\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^6} \, dx\) [73]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {c^3 (6 B d-A e) x}{e^7}+\frac {B c^3 x^2}{2 e^6}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{5 e^8 (d+e x)^5}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{4 e^8 (d+e x)^4}+\frac {c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^8 (d+e x)^3}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{2 e^8 (d+e x)^2}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{e^8 (d+e x)}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) \log (d+e x)}{e^8} \] Output:

-c^3*(-A*e+6*B*d)*x/e^7+1/2*B*c^3*x^2/e^6+1/5*(-A*e+B*d)*(a*e^2+c*d^2)^3/e 
^8/(e*x+d)^5-1/4*(a*e^2+c*d^2)^2*(-6*A*c*d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d 
)^4+c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c*d^2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+ 
d)^3+1/2*c*(4*A*c*d*e*(3*a*e^2+5*c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2 
*d^4))/e^8/(e*x+d)^2+c^2*(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3) 
/e^8/(e*x+d)+3*c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*ln(e*x+d)/e^8
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {-2 A e \left (2 a^3 e^6+a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+B \left (-a^3 e^6 (d+5 e x)-3 a^2 c e^4 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+a c^2 d e^2 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+c^3 \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5 \log (d+e x)}{20 e^8 (d+e x)^5} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]
 

Output:

(-2*A*e*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2* 
(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^ 
6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50* 
d*e^5*x^5 - 10*e^6*x^6)) + B*(-(a^3*e^6*(d + 5*e*x)) - 3*a^2*c*e^4*(d^3 + 
5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + a*c^2*d*e^2*(137*d^4 + 625*d^3*e* 
x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + c^3*(459*d^7 + 1875* 
d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2* 
e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x^7)) + 60*c^2*(7*B*c*d^2 - 2*A*c*d*e + a* 
B*e^2)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^3}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^6}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^4}+\frac {c^3 (A e-6 B d)}{e^7}+\frac {B c^3 x}{e^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac {c^3 x (6 B d-A e)}{e^7}+\frac {B c^3 x^2}{2 e^6}\)

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]
 

Output:

