3.1.31 \(\int \frac {(a+c x^2)^2 (A+B x+C x^2)}{(d+e x)^3} \, dx\) [31]

3.1.31.1 Optimal result
3.1.31.2 Mathematica [A] (verified)
3.1.31.3 Rubi [A] (verified)
3.1.31.4 Maple [A] (verified)
3.1.31.5 Fricas [B] (verification not implemented)
3.1.31.6 Sympy [A] (verification not implemented)
3.1.31.7 Maxima [A] (verification not implemented)
3.1.31.8 Giac [A] (verification not implemented)
3.1.31.9 Mupad [B] (verification not implemented)

3.1.31.1 Optimal result

Integrand size = 27, antiderivative size = 295 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=-\frac {c \left (2 a e^2 (3 C d-B e)+c d \left (10 C d^2-3 e (2 B d-A e)\right )\right ) x}{e^6}+\frac {c \left (2 a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right ) x^2}{2 e^5}-\frac {c^2 (3 C d-B e) x^3}{3 e^4}+\frac {c^2 C x^4}{4 e^3}-\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{2 e^7 (d+e x)^2}+\frac {\left (c d^2+a e^2\right ) \left (a e^2 (2 C d-B e)+c d \left (6 C d^2-e (5 B d-4 A e)\right )\right )}{e^7 (d+e x)}+\frac {\left (a^2 C e^4+c^2 d^2 \left (15 C d^2-2 e (5 B d-3 A e)\right )+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) \log (d+e x)}{e^7} \]

output
-c*(2*a*e^2*(-B*e+3*C*d)+c*d*(10*C*d^2-3*e*(-A*e+2*B*d)))*x/e^6+1/2*c*(2*a 
*C*e^2+c*(6*C*d^2-e*(-A*e+3*B*d)))*x^2/e^5-1/3*c^2*(-B*e+3*C*d)*x^3/e^4+1/ 
4*c^2*C*x^4/e^3-1/2*(a*e^2+c*d^2)^2*(A*e^2-B*d*e+C*d^2)/e^7/(e*x+d)^2+(a*e 
^2+c*d^2)*(a*e^2*(-B*e+2*C*d)+c*d*(6*C*d^2-e*(-4*A*e+5*B*d)))/e^7/(e*x+d)+ 
(a^2*C*e^4+c^2*d^2*(15*C*d^2-2*e*(-3*A*e+5*B*d))+2*a*c*e^2*(6*C*d^2-e*(-A* 
e+3*B*d)))*ln(e*x+d)/e^7
 
3.1.31.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {-12 c e \left (10 c C d^3+3 c d e (-2 B d+A e)-2 a e^2 (-3 C d+B e)\right ) x+6 c e^2 \left (6 c C d^2+2 a C e^2+c e (-3 B d+A e)\right ) x^2+4 c^2 e^3 (-3 C d+B e) x^3+3 c^2 C e^4 x^4-\frac {6 \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right )}{(d+e x)^2}+\frac {12 \left (c d^2+a e^2\right ) \left (6 c C d^3+c d e (-5 B d+4 A e)+a e^2 (2 C d-B e)\right )}{d+e x}+12 \left (a^2 C e^4+2 a c e^2 \left (6 C d^2+e (-3 B d+A e)\right )+c^2 \left (15 C d^4+2 d^2 e (-5 B d+3 A e)\right )\right ) \log (d+e x)}{12 e^7} \]

input
Integrate[((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x)^3,x]
 
output
(-12*c*e*(10*c*C*d^3 + 3*c*d*e*(-2*B*d + A*e) - 2*a*e^2*(-3*C*d + B*e))*x 
+ 6*c*e^2*(6*c*C*d^2 + 2*a*C*e^2 + c*e*(-3*B*d + A*e))*x^2 + 4*c^2*e^3*(-3 
*C*d + B*e)*x^3 + 3*c^2*C*e^4*x^4 - (6*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d 
) + A*e)))/(d + e*x)^2 + (12*(c*d^2 + a*e^2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 
4*A*e) + a*e^2*(2*C*d - B*e)))/(d + e*x) + 12*(a^2*C*e^4 + 2*a*c*e^2*(6*C* 
d^2 + e*(-3*B*d + A*e)) + c^2*(15*C*d^4 + 2*d^2*e*(-5*B*d + 3*A*e)))*Log[d 
 + e*x])/(12*e^7)
 
3.1.31.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2159, 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.

