7*D*c)/d^2*x^6)/(d*x+c)^2+1/d^8*(2*A*a*b*d^5+6*A*b^2*c^2*d^3- 6*B*a*b*c*d^4-10*B*b^2*c^3*d^2+C*a^2*d^5+12*C*a*b*c^2*d^3+15*C*b^2*c^4*d-3 *D*a^2*c*d^4-20*D*a*b*c^3*d^2-21*D*b^2*c^5)*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 863 vs. \(2 (415) = 830\).
Time = 0.08 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="fricas")
Output:
1/60*(12*D*b^2*d^7*x^7 - 390*D*b^2*c^7 + 330*C*b^2*c^6*d - 30*B*a^2*c*d^6 - 30*A*a^2*d^7 - 270*(2*D*a*b + B*b^2)*c^5*d^2 + 210*(2*C*a*b + A*b^2)*c^4 *d^3 - 150*(D*a^2 + 2*B*a*b)*c^3*d^4 + 90*(C*a^2 + 2*A*a*b)*c^2*d^5 - 3*(7 *D*b^2*c*d^6 - 5*C*b^2*d^7)*x^6 + 2*(21*D*b^2*c^2*d^5 - 15*C*b^2*c*d^6 + 1 0*(2*D*a*b + B*b^2)*d^7)*x^5 - 5*(21*D*b^2*c^3*d^4 - 15*C*b^2*c^2*d^5 + 10 *(2*D*a*b + B*b^2)*c*d^6 - 6*(2*C*a*b + A*b^2)*d^7)*x^4 + 20*(21*D*b^2*c^4 *d^3 - 15*C*b^2*c^3*d^4 + 10*(2*D*a*b + B*b^2)*c^2*d^5 - 6*(2*C*a*b + A*b^ 2)*c*d^6 + 3*(D*a^2 + 2*B*a*b)*d^7)*x^3 + 30*(50*D*b^2*c^5*d^2 - 34*C*b^2* c^4*d^3 + 21*(2*D*a*b + B*b^2)*c^3*d^4 - 11*(2*C*a*b + A*b^2)*c^2*d^5 + 4* (D*a^2 + 2*B*a*b)*c*d^6)*x^2 + 60*(8*D*b^2*c^6*d - 4*C*b^2*c^5*d^2 - B*a^2 *d^7 + (2*D*a*b + B*b^2)*c^4*d^3 + (2*C*a*b + A*b^2)*c^3*d^4 - 2*(D*a^2 + 2*B*a*b)*c^2*d^5 + 2*(C*a^2 + 2*A*a*b)*c*d^6)*x - 60*(21*D*b^2*c^7 - 15*C* b^2*c^6*d + 10*(2*D*a*b + B*b^2)*c^5*d^2 - 6*(2*C*a*b + A*b^2)*c^4*d^3 + 3 *(D*a^2 + 2*B*a*b)*c^3*d^4 - (C*a^2 + 2*A*a*b)*c^2*d^5 + (21*D*b^2*c^5*d^2 - 15*C*b^2*c^4*d^3 + 10*(2*D*a*b + B*b^2)*c^3*d^4 - 6*(2*C*a*b + A*b^2)*c ^2*d^5 + 3*(D*a^2 + 2*B*a*b)*c*d^6 - (C*a^2 + 2*A*a*b)*d^7)*x^2 + 2*(21*D* b^2*c^6*d - 15*C*b^2*c^5*d^2 + 10*(2*D*a*b + B*b^2)*c^4*d^3 - 6*(2*C*a*b + A*b^2)*c^3*d^4 + 3*(D*a^2 + 2*B*a*b)*c^2*d^5 - (C*a^2 + 2*A*a*b)*c*d^6)*x )*log(d*x + c))/(d^10*x^2 + 2*c*d^9*x + c^2*d^8)
Time = 16.64 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {D b^{2} x^{5}}{5 d^{3}} + x^{4} \left (\frac {C b^{2}}{4 d^{3}} - \frac {3 D b^{2} c}{4 d^{4}}\right ) + x^{3} \left (\frac {B b^{2}}{3 d^{3}} - \frac {C b^{2} c}{d^{4}} + \frac {2 D a b}{3 d^{3}} + \frac {2 D b^{2} c^{2}}{d^{5}}\right ) + x^{2} \left (\frac {A b^{2}}{2 d^{3}} - \frac {3 B b^{2} c}{2 d^{4}} + \frac {C a b}{d^{3}} + \frac {3 C b^{2} c^{2}}{d^{5}} - \frac {3 D a b c}{d^{4}} - \frac {5 D b^{2} c^{3}}{d^{6}}\right ) + x \left (- \frac {3 A b^{2} c}{d^{4}} + \frac {2 B a b}{d^{3}} + \frac {6 B b^{2} c^{2}}{d^{5}} - \frac {6 C a b c}{d^{4}} - \frac {10 C