-((c^3*(6*B*d - A*e)*x)/e^7) + (B*c^3*x^2)/(2*e^6) + ((B*d - A*e)*(c*d^2 + 
 a*e^2)^3)/(5*e^8*(d + e*x)^5) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e 
 + a*B*e^2))/(4*e^8*(d + e*x)^4) + (c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d 
^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^8*(d + e*x)^3) + (c*(4*A*c*d*e*(5*c*d^2 
+ 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(2*e^8*(d + e*x 
)^2) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(e^8*( 
d + e*x)) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Log[d + e*x])/e^8
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {c^{3} \left (\frac {1}{2} B e \,x^{2}+A e x -6 B d x \right )}{e^{7}}-\frac {c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{e^{8} \left (e x +d \right )^{3}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{4 e^{8} \left (e x +d \right )^{4}}-\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{e^{8} \left (e x +d \right )}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{2}}\)	\(433\)
norman	\(\frac {-\frac {4 A \,a^{3} e^{7}+2 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+274 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}-137 B a \,c^{2} d^{5} e^{2}-959 B \,c^{3} d^{7}}{20 e^{8}}-\frac {\left (3 A a \,c^{2} e^{3}+30 A \,c^{3} d^{2} e -15 B a \,c^{2} d \,e^{2}-105 B \,c^{3} d^{3}\right ) x^{4}}{e^{4}}-\frac {\left (12 A a \,c^{2} d \,e^{3}+180 A \,c^{3} d^{3} e +3 B \,e^{4} a^{2} c -90 B a \,c^{2} d^{2} e^{2}-630 B \,c^{3} d^{4}\right ) x^{3}}{2 e^{5}}-\frac {\left (2 A \,a^{2} c \,e^{5}+12 A a \,c^{2} d^{2} e^{3}+220 A \,c^{3} d^{4} e +3 B \,a^{2} c d \,e^{4}-110 B a \,c^{2} d^{3} e^{2}-770 B \,c^{3} d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (2 A \,a^{2} c d \,e^{5}+12 A a \,c^{2} d^{3} e^{3}+250 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+3 B \,a^{2} c \,d^{2} e^{4}-125 B a \,c^{2} d^{4} e^{2}-875 B \,c^{3} d^{6}\right ) x}{4 e^{7}}+\frac {B \,c^{3} x^{7}}{2 e}+\frac {c^{3} \left (2 A e -7 B d \right ) x^{6}}{2 e^{2}}}{\left (e x +d \right )^{5}}-\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(443\)
risch	\(\frac {B \,c^{3} x^{2}}{2 e^{6}}+\frac {c^{3} A x}{e^{6}}-\frac {6 c^{3} B d x}{e^{7}}+\frac {\left (-3 A a \,c^{2} e^{6}-15 A \,c^{3} d^{2} e^{4}+15 B a \,c^{2} d \,e^{5}+35 B \,c^{3} d^{3} e^{3}\right ) x^{4}-\frac {e^{2} c \left (12 A a c d \,e^{3}+100 A \,c^{2} d^{3} e +3 B \,e^{4} a^{2}-90 B a c \,d^{2} e^{2}-245 B \,c^{2} d^{4}\right ) x^{3}}{2}-\frac {c e \left (2 A \,a^{2} e^{5}+12 A a c \,d^{2} e^{3}+130 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}-110 B a c \,d^{3} e^{2}-329 B \,c^{2} d^{5}\right ) x^{2}}{2}+\left (-\frac {1}{2} A \,a^{2} c d \,e^{5}-3 A a \,c^{2} d^{3} e^{3}-\frac {77}{2} A \,c^{3} d^{5} e -\frac {1}{4} B \,a^{3} e^{6}-\frac {3}{4} B \,a^{2} c \,d^{2} e^{4}+\frac {125}{4} B a \,c^{2} d^{4} e^{2}+\frac {399}{4} B \,c^{3} d^{6}\right ) x -\frac {4 A \,a^{3} e^{7}+2 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+174 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}-137 B a \,c^{2} d^{5} e^{2}-459 B \,c^{3} d^{7}}{20 e}}{e^{7} \left (e x +d \right )^{5}}-\frac {6 c^{3} \ln \left (e x +d \right ) A d}{e^{7}}+\frac {3 c^{2} \ln \left (e x +d \right ) B a}{e^{6}}+\frac {21 c^{3} \ln \left (e x +d \right ) B \,d^{2}}{e^{8}}\)	\(448\)
parallelrisch	\(-\frac {3 B \,a^{2} c \,d^{3} e^{4}+2 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}-20 A \,x^{6} c^{3} e^{7}-10 B \,x^{7} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}+5 B x \,a^{3} e^{7}+120 A \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{6}-60 B \ln \left (e x +d \right ) x^{5} a \,c^{2} e^{7}-420 B \ln \left (e x +d \right ) x^{5} c^{3} d^{2} e^{5}-625 B x a \,c^{2} d^{4} e^{3}-300 B \ln \left (e x +d \right ) x^{4} a \,c^{2} d \,e^{6}-137 B a \,c^{2} d^{5} e^{2}+1200 A \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{3}-4200 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}+600 A \,x^{4} c^{3} d^{2} e^{5}-2100 B \,x^{4} c^{3} d^{3} e^{4}+1800 A \,x^{3} c^{3} d^{3} e^{4}-959 B \,c^{3} d^{7}-300 B \ln \left (e x +d \right ) x a \,c^{2} d^{4} e^{3}-600 B \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{3} e^{4}-600 B \ln \left (e x +d \right ) x^{3} a \,c^{2} d^{2} e^{5}+1200 A \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{4}-4200 B \ln \left (e x +d \right ) x^{3} c^{3} d^{4} e^{3}+600 A \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{5}-2100 B \ln \left (e x +d \right ) x^{4} c^{3} d^{3} e^{4}+70 B \,x^{6} c^{3} d \,e^{6}+120 A \ln \left (e x +d \right ) c^{3} d^{6} e +600 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-2100 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e +30 B \,x^{3} a^{2} c \,e^{7}-6300 B \,x^{3} c^{3} d^{4} e^{3}+20 A \,x^{2} a^{2} c \,e^{7}+2200 A \,x^{2} c^{3} d^{4} e^{3}-7700 B \,x^{2} c^{3} d^{5} e^{2}+1250 A x \,c^{3} d^{5} e^{2}-4375 B x \,c^{3} d^{6} e +60 A \,x^{4} a \,c^{2} e^{7}+274 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}-60 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{2}-900 B \,x^{3} a \,c^{2} d^{2} e^{5}+120 A \,x^{2} a \,c^{2} d^{2} e^{5}+30 B \,x^{2} a^{2} c d \,e^{6}-1100 B \,x^{2} a \,c^{2} d^{3} e^{4}+10 A x \,a^{2} c d \,e^{6}+60 A x a \,c^{2} d^{3} e^{4}+15 B x \,a^{2} c \,d^{2} e^{5}-300 B \,x^{4} a \,c^{2} d \,e^{6}+120 A \,x^{3} a \,c^{2} d \,e^{6}+4 A \,a^{3} e^{7}}{20 e^{8} \left (e x +d \right )^{5}}\)	\(802\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