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

\(\Big \downarrow \) 2159

\(\displaystyle \int \left (\frac {a^2 C e^4+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )}{e^6 (d+e x)}+\frac {c \left (-2 a e^2 (3 C d-B e)+3 c d e (2 B d-A e)-10 c C d^3\right )}{e^6}+\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^6 (d+e x)^3}+\frac {c x \left (2 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{e^5}+\frac {\left (a e^2+c d^2\right ) \left (-a e^2 (2 C d-B e)+c d e (5 B d-4 A e)-6 c C d^3\right )}{e^6 (d+e x)^2}+\frac {c^2 x^2 (B e-3 C d)}{e^4}+\frac {c^2 C x^3}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (d+e x) \left (a^2 C e^4+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{e^7}-\frac {c x \left (2 a e^2 (3 C d-B e)-3 c d e (2 B d-A e)+10 c C d^3\right )}{e^6}-\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{2 e^7 (d+e x)^2}+\frac {c x^2 \left (2 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{2 e^5}+\frac {\left (a e^2+c d^2\right ) \left (a e^2 (2 C d-B e)-c d e (5 B d-4 A e)+6 c C d^3\right )}{e^7 (d+e x)}-\frac {c^2 x^3 (3 C d-B e)}{3 e^4}+\frac {c^2 C x^4}{4 e^3}\)

input
Int[((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x)^3,x]
 
output
-((c*(10*c*C*d^3 - 3*c*d*e*(2*B*d - A*e) + 2*a*e^2*(3*C*d - B*e))*x)/e^6) 
+ (c*(6*c*C*d^2 + 2*a*C*e^2 - c*e*(3*B*d - A*e))*x^2)/(2*e^5) - (c^2*(3*C* 
d - B*e)*x^3)/(3*e^4) + (c^2*C*x^4)/(4*e^3) - ((c*d^2 + a*e^2)^2*(C*d^2 - 
B*d*e + A*e^2))/(2*e^7*(d + e*x)^2) + ((c*d^2 + a*e^2)*(6*c*C*d^3 - c*d*e* 
(5*B*d - 4*A*e) + a*e^2*(2*C*d - B*e)))/(e^7*(d + e*x)) + ((a^2*C*e^4 + c^ 
2*(15*C*d^4 - 2*d^2*e*(5*B*d - 3*A*e)) + 2*a*c*e^2*(6*C*d^2 - e*(3*B*d - A 
*e)))*Log[d + e*x])/e^7
 

3.1.31.3.1 Defintions of rubi rules used

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2159
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 
3.1.31.4 Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.33