b^{2} c^{3}}{d^{6}} + \frac {D a^{2}}{d^{3}} + \frac {12 D a b c^{2}}{d^{5}} + \frac {15 D b^{2} c^{4}}{d^{7}}\right ) + \frac {- A a^{2} d^{7} + 6 A a b c^{2} d^{5} + 7 A b^{2} c^{4} d^{3} - B a^{2} c d^{6} - 10 B a b c^{3} d^{4} - 9 B b^{2} c^{5} d^{2} + 3 C a^{2} c^{2} d^{5} + 14 C a b c^{4} d^{3} + 11 C b^{2} c^{6} d - 5 D a^{2} c^{3} d^{4} - 18 D a b c^{5} d^{2} - 13 D b^{2} c^{7} + x \left (8 A a b c d^{6} + 8 A b^{2} c^{3} d^{4} - 2 B a^{2} d^{7} - 12 B a b c^{2} d^{5} - 10 B b^{2} c^{4} d^{3} + 4 C a^{2} c d^{6} + 16 C a b c^{3} d^{4} + 12 C b^{2} c^{5} d^{2} - 6 D a^{2} c^{2} d^{5} - 20 D a b c^{4} d^{3} - 14 D b^{2} c^{6} d\right )}{2 c^{2} d^{8} + 4 c d^{9} x + 2 d^{10} x^{2}} - \frac {\left (- 2 A a b d^{5} - 6 A b^{2} c^{2} d^{3} + 6 B a b c d^{4} + 10 B b^{2} c^{3} d^{2} - C a^{2} d^{5} - 12 C a b c^{2} d^{3} - 15 C b^{2} c^{4} d + 3 D a^{2} c d^{4} + 20 D a b c^{3} d^{2} + 21 D b^{2} c^{5}\right ) \log {\left (c + d x \right )}}{d^{8}} \] Input:
integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**3,x)
Output:
D*b**2*x**5/(5*d**3) + x**4*(C*b**2/(4*d**3) - 3*D*b**2*c/(4*d**4)) + x**3 *(B*b**2/(3*d**3) - C*b**2*c/d**4 + 2*D*a*b/(3*d**3) + 2*D*b**2*c**2/d**5) + x**2*(A*b**2/(2*d**3) - 3*B*b**2*c/(2*d**4) + C*a*b/d**3 + 3*C*b**2*c** 2/d**5 - 3*D*a*b*c/d**4 - 5*D*b**2*c**3/d**6) + x*(-3*A*b**2*c/d**4 + 2*B* a*b/d**3 + 6*B*b**2*c**2/d**5 - 6*C*a*b*c/d**4 - 10*C*b**2*c**3/d**6 + D*a **2/d**3 + 12*D*a*b*c**2/d**5 + 15*D*b**2*c**4/d**7) + (-A*a**2*d**7 + 6*A *a*b*c**2*d**5 + 7*A*b**2*c**4*d**3 - B*a**2*c*d**6 - 10*B*a*b*c**3*d**4 - 9*B*b**2*c**5*d**2 + 3*C*a**2*c**2*d**5 + 14*C*a*b*c**4*d**3 + 11*C*b**2* c**6*d - 5*D*a**2*c**3*d**4 - 18*D*a*b*c**5*d**2 - 13*D*b**2*c**7 + x*(8*A *a*b*c*d**6 + 8*A*b**2*c**3*d**4 - 2*B*a**2*d**7 - 12*B*a*b*c**2*d**5 - 10 *B*b**2*c**4*d**3 + 4*C*a**2*c*d**6 + 16*C*a*b*c**3*d**4 + 12*C*b**2*c**5* d**2 - 6*D*a**2*c**2*d**5 - 20*D*a*b*c**4*d**3 - 14*D*b**2*c**6*d))/(2*c** 2*d**8 + 4*c*d**9*x + 2*d**10*x**2) - (-2*A*a*b*d**5 - 6*A*b**2*c**2*d**3 + 6*B*a*b*c*d**4 + 10*B*b**2*c**3*d**2 - C*a**2*d**5 - 12*C*a*b*c**2*d**3 - 15*C*b**2*c**4*d + 3*D*a**2*c*d**4 + 20*D*a*b*c**3*d**2 + 21*D*b**2*c**5 )*log(c + d*x)/d**8