c^3/e^7*(1/2*B*e*x^2+A*e*x-6*B*d*x)-c/e^8*(A*a^2*e^5+6*A*a*c*d^2*e^3+5*A*c 
^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7*B*c^2*d^5)/(e*x+d)^3-1/4*(-6*A*a 
^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a^3*e^6+9*B*a^2*c*d^2*e^4+15 
*B*a*c^2*d^4*e^2+7*B*c^3*d^6)/e^8/(e*x+d)^4-3*c^2/e^8*(2*A*c*d*e-B*a*e^2-7 
*B*c*d^2)*ln(e*x+d)-1/5*(A*a^3*e^7+3*A*a^2*c*d^2*e^5+3*A*a*c^2*d^4*e^3+A*c 
^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3*e^4-3*B*a*c^2*d^5*e^2-B*c^3*d^7)/e^8/(e 
*x+d)^5-c^2/e^8*(3*A*a*e^3+15*A*c*d^2*e-15*B*a*d*e^2-35*B*c*d^3)/(e*x+d)+1 
/2/e^8*c*(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a^2*e^4-30*B*a*c*d^2*e^2-35*B* 
c^2*d^4)/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (305) = 610\).

Time = 0.09 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {10 \, B c^{3} e^{7} x^{7} + 459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - 10 \, {\left (7 \, B c^{3} d e^{6} - 2 \, A c^{3} e^{7}\right )} x^{6} - 100 \, {\left (5 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} - 20 \, {\left (20 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (130 \, B c^{3} d^{4} e^{3} - 80 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (270 \, B c^{3} d^{5} e^{2} - 120 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (375 \, B c^{3} d^{6} e - 150 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 60 \, {\left (7 \, B c^{3} d^{7} - 2 \, A c^{3} d^{6} e + B a c^{2} d^{5} e^{2} + {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 5 \, {\left (7 \, B c^{3} d^{3} e^{4} - 2 \, A c^{3} d^{2} e^{5} + B a c^{2} d e^{6}\right )} x^{4} + 10 \, {\left (7 \, B c^{3} d^{4} e^{3} - 2 \, A c^{3} d^{3} e^{4} + B a c^{2} d^{2} e^{5}\right )} x^{3} + 10 \, {\left (7 \, B c^{3} d^{5} e^{2} - 2 \, A c^{3} d^{4} e^{3} + B a c^{2} d^{3} e^{4}\right )} x^{2} + 5 \, {\left (7 \, B c^{3} d^{6} e - 2 \, A c^{3} d^{5} e^{2} + B a c^{2} d^{4} e^{3}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x, algorithm="fricas")
 

Output:

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5 
*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3* 
d*e^6 - 4*A*a^3*e^7 - 10*(7*B*c^3*d*e^6 - 2*A*c^3*e^7)*x^6 - 100*(5*B*c^3* 
d^2*e^5 - A*c^3*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 - 15*B 
*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 10*(130*B*c^3*d^4*e^3 - 80*A*c^3*d^3*e 
^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(270* 
B*c^3*d^5*e^2 - 120*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e 
^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 + 5*(375*B*c^3*d^6*e - 150*A*c^3 
*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 
2*A*a^2*c*d*e^6 - B*a^3*e^7)*x + 60*(7*B*c^3*d^7 - 2*A*c^3*d^6*e + B*a*c^2 
*d^5*e^2 + (7*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 5*(7*B*c^ 
3*d^3*e^4 - 2*A*c^3*d^2*e^5 + B*a*c^2*d*e^6)*x^4 + 10*(7*B*c^3*d^4*e^3 - 2 
*A*c^3*d^3*e^4 + B*a*c^2*d^2*e^5)*x^3 + 10*(7*B*c^3*d^5*e^2 - 2*A*c^3*d^4* 
e^3 + B*a*c^2*d^3*e^4)*x^2 + 5*(7*B*c^3*d^6*e - 2*A*c^3*d^5*e^2 + B*a*c^2* 
d^4*e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10* 
d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**6,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac {B c^{3} e x^{2} - 2 \, {\left (6 \, B c^{3} d - A c^{3} e\right )} x}{2 \, e^{7}} + \frac {3 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x, algorithm="maxima")
 