method result size
norman \(\frac {\frac {\left (4 A a c d \,e^{4}+12 A \,c^{2} d^{3} e^{2}-B \,e^{5} a^{2}-12 B a c \,d^{2} e^{3}-20 B \,c^{2} d^{4} e +2 C \,a^{2} d \,e^{4}+24 C a c \,d^{3} e^{2}+30 C \,c^{2} d^{5}\right ) x}{e^{6}}-\frac {A \,a^{2} e^{6}-6 A a c \,d^{2} e^{4}-18 A \,c^{2} d^{4} e^{2}+B \,a^{2} d \,e^{5}+18 B a c \,d^{3} e^{3}+30 B \,c^{2} d^{5} e -3 C \,a^{2} d^{2} e^{4}-36 C a c \,d^{4} e^{2}-45 C \,c^{2} d^{6}}{2 e^{7}}+\frac {c \left (6 A c \,e^{2}-10 B c d e +12 a C \,e^{2}+15 C c \,d^{2}\right ) x^{4}}{12 e^{3}}-\frac {c \left (6 A c d \,e^{2}-6 B \,e^{3} a -10 B c \,d^{2} e +12 C a d \,e^{2}+15 C c \,d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c^{2} C \,x^{6}}{4 e}+\frac {c^{2} \left (2 B e -3 C d \right ) x^{5}}{6 e^{2}}}{\left (e x +d \right )^{2}}+\frac {\left (2 A a c \,e^{4}+6 A \,c^{2} d^{2} e^{2}-6 B a c d \,e^{3}-10 B \,c^{2} d^{3} e +a^{2} C \,e^{4}+12 C a c \,d^{2} e^{2}+15 C \,c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\) \(392\)
default \(-\frac {c \left (-\frac {1}{4} c C \,x^{4} e^{3}-\frac {1}{3} B c \,x^{3} e^{3}+C c d \,e^{2} x^{3}-\frac {1}{2} A c \,e^{3} x^{2}+\frac {3}{2} B \,x^{2} c d \,e^{2}-C a \,e^{3} x^{2}-3 C c \,d^{2} e \,x^{2}+3 A c d \,e^{2} x -2 B x a \,e^{3}-6 B c \,d^{2} e x +6 C a d \,e^{2} x +10 C c \,d^{3} x \right )}{e^{6}}-\frac {-4 A a c d \,e^{4}-4 A \,c^{2} d^{3} e^{2}+B \,e^{5} a^{2}+6 B a c \,d^{2} e^{3}+5 B \,c^{2} d^{4} e -2 C \,a^{2} d \,e^{4}-8 C a c \,d^{3} e^{2}-6 C \,c^{2} d^{5}}{e^{7} \left (e x +d \right )}+\frac {\left (2 A a c \,e^{4}+6 A \,c^{2} d^{2} e^{2}-6 B a c d \,e^{3}-10 B \,c^{2} d^{3} e +a^{2} C \,e^{4}+12 C a c \,d^{2} e^{2}+15 C \,c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}}{2 e^{7} \left (e x +d \right )^{2}}\) \(399\)
risch \(\frac {c^{2} C \,x^{4}}{4 e^{3}}+\frac {c^{2} B \,x^{3}}{3 e^{3}}-\frac {c^{2} C d \,x^{3}}{e^{4}}+\frac {c^{2} A \,x^{2}}{2 e^{3}}-\frac {3 c^{2} B \,x^{2} d}{2 e^{4}}+\frac {c C a \,x^{2}}{e^{3}}+\frac {3 c^{2} C \,d^{2} x^{2}}{e^{5}}-\frac {3 c^{2} A d x}{e^{4}}+\frac {2 c B x a}{e^{3}}+\frac {6 c^{2} B \,d^{2} x}{e^{5}}-\frac {6 c C a d x}{e^{4}}-\frac {10 c^{2} C \,d^{3} x}{e^{6}}+\frac {\left (4 A a c d \,e^{4}+4 A \,c^{2} d^{3} e^{2}-B \,e^{5} a^{2}-6 B a c \,d^{2} e^{3}-5 B \,c^{2} d^{4} e +2 C \,a^{2} d \,e^{4}+8 C a c \,d^{3} e^{2}+6 C \,c^{2} d^{5}\right ) x -\frac {A \,a^{2} e^{6}-6 A a c \,d^{2} e^{4}-7 A \,c^{2} d^{4} e^{2}+B \,a^{2} d \,e^{5}+10 B a c \,d^{3} e^{3}+9 B \,c^{2} d^{5} e -3 C \,a^{2} d^{2} e^{4}-14 C a c \,d^{4} e^{2}-11 C \,c^{2} d^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {2 \ln \left (e x +d \right ) A a c}{e^{3}}+\frac {6 \ln \left (e x +d \right ) A \,c^{2} d^{2}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) B a c d}{e^{4}}-\frac {10 \ln \left (e x +d \right ) B \,c^{2} d^{3}}{e^{6}}+\frac {\ln \left (e x +d \right ) a^{2} C}{e^{3}}+\frac {12 \ln \left (e x +d \right ) C a c \,d^{2}}{e^{5}}+\frac {15 \ln \left (e x +d \right ) C \,c^{2} d^{4}}{e^{7}}\) \(454\)
parallelrisch \(\frac {12 C \ln \left (e x +d \right ) x^{2} a^{2} e^{6}-6 C \,x^{5} c^{2} d \,e^{5}-10 B \,x^{4} c^{2} d \,e^{5}+12 C \ln \left (e x +d \right ) a^{2} d^{2} e^{4}+72 A \ln \left (e x +d \right ) c^{2} d^{4} e^{2}-120 B \ln \left (e x +d \right ) c^{2} d^{5} e +12 C \,x^{4} a c \,e^{6}+15 C \,x^{4} c^{2} d^{2} e^{4}+144 A x \,c^{2} d^{3} e^{3}-240 B x \,c^{2} d^{4} e^{2}+24 C x \,a^{2} d \,e^{5}+72 A \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{4}-120 B \ln \left (e x +d \right ) x^{2} c^{2} d^{3} e^{3}+180 C \ln \left (e x +d \right ) x^{2} c^{2} d^{4} e^{2}+36 A a c \,d^{2} e^{4}-108 B a c \,d^{3} e^{3}+216 C a c \,d^{4} e^{2}-6 A \,a^{2} e^{6}+270 C \,c^{2} d^{6}+4 B \,x^{5} c^{2} e^{6}+6 A \,x^{4} c^{2} e^{6}+3 C \,x^{6} c^{2} e^{6}+180 C \ln \left (e x +d \right ) c^{2} d^{6}-12 B x \,a^{2} e^{6}-180 B \,c^{2} d^{5} e +18 C \,a^{2} d^{2} e^{4}-6 B \,a^{2} d \,e^{5}+108 A \,c^{2} d^{4} e^{2}+288 C x a c \,d^{3} e^{3}-48 C \,x^{3} a c d \,e^{5}+40 B \,x^{3} c^{2} d^{2} e^{4}-60 C \,x^{3} c^{2} d^{3} e^{3}+144 C \ln \left (e x +d \right ) a c \,d^{4} e^{2}+48 A \ln \left (e x +d \right ) x a c d \,e^{5}-144 B \ln \left (e x +d \right ) x a c \,d^{2} e^{4}+288 C \ln \left (e x +d \right ) x a c \,d^{3} e^{3}+144 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{3}+144 C \ln \left (e x +d \right ) x^{2} a c \,d^{2} e^{4}-72 B \ln \left (e x +d \right ) a c \,d^{3} e^{3}+360 C x \,c^{2} d^{5} e +24 B \,x^{3} a c \,e^{6}-24 A \,x^{3} c^{2} d \,e^{5}-72 B \ln \left (e x +d \right ) x^{2} a c d \,e^{5}-240 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e^{2}+24 C \ln \left (e x +d \right ) x \,a^{2} d \,e^{5}+360 C \ln \left (e x +d \right ) x \,c^{2} d^{5} e +24 A \ln \left (e x +d \right ) x^{2} a c \,e^{6}+48 A x a c d \,e^{5}-144 B x a c \,d^{2} e^{4}+24 A \ln \left (e x +d \right ) a c \,d^{2} e^{4}}{12 e^{7} \left (e x +d \right )^{2}}\) \(733\)