Time = 0.04 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=-\frac {13 \, D b^{2} c^{7} - 11 \, C b^{2} c^{6} d + B a^{2} c d^{6} + A a^{2} d^{7} + 9 \, {\left (2 \, D a b + B b^{2}\right )} c^{5} d^{2} - 7 \, {\left (2 \, C a b + A b^{2}\right )} c^{4} d^{3} + 5 \, {\left (D a^{2} + 2 \, B a b\right )} c^{3} d^{4} - 3 \, {\left (C a^{2} + 2 \, A a b\right )} c^{2} d^{5} + 2 \, {\left (7 \, D b^{2} c^{6} d - 6 \, C b^{2} c^{5} d^{2} + B a^{2} d^{7} + 5 \, {\left (2 \, D a b + B b^{2}\right )} c^{4} d^{3} - 4 \, {\left (2 \, C a b + A b^{2}\right )} c^{3} d^{4} + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c^{2} d^{5} - 2 \, {\left (C a^{2} + 2 \, A a b\right )} c d^{6}\right )} x}{2 \, {\left (d^{10} x^{2} + 2 \, c d^{9} x + c^{2} d^{8}\right )}} + \frac {12 \, D b^{2} d^{4} x^{5} - 15 \, {\left (3 \, D b^{2} c d^{3} - C b^{2} d^{4}\right )} x^{4} + 20 \, {\left (6 \, D b^{2} c^{2} d^{2} - 3 \, C b^{2} c d^{3} + {\left (2 \, D a b + B b^{2}\right )} d^{4}\right )} x^{3} - 30 \, {\left (10 \, D b^{2} c^{3} d - 6 \, C b^{2} c^{2} d^{2} + 3 \, {\left (2 \, D a b + B b^{2}\right )} c d^{3} - {\left (2 \, C a b + A b^{2}\right )} d^{4}\right )} x^{2} + 60 \, {\left (15 \, D b^{2} c^{4} - 10 \, C b^{2} c^{3} d + 6 \, {\left (2 \, D a b + B b^{2}\right )} c^{2} d^{2} - 3 \, {\left (2 \, C a b + A b^{2}\right )} c d^{3} + {\left (D a^{2} + 2 \, B a b\right )} d^{4}\right )} x}{60 \, d^{7}} - \frac {{\left (21 \, D b^{2} c^{5} - 15 \, C b^{2} c^{4} d + 10 \, {\left (2 \, D a b + B b^{2}\right )} c^{3} d^{2} - 6 \, {\left (2 \, C a b + A b^{2}\right )} c^{2} d^{3} + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c d^{4} - {\left (C a^{2} + 2 \, A a b\right )} d^{5}\right )} \log \left (d x + c\right )}{d^{8}} \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="maxima")
Output:
-1/2*(13*D*b^2*c^7 - 11*C*b^2*c^6*d + B*a^2*c*d^6 + A*a^2*d^7 + 9*(2*D*a*b + B*b^2)*c^5*d^2 - 7*(2*C*a*b + A*b^2)*c^4*d^3 + 5*(D*a^2 + 2*B*a*b)*c^3* d^4 - 3*(C*a^2 + 2*A*a*b)*c^2*d^5 + 2*(7*D*b^2*c^6*d - 6*C*b^2*c^5*d^2 + B *a^2*d^7 + 5*(2*D*a*b + B*b^2)*c^4*d^3 - 4*(2*C*a*b + A*b^2)*c^3*d^4 + 3*( D*a^2 + 2*B*a*b)*c^2*d^5 - 2*(C*a^2 + 2*A*a*b)*c*d^6)*x)/(d^10*x^2 + 2*c*d ^9*x + c^2*d^8) + 1/60*(12*D*b^2*d^4*x^5 - 15*(3*D*b^2*c*d^3 - C*b^2*d^4)* x^4 + 20*(6*D*b^2*c^2*d^2 - 3*C*b^2*c*d^3 + (2*D*a*b + B*b^2)*d^4)*x^3 - 3 0*(10*D*b^2*c^3*d - 6*C*b^2*c^2*d^2 + 3*(2*D*a*b + B*b^2)*c*d^3 - (2*C*a*b + A*b^2)*d^4)*x^2 + 60*(15*D*b^2*c^4 - 10*C*b^2*c^3*d + 6*(2*D*a*b + B*b^ 2)*c^2*d^2 - 3*(2*C*a*b + A*b^2)*c*d^3 + (D*a^2 + 2*B*a*b)*d^4)*x)/d^7 - ( 21*D*b^2*c^5 - 15*C*b^2*c^4*d + 10*(2*D*a*b + B*b^2)*c^3*d^2 - 6*(2*C*a*b + A*b^2)*c^2*d^3 + 3*(D*a^2 + 2*B*a*b)*c*d^4 - (C*a^2 + 2*A*a*b)*d^5)*log( d*x + c)/d^8