Output:

1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d 
^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 
 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^2* 
e^7)*x^4 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 
- 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^ 
3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2 
*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d 
^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3* 
e^7)*x)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d 
^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d - A*c^3*e)*x)/e^7 + 
3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*log(e*x + d)/e^8
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {3 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {B c^{3} e^{6} x^{2} - 12 \, B c^{3} d e^{5} x + 2 \, A c^{3} e^{6} x}{2 \, e^{12}} + \frac {459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x, algorithm="giac")
 

Output:

3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*log(abs(e*x + d))/e^8 + 1/2*(B 
*c^3*e^6*x^2 - 12*B*c^3*d*e^5*x + 2*A*c^3*e^6*x)/e^12 + 1/20*(459*B*c^3*d^ 
7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*c 
*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*c^3*d^ 
3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 10*(245 
*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 
 - 3*B*a^2*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B* 
a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 
+ 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^ 
2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/((e*x + d) 
^5*e^8)
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.58 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx=x\,\left (\frac {A\,c^3}{e^6}-\frac {6\,B\,c^3\,d}{e^7}\right )-\frac {\frac {B\,a^3\,d\,e^6+4\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+2\,A\,a^2\,c\,d^2\,e^5-137\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3-459\,B\,c^3\,d^7+174\,A\,c^3\,d^6\,e}{20\,e}+x^2\,\left (\frac {3\,B\,a^2\,c\,d\,e^5}{2}+A\,a^2\,c\,e^6-55\,B\,a\,c^2\,d^3\,e^3+6\,A\,a\,c^2\,d^2\,e^4-\frac {329\,B\,c^3\,d^5\,e}{2}+65\,A\,c^3\,d^4\,e^2\right )+x^3\,\left (\frac {3\,B\,a^2\,c\,e^6}{2}-45\,B\,a\,c^2\,d^2\,e^4+6\,A\,a\,c^2\,d\,e^5-\frac {245\,B\,c^3\,d^4\,e^2}{2}+50\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{4}+\frac {3\,B\,a^2\,c\,d^2\,e^4}{4}+\frac {A\,a^2\,c\,d\,e^5}{2}-\frac {125\,B\,a\,c^2\,d^4\,e^2}{4}+3\,A\,a\,c^2\,d^3\,e^3-\frac {399\,B\,c^3\,d^6}{4}+\frac {77\,A\,c^3\,d^5\,e}{2}\right )+x^4\,\left (-35\,B\,c^3\,d^3\,e^3+15\,A\,c^3\,d^2\,e^4-15\,B\,a\,c^2\,d\,e^5+3\,A\,a\,c^2\,e^6\right )}{d^5\,e^7+5\,d^4\,e^8\,x+10\,d^3\,e^9\,x^2+10\,d^2\,e^{10}\,x^3+5\,d\,e^{11}\,x^4+e^{12}\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (21\,B\,c^3\,d^2-6\,A\,c^3\,d\,e+3\,B\,a\,c^2\,e^2\right )}{e^8}+\frac {B\,c^3\,x^2}{2\,e^6} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^6,x)
 

Output:

x*((A*c^3)/e^6 - (6*B*c^3*d)/e^7) - ((4*A*a^3*e^7 - 459*B*c^3*d^7 + B*a^3* 
d*e^6 + 174*A*c^3*d^6*e + 12*A*a*c^2*d^4*e^3 + 2*A*a^2*c*d^2*e^5 - 137*B*a 
*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4)/(20*e) + x^2*(A*a^2*c*e^6 - (329*B*c^3*d 
^5*e)/2 + 65*A*c^3*d^4*e^2 + 6*A*a*c^2*d^2*e^4 - 55*B*a*c^2*d^3*e^3 + (3*B 
*a^2*c*d*e^5)/2) + x^3*((3*B*a^2*c*e^6)/2 + 50*A*c^3*d^3*e^3 - (245*B*c^3* 
d^4*e^2)/2 - 45*B*a*c^2*d^2*e^4 + 6*A*a*c^2*d*e^5) + x*((B*a^3*e^6)/4 - (3 
99*B*c^3*d^6)/4 + (77*A*c^3*d^5*e)/2 + 3*A*a*c^2*d^3*e^3 - (125*B*a*c^2*d^ 
4*e^2)/4 + (3*B*a^2*c*d^2*e^4)/4 + (A*a^2*c*d*e^5)/2) + x^4*(3*A*a*c^2*e^6 
 + 15*A*c^3*d^2*e^4 - 35*B*c^3*d^3*e^3 - 15*B*a*c^2*d*e^5))/(d^5*e^7 + e^1 
2*x^5 + 5*d^4*e^8*x + 5*d*e^11*x^4 + 10*d^3*e^9*x^2 + 10*d^2*e^10*x^3) + ( 
log(d + e*x)*(21*B*c^3*d^2 - 6*A*c^3*d*e + 3*B*a*c^2*e^2))/e^8 + (B*c^3*x^ 
2)/(2*e^6)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 808, normalized size of antiderivative = 2.58 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x)
 

Output:

(60*log(d + e*x)*a*b*c**2*d**6*e**2 + 300*log(d + e*x)*a*b*c**2*d**5*e**3* 
x + 600*log(d + e*x)*a*b*c**2*d**4*e**4*x**2 + 600*log(d + e*x)*a*b*c**2*d 
**3*e**5*x**3 + 300*log(d + e*x)*a*b*c**2*d**2*e**6*x**4 + 60*log(d + e*x) 
*a*b*c**2*d*e**7*x**5 - 120*log(d + e*x)*a*c**3*d**7*e - 600*log(d + e*x)* 
a*c**3*d**6*e**2*x - 1200*log(d + e*x)*a*c**3*d**5*e**3*x**2 - 1200*log(d 
+ e*x)*a*c**3*d**4*e**4*x**3 - 600*log(d + e*x)*a*c**3*d**3*e**5*x**4 - 12 
0*log(d + e*x)*a*c**3*d**2*e**6*x**5 + 420*log(d + e*x)*b*c**3*d**8 + 2100 
*log(d + e*x)*b*c**3*d**7*e*x + 4200*log(d + e*x)*b*c**3*d**6*e**2*x**2 + 
4200*log(d + e*x)*b*c**3*d**5*e**3*x**3 + 2100*log(d + e*x)*b*c**3*d**4*e* 
*4*x**4 + 420*log(d + e*x)*b*c**3*d**3*e**5*x**5 - 4*a**4*d*e**7 - a**3*b* 
d**2*e**6 - 5*a**3*b*d*e**7*x - 2*a**3*c*d**3*e**5 - 10*a**3*c*d**2*e**6*x 
 - 20*a**3*c*d*e**7*x**2 - 3*a**2*b*c*d**4*e**4 - 15*a**2*b*c*d**3*e**5*x 
- 30*a**2*b*c*d**2*e**6*x**2 - 30*a**2*b*c*d*e**7*x**3 + 12*a**2*c**2*e**8 
*x**5 + 77*a*b*c**2*d**6*e**2 + 325*a*b*c**2*d**5*e**3*x + 500*a*b*c**2*d* 
*4*e**4*x**2 + 300*a*b*c**2*d**3*e**5*x**3 - 60*a*b*c**2*d*e**7*x**5 - 154 
*a*c**3*d**7*e - 650*a*c**3*d**6*e**2*x - 1000*a*c**3*d**5*e**3*x**2 - 600 
*a*c**3*d**4*e**4*x**3 + 120*a*c**3*d**2*e**6*x**5 + 20*a*c**3*d*e**7*x**6 
 + 539*b*c**3*d**8 + 2275*b*c**3*d**7*e*x + 3500*b*c**3*d**6*e**2*x**2 + 2 
100*b*c**3*d**5*e**3*x**3 - 420*b*c**3*d**3*e**5*x**5 - 70*b*c**3*d**2*e** 
6*x**6 + 10*b*c**3*d*e**7*x**7)/(20*d*e**8*(d**5 + 5*d**4*e*x + 10*d**3...