input
int((c*x^2+a)^2*(C*x^2+B*x+A)/(e*x+d)^3,x,method=_RETURNVERBOSE)
 
output
((4*A*a*c*d*e^4+12*A*c^2*d^3*e^2-B*a^2*e^5-12*B*a*c*d^2*e^3-20*B*c^2*d^4*e 
+2*C*a^2*d*e^4+24*C*a*c*d^3*e^2+30*C*c^2*d^5)/e^6*x-1/2*(A*a^2*e^6-6*A*a*c 
*d^2*e^4-18*A*c^2*d^4*e^2+B*a^2*d*e^5+18*B*a*c*d^3*e^3+30*B*c^2*d^5*e-3*C* 
a^2*d^2*e^4-36*C*a*c*d^4*e^2-45*C*c^2*d^6)/e^7+1/12*c*(6*A*c*e^2-10*B*c*d* 
e+12*C*a*e^2+15*C*c*d^2)/e^3*x^4-1/3*c*(6*A*c*d*e^2-6*B*a*e^3-10*B*c*d^2*e 
+12*C*a*d*e^2+15*C*c*d^3)/e^4*x^3+1/4*c^2*C*x^6/e+1/6*c^2*(2*B*e-3*C*d)/e^ 
2*x^5)/(e*x+d)^2+1/e^7*(2*A*a*c*e^4+6*A*c^2*d^2*e^2-6*B*a*c*d*e^3-10*B*c^2 
*d^3*e+C*a^2*e^4+12*C*a*c*d^2*e^2+15*C*c^2*d^4)*ln(e*x+d)
 
3.1.31.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (287) = 574\).