Time = 0.31 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=-\frac {{\left (21 \, D b^{2} c^{5} - 15 \, C b^{2} c^{4} d + 20 \, D a b c^{3} d^{2} + 10 \, B b^{2} c^{3} d^{2} - 12 \, C a b c^{2} d^{3} - 6 \, A b^{2} c^{2} d^{3} + 3 \, D a^{2} c d^{4} + 6 \, B a b c d^{4} - C a^{2} d^{5} - 2 \, A a b d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{8}} - \frac {13 \, D b^{2} c^{7} - 11 \, C b^{2} c^{6} d + 18 \, D a b c^{5} d^{2} + 9 \, B b^{2} c^{5} d^{2} - 14 \, C a b c^{4} d^{3} - 7 \, A b^{2} c^{4} d^{3} + 5 \, D a^{2} c^{3} d^{4} + 10 \, B a b c^{3} d^{4} - 3 \, C a^{2} c^{2} d^{5} - 6 \, A a b c^{2} d^{5} + B a^{2} c d^{6} + A a^{2} d^{7} + 2 \, {\left (7 \, D b^{2} c^{6} d - 6 \, C b^{2} c^{5} d^{2} + 10 \, D a b c^{4} d^{3} + 5 \, B b^{2} c^{4} d^{3} - 8 \, C a b c^{3} d^{4} - 4 \, A b^{2} c^{3} d^{4} + 3 \, D a^{2} c^{2} d^{5} + 6 \, B a b c^{2} d^{5} - 2 \, C a^{2} c d^{6} - 4 \, A a b c d^{6} + B a^{2} d^{7}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{8}} + \frac {12 \, D b^{2} d^{12} x^{5} - 45 \, D b^{2} c d^{11} x^{4} + 15 \, C b^{2} d^{12} x^{4} + 120 \, D b^{2} c^{2} d^{10} x^{3} - 60 \, C b^{2} c d^{11} x^{3} + 40 \, D a b d^{12} x^{3} + 20 \, B b^{2} d^{12} x^{3} - 300 \, D b^{2} c^{3} d^{9} x^{2} + 180 \, C b^{2} c^{2} d^{10} x^{2} - 180 \, D a b c d^{11} x^{2} - 90 \, B b^{2} c d^{11} x^{2} + 60 \, C a b d^{12} x^{2} + 30 \, A b^{2} d^{12} x^{2} + 900 \, D b^{2} c^{4} d^{8} x - 600 \, C b^{2} c^{3} d^{9} x + 720 \, D a b c^{2} d^{10} x + 360 \, B b^{2} c^{2} d^{10} x - 360 \, C a b c d^{11} x - 180 \, A b^{2} c d^{11} x + 60 \, D a^{2} d^{12} x + 120 \, B a b d^{12} x}{60 \, d^{15}} \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="giac")
Output:
-(21*D*b^2*c^5 - 15*C*b^2*c^4*d + 20*D*a*b*c^3*d^2 + 10*B*b^2*c^3*d^2 - 12 *C*a*b*c^2*d^3 - 6*A*b^2*c^2*d^3 + 3*D*a^2*c*d^4 + 6*B*a*b*c*d^4 - C*a^2*d ^5 - 2*A*a*b*d^5)*log(abs(d*x + c))/d^8 - 1/2*(13*D*b^2*c^7 - 11*C*b^2*c^6 *d + 18*D*a*b*c^5*d^2 + 9*B*b^2*c^5*d^2 - 14*C*a*b*c^4*d^3 - 7*A*b^2*c^4*d ^3 + 5*D*a^2*c^3*d^4 + 10*B*a*b*c^3*d^4 - 3*C*a^2*c^2*d^5 - 6*A*a*b*c^2*d^ 5 + B*a^2*c*d^6 + A*a^2*d^7 + 2*(7*D*b^2*c^6*d - 6*C*b^2*c^5*d^2 + 10*D*a* b*c^4*d^3 + 5*B*b^2*c^4*d^3 - 8*C*a*b*c^3*d^4 - 4*A*b^2*c^3*d^4 + 3*D*a^2* c^2*d^5 + 6*B*a*b*c^2*d^5 - 2*C*a^2*c*d^6 - 4*A*a*b*c*d^6 + B*a^2*d^7)*x)/ ((d*x + c)^2*d^8) + 1/60*(12*D*b^2*d^12*x^5 - 45*D*b^2*c*d^11*x^4 + 15*C*b ^2*d^12*x^4 + 120*D*b^2*c^2*d^10*x^3 - 60*C*b^2*c*d^11*x^3 + 40*D*a*b*d^12 *x^3 + 20*B*b^2*d^12*x^3 - 300*D*b^2*c^3*d^9*x^2 + 180*C*b^2*c^2*d^10*x^2 - 180*D*a*b*c*d^11*x^2 - 90*B*b^2*c*d^11*x^2 + 60*C*a*b*d^12*x^2 + 30*A*b^ 2*d^12*x^2 + 900*D*b^2*c^4*d^8*x - 600*C*b^2*c^3*d^9*x + 720*D*a*b*c^2*d^1 0*x + 360*B*b^2*c^2*d^10*x - 360*C*a*b*c*d^11*x - 180*A*b^2*c*d^11*x + 60* D*a^2*d^12*x + 120*B*a*b*d^12*x)/d^15
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b*x^2)^2*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3,x)
Output:
int(((a + b*x^2)^2*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3, x)
Time = 174.81 (sec) , antiderivative size = 800, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x)
Output:
(60*log(c + d*x)*a**2*b*c**3*d**4 + 120*log(c + d*x)*a**2*b*c**2*d**5*x + 60*log(c + d*x)*a**2*b*c*d**6*x**2 - 60*log(c + d*x)*a**2*c**4*d**4 - 120* log(c + d*x)*a**2*c**3*d**5*x - 60*log(c + d*x)*a**2*c**2*d**6*x**2 + 180* log(c + d*x)*a*b**2*c**5*d**2 + 360*log(c + d*x)*a*b**2*c**4*d**3*x - 180* log(c + d*x)*a*b**2*c**4*d**3 + 180*log(c + d*x)*a*b**2*c**3*d**4*x**2 - 3 60*log(c + d*x)*a*b**2*c**3*d**4*x - 180*log(c + d*x)*a*b**2*c**2*d**5*x** 2 - 240*log(c + d*x)*a*b*c**6*d**2 - 480*log(c + d*x)*a*b*c**5*d**3*x - 24 0*log(c + d*x)*a*b*c**4*d**4*x**2 - 300*log(c + d*x)*b**3*c**6*d - 600*log (c + d*x)*b**3*c**5*d**2*x - 300*log(c + d*x)*b**3*c**4*d**3*x**2 - 180*lo g(c + d*x)*b**2*c**8 - 360*log(c + d*x)*b**2*c**7*d*x - 180*log(c + d*x)*b **2*c**6*d**2*x**2 - 15*a**3*c*d**6 + 30*a**2*b*c**3*d**4 - 60*a**2*b*c*d* *6*x**2 + 15*a**2*b*d**7*x**2 - 30*a**2*c**4*d**4 + 60*a**2*c**2*d**6*x**2 + 30*a**2*c*d**7*x**3 + 90*a*b**2*c**5*d**2 - 90*a*b**2*c**4*d**3 - 180*a *b**2*c**3*d**4*x**2 - 60*a*b**2*c**2*d**5*x**3 + 180*a*b**2*c**2*d**5*x** 2 + 15*a*b**2*c*d**6*x**4 + 60*a*b**2*c*d**6*x**3 - 120*a*b*c**6*d**2 + 24 0*a*b*c**4*d**4*x**2 + 80*a*b*c**3*d**5*x**3 - 20*a*b*c**2*d**6*x**4 + 20* a*b*c*d**7*x**5 - 150*b**3*c**6*d + 300*b**3*c**4*d**3*x**2 + 100*b**3*c** 3*d**4*x**3 - 25*b**3*c**2*d**5*x**4 + 10*b**3*c*d**6*x**5 - 90*b**2*c**8 + 180*b**2*c**6*d**2*x**2 + 60*b**2*c**5*d**3*x**3 - 15*b**2*c**4*d**4*x** 4 + 6*b**2*c**3*d**5*x**5 - 3*b**2*c**2*d**6*x**6 + 6*b**2*c*d**7*x**7)...