Time = 0.29 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {3 \, C c^{2} e^{6} x^{6} + 66 \, C c^{2} d^{6} - 54 \, B c^{2} d^{5} e - 60 \, B a c d^{3} e^{3} - 6 \, B a^{2} d e^{5} - 6 \, A a^{2} e^{6} + 42 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 18 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} - 2 \, {\left (3 \, C c^{2} d e^{5} - 2 \, B c^{2} e^{6}\right )} x^{5} + {\left (15 \, C c^{2} d^{2} e^{4} - 10 \, B c^{2} d e^{5} + 6 \, {\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 4 \, {\left (15 \, C c^{2} d^{3} e^{3} - 10 \, B c^{2} d^{2} e^{4} - 6 \, B a c e^{6} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} - 6 \, {\left (34 \, C c^{2} d^{4} e^{2} - 21 \, B c^{2} d^{3} e^{3} - 8 \, B a c d e^{5} + 11 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4}\right )} x^{2} - 12 \, {\left (4 \, C c^{2} d^{5} e - B c^{2} d^{4} e^{2} + 4 \, B a c d^{2} e^{4} + B a^{2} e^{6} - {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} - 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x + 12 \, {\left (15 \, C c^{2} d^{6} - 10 \, B c^{2} d^{5} e - 6 \, B a c d^{3} e^{3} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} + {\left (15 \, C c^{2} d^{4} e^{2} - 10 \, B c^{2} d^{3} e^{3} - 6 \, B a c d e^{5} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4} + {\left (C a^{2} + 2 \, A a c\right )} e^{6}\right )} x^{2} + 2 \, {\left (15 \, C c^{2} d^{5} e - 10 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

input
integrate((c*x^2+a)^2*(C*x^2+B*x+A)/(e*x+d)^3,x, algorithm="fricas")
 
output
1/12*(3*C*c^2*e^6*x^6 + 66*C*c^2*d^6 - 54*B*c^2*d^5*e - 60*B*a*c*d^3*e^3 - 
 6*B*a^2*d*e^5 - 6*A*a^2*e^6 + 42*(2*C*a*c + A*c^2)*d^4*e^2 + 18*(C*a^2 + 
2*A*a*c)*d^2*e^4 - 2*(3*C*c^2*d*e^5 - 2*B*c^2*e^6)*x^5 + (15*C*c^2*d^2*e^4 
 - 10*B*c^2*d*e^5 + 6*(2*C*a*c + A*c^2)*e^6)*x^4 - 4*(15*C*c^2*d^3*e^3 - 1 
0*B*c^2*d^2*e^4 - 6*B*a*c*e^6 + 6*(2*C*a*c + A*c^2)*d*e^5)*x^3 - 6*(34*C*c 
^2*d^4*e^2 - 21*B*c^2*d^3*e^3 - 8*B*a*c*d*e^5 + 11*(2*C*a*c + A*c^2)*d^2*e 
^4)*x^2 - 12*(4*C*c^2*d^5*e - B*c^2*d^4*e^2 + 4*B*a*c*d^2*e^4 + B*a^2*e^6 
- (2*C*a*c + A*c^2)*d^3*e^3 - 2*(C*a^2 + 2*A*a*c)*d*e^5)*x + 12*(15*C*c^2* 
d^6 - 10*B*c^2*d^5*e - 6*B*a*c*d^3*e^3 + 6*(2*C*a*c + A*c^2)*d^4*e^2 + (C* 
a^2 + 2*A*a*c)*d^2*e^4 + (15*C*c^2*d^4*e^2 - 10*B*c^2*d^3*e^3 - 6*B*a*c*d* 
e^5 + 6*(2*C*a*c + A*c^2)*d^2*e^4 + (C*a^2 + 2*A*a*c)*e^6)*x^2 + 2*(15*C*c 
^2*d^5*e - 10*B*c^2*d^4*e^2 - 6*B*a*c*d^2*e^4 + 6*(2*C*a*c + A*c^2)*d^3*e^ 
3 + (C*a^2 + 2*A*a*c)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e 
^7)
 
3.1.31.6 Sympy [A] (verification not implemented)

Time = 4.59 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {C c^{2} x^{4}}{4 e^{3}} + x^{3} \left (\frac {B c^{2}}{3 e^{3}} - \frac {C c^{2} d}{e^{4}}\right ) + x^{2} \left (\frac {A c^{2}}{2 e^{3}} - \frac {3 B c^{2} d}{2 e^{4}} + \frac {C a c}{e^{3}} + \frac {3 C c^{2} d^{2}}{e^{5}}\right ) + x \left (- \frac {3 A c^{2} d}{e^{4}} + \frac {2 B a c}{e^{3}} + \frac {6 B c^{2} d^{2}}{e^{5}} - \frac {6 C a c d}{e^{4}} - \frac {10 C c^{2} d^{3}}{e^{6}}\right ) + \frac {- A a^{2} e^{6} + 6 A a c d^{2} e^{4} + 7 A c^{2} d^{4} e^{2} - B a^{2} d e^{5} - 10 B a c d^{3} e^{3} - 9 B c^{2} d^{5} e + 3 C a^{2} d^{2} e^{4} + 14 C a c d^{4} e^{2} + 11 C c^{2} d^{6} + x \left (8 A a c d e^{5} + 8 A c^{2} d^{3} e^{3} - 2 B a^{2} e^{6} - 12 B a c d^{2} e^{4} - 10 B c^{2} d^{4} e^{2} + 4 C a^{2} d e^{5} + 16 C a c d^{3} e^{3} + 12 C c^{2} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac {\left (2 A a c e^{4} + 6 A c^{2} d^{2} e^{2} - 6 B a c d e^{3} - 10 B c^{2} d^{3} e + C a^{2} e^{4} + 12 C a c d^{2} e^{2} + 15 C c^{2} d^{4}\right ) \log {\left (d + e x \right )}}{e^{7}} \]

input
integrate((c*x**2+a)**2*(C*x**2+B*x+A)/(e*x+d)**3,x)
 
output
C*c**2*x**4/(4*e**3) + x**3*(B*c**2/(3*e**3) - C*c**2*d/e**4) + x**2*(A*c* 
*2/(2*e**3) - 3*B*c**2*d/(2*e**4) + C*a*c/e**3 + 3*C*c**2*d**2/e**5) + x*( 
-3*A*c**2*d/e**4 + 2*B*a*c/e**3 + 6*B*c**2*d**2/e**5 - 6*C*a*c*d/e**4 - 10 
*C*c**2*d**3/e**6) + (-A*a**2*e**6 + 6*A*a*c*d**2*e**4 + 7*A*c**2*d**4*e** 
2 - B*a**2*d*e**5 - 10*B*a*c*d**3*e**3 - 9*B*c**2*d**5*e + 3*C*a**2*d**2*e 
**4 + 14*C*a*c*d**4*e**2 + 11*C*c**2*d**6 + x*(8*A*a*c*d*e**5 + 8*A*c**2*d 
**3*e**3 - 2*B*a**2*e**6 - 12*B*a*c*d**2*e**4 - 10*B*c**2*d**4*e**2 + 4*C* 
a**2*d*e**5 + 16*C*a*c*d**3*e**3 + 12*C*c**2*d**5*e))/(2*d**2*e**7 + 4*d*e 
**8*x + 2*e**9*x**2) + (2*A*a*c*e**4 + 6*A*c**2*d**2*e**2 - 6*B*a*c*d*e**3 
 - 10*B*c**2*d**3*e + C*a**2*e**4 + 12*C*a*c*d**2*e**2 + 15*C*c**2*d**4)*l 
og(d + e*x)/e**7
 
3.1.31.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {11 \, C c^{2} d^{6} - 9 \, B c^{2} d^{5} e - 10 \, B a c d^{3} e^{3} - B a^{2} d e^{5} - A a^{2} e^{6} + 7 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 3 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} + 2 \, {\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} - B a^{2} e^{6} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {3 \, C c^{2} e^{3} x^{4} - 4 \, {\left (3 \, C c^{2} d e^{2} - B c^{2} e^{3}\right )} x^{3} + 6 \, {\left (6 \, C c^{2} d^{2} e - 3 \, B c^{2} d e^{2} + {\left (2 \, C a c + A c^{2}\right )} e^{3}\right )} x^{2} - 12 \, {\left (10 \, C c^{2} d^{3} - 6 \, B c^{2} d^{2} e - 2 \, B a c e^{3} + 3 \, {\left (2 \, C a c + A c^{2}\right )} d e^{2}\right )} x}{12 \, e^{6}} + \frac {{\left (15 \, C c^{2} d^{4} - 10 \, B c^{2} d^{3} e - 6 \, B a c d e^{3} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

input
integrate((c*x^2+a)^2*(C*x^2+B*x+A)/(e*x+d)^3,x, algorithm="maxima")
 
output
1/2*(11*C*c^2*d^6 - 9*B*c^2*d^5*e - 10*B*a*c*d^3*e^3 - B*a^2*d*e^5 - A*a^2 
*e^6 + 7*(2*C*a*c + A*c^2)*d^4*e^2 + 3*(C*a^2 + 2*A*a*c)*d^2*e^4 + 2*(6*C* 
c^2*d^5*e - 5*B*c^2*d^4*e^2 - 6*B*a*c*d^2*e^4 - B*a^2*e^6 + 4*(2*C*a*c + A 
*c^2)*d^3*e^3 + 2*(C*a^2 + 2*A*a*c)*d*e^5)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e 
^7) + 1/12*(3*C*c^2*e^3*x^4 - 4*(3*C*c^2*d*e^2 - B*c^2*e^3)*x^3 + 6*(6*C*c 
^2*d^2*e - 3*B*c^2*d*e^2 + (2*C*a*c + A*c^2)*e^3)*x^2 - 12*(10*C*c^2*d^3 - 
 6*B*c^2*d^2*e - 2*B*a*c*e^3 + 3*(2*C*a*c + A*c^2)*d*e^2)*x)/e^6 + (15*C*c 
^2*d^4 - 10*B*c^2*d^3*e - 6*B*a*c*d*e^3 + 6*(2*C*a*c + A*c^2)*d^2*e^2 + (C 
*a^2 + 2*A*a*c)*e^4)*log(e*x + d)/e^7
 
3.1.31.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {{\left (15 \, C c^{2} d^{4} - 10 \, B c^{2} d^{3} e + 12 \, C a c d^{2} e^{2} + 6 \, A c^{2} d^{2} e^{2} - 6 \, B a c d e^{3} + C a^{2} e^{4} + 2 \, A a c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, C c^{2} d^{6} - 9 \, B c^{2} d^{5} e + 14 \, C a c d^{4} e^{2} + 7 \, A c^{2} d^{4} e^{2} - 10 \, B a c d^{3} e^{3} + 3 \, C a^{2} d^{2} e^{4} + 6 \, A a c d^{2} e^{4} - B a^{2} d e^{5} - A a^{2} e^{6} + 2 \, {\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} + 8 \, C a c d^{3} e^{3} + 4 \, A c^{2} d^{3} e^{3} - 6 \, B a c d^{2} e^{4} + 2 \, C a^{2} d e^{5} + 4 \, A a c d e^{5} - B a^{2} e^{6}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {3 \, C c^{2} e^{9} x^{4} - 12 \, C c^{2} d e^{8} x^{3} + 4 \, B c^{2} e^{9} x^{3} + 36 \, C c^{2} d^{2} e^{7} x^{2} - 18 \, B c^{2} d e^{8} x^{2} + 12 \, C a c e^{9} x^{2} + 6 \, A c^{2} e^{9} x^{2} - 120 \, C c^{2} d^{3} e^{6} x + 72 \, B c^{2} d^{2} e^{7} x - 72 \, C a c d e^{8} x - 36 \, A c^{2} d e^{8} x + 24 \, B a c e^{9} x}{12 \, e^{12}} \]

input
integrate((c*x^2+a)^2*(C*x^2+B*x+A)/(e*x+d)^3,x, algorithm="giac")
 
output
(15*C*c^2*d^4 - 10*B*c^2*d^3*e + 12*C*a*c*d^2*e^2 + 6*A*c^2*d^2*e^2 - 6*B* 
a*c*d*e^3 + C*a^2*e^4 + 2*A*a*c*e^4)*log(abs(e*x + d))/e^7 + 1/2*(11*C*c^2 
*d^6 - 9*B*c^2*d^5*e + 14*C*a*c*d^4*e^2 + 7*A*c^2*d^4*e^2 - 10*B*a*c*d^3*e 
^3 + 3*C*a^2*d^2*e^4 + 6*A*a*c*d^2*e^4 - B*a^2*d*e^5 - A*a^2*e^6 + 2*(6*C* 
c^2*d^5*e - 5*B*c^2*d^4*e^2 + 8*C*a*c*d^3*e^3 + 4*A*c^2*d^3*e^3 - 6*B*a*c* 
d^2*e^4 + 2*C*a^2*d*e^5 + 4*A*a*c*d*e^5 - B*a^2*e^6)*x)/((e*x + d)^2*e^7) 
+ 1/12*(3*C*c^2*e^9*x^4 - 12*C*c^2*d*e^8*x^3 + 4*B*c^2*e^9*x^3 + 36*C*c^2* 
d^2*e^7*x^2 - 18*B*c^2*d*e^8*x^2 + 12*C*a*c*e^9*x^2 + 6*A*c^2*e^9*x^2 - 12 
0*C*c^2*d^3*e^6*x + 72*B*c^2*d^2*e^7*x - 72*C*a*c*d*e^8*x - 36*A*c^2*d*e^8 
*x + 24*B*a*c*e^9*x)/e^12
 
3.1.31.9 Mupad [B] (verification not implemented)

Time = 12.32 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=x\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {B\,c^2}{e^3}-\frac {3\,C\,c^2\,d}{e^4}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^3}+\frac {3\,C\,c^2\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {B\,c^2}{e^3}-\frac {3\,C\,c^2\,d}{e^4}\right )}{e^2}+\frac {2\,B\,a\,c}{e^3}-\frac {C\,c^2\,d^3}{e^6}\right )+x^3\,\left (\frac {B\,c^2}{3\,e^3}-\frac {C\,c^2\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {B\,c^2}{e^3}-\frac {3\,C\,c^2\,d}{e^4}\right )}{2\,e}-\frac {A\,c^2+2\,C\,a\,c}{2\,e^3}+\frac {3\,C\,c^2\,d^2}{2\,e^5}\right )+\frac {\frac {3\,C\,a^2\,d^2\,e^4-B\,a^2\,d\,e^5-A\,a^2\,e^6+14\,C\,a\,c\,d^4\,e^2-10\,B\,a\,c\,d^3\,e^3+6\,A\,a\,c\,d^2\,e^4+11\,C\,c^2\,d^6-9\,B\,c^2\,d^5\,e+7\,A\,c^2\,d^4\,e^2}{2\,e}+x\,\left (2\,C\,a^2\,d\,e^4-B\,a^2\,e^5+8\,C\,a\,c\,d^3\,e^2-6\,B\,a\,c\,d^2\,e^3+4\,A\,a\,c\,d\,e^4+6\,C\,c^2\,d^5-5\,B\,c^2\,d^4\,e+4\,A\,c^2\,d^3\,e^2\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (C\,a^2\,e^4+12\,C\,a\,c\,d^2\,e^2-6\,B\,a\,c\,d\,e^3+2\,A\,a\,c\,e^4+15\,C\,c^2\,d^4-10\,B\,c^2\,d^3\,e+6\,A\,c^2\,d^2\,e^2\right )}{e^7}+\frac {C\,c^2\,x^4}{4\,e^3} \]

input
int(((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x)^3,x)
 
output
x*((3*d*((3*d*((B*c^2)/e^3 - (3*C*c^2*d)/e^4))/e - (A*c^2 + 2*C*a*c)/e^3 + 
 (3*C*c^2*d^2)/e^5))/e - (3*d^2*((B*c^2)/e^3 - (3*C*c^2*d)/e^4))/e^2 + (2* 
B*a*c)/e^3 - (C*c^2*d^3)/e^6) + x^3*((B*c^2)/(3*e^3) - (C*c^2*d)/e^4) - x^ 
2*((3*d*((B*c^2)/e^3 - (3*C*c^2*d)/e^4))/(2*e) - (A*c^2 + 2*C*a*c)/(2*e^3) 
 + (3*C*c^2*d^2)/(2*e^5)) + ((11*C*c^2*d^6 - A*a^2*e^6 - B*a^2*d*e^5 - 9*B 
*c^2*d^5*e + 7*A*c^2*d^4*e^2 + 3*C*a^2*d^2*e^4 + 6*A*a*c*d^2*e^4 - 10*B*a* 
c*d^3*e^3 + 14*C*a*c*d^4*e^2)/(2*e) + x*(6*C*c^2*d^5 - B*a^2*e^5 + 2*C*a^2 
*d*e^4 - 5*B*c^2*d^4*e + 4*A*c^2*d^3*e^2 + 4*A*a*c*d*e^4 - 6*B*a*c*d^2*e^3 
 + 8*C*a*c*d^3*e^2))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) + (log(d + e*x)*(C*a^ 
2*e^4 + 15*C*c^2*d^4 + 2*A*a*c*e^4 - 10*B*c^2*d^3*e + 6*A*c^2*d^2*e^2 - 6* 
B*a*c*d*e^3 + 12*C*a*c*d^2*e^2))/e^7 + (C*c^2*x^4)/(4*e^3)