\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^5} \, dx\) [72]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 \left (5 A c d e-3 B \left (5 c d^2+a e^2\right )\right ) x}{e^7}-\frac {c^3 (5 B d-A e) x^2}{2 e^6}+\frac {B c^3 x^3}{3 e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{4 e^8 (d+e x)^4}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^8 (d+e x)^2}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{e^8 (d+e x)}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {3 A e \left (-a^3 e^6-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+a c^2 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )-B \left (a^3 e^6 (d+4 e x)+9 a^2 c e^4 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+3 a c^2 e^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )+c^3 \left (319 d^7+856 d^6 e x+444 d^5 e^2 x^2-544 d^4 e^3 x^3-556 d^3 e^4 x^4-84 d^2 e^5 x^5+14 d e^6 x^6-4 e^7 x^7\right )\right )+12 c^2 \left (3 A e \left (5 c d^2+a e^2\right )-5 B \left (7 c d^3+3 a d e^2\right )\right ) (d+e x)^4 \log (d+e x)}{12 e^8 (d+e x)^4} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^5,x]
 

Output:

(3*A*e*(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*( 
25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + c^3*(57*d^6 + 168*d^5* 
e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2 
*e^6*x^6)) - B*(a^3*e^6*(d + 4*e*x) + 9*a^2*c*e^4*(d^3 + 4*d^2*e*x + 6*d*e 
^2*x^2 + 4*e^3*x^3) + 3*a*c^2*e^2*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 
+ 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + c^3*(319*d^7 + 856*d^6*e*x 
 + 444*d^5*e^2*x^2 - 544*d^4*e^3*x^3 - 556*d^3*e^4*x^4 - 84*d^2*e^5*x^5 + 
14*d*e^6*x^6 - 4*e^7*x^7)) + 12*c^2*(3*A*e*(5*c*d^2 + a*e^2) - 5*B*(7*c*d^ 
3 + 3*a*d*e^2))*(d + e*x)^4*Log[d + e*x])/(12*e^8*(d + e*x)^4)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 314, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^5,x]
 

Output:

-((c^2*(5*A*c*d*e - 3*B*(5*c*d^2 + a*e^2))*x)/e^7) - (c^3*(5*B*d - A*e)*x^ 
2)/(2*e^6) + (B*c^3*x^3)/(3*e^5) + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(4*e^8* 
(d + e*x)^4) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(3*e^ 
8*(d + e*x)^3) + (3*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))/(2*e^8*(d + e*x)^2) + (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)))/(e^8*(d + e*x)) - (c^2*(35*B* 
c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*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 = 0.66 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.38

method	result	size
default	\(-\frac {c^{2} \left (-\frac {1}{3} B c \,x^{3} e^{2}-\frac {1}{2} A c \,e^{2} x^{2}+\frac {5}{2} B c d e \,x^{2}+5 A c d e x -3 B a \,e^{2} x -15 B c \,d^{2} x \right )}{e^{7}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{4 e^{8} \left (e x +d \right )^{4}}+\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right ) \ln \left (e x +d \right )}{e^{8}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{e^{8} \left (e x +d \right )}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{2 e^{8} \left (e x +d \right )^{2}}\)	\(432\)
norman	\(\frac {\frac {\left (12 A a \,c^{2} d \,e^{3}+60 A \,c^{3} d^{3} e -3 B \,e^{4} a^{2} c -60 B a \,c^{2} d^{2} e^{2}-140 B \,c^{3} d^{4}\right ) x^{3}}{e^{5}}-\frac {3 A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}-75 A a \,c^{2} d^{4} e^{3}-375 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+375 B a \,c^{2} d^{5} e^{2}+875 B \,c^{3} d^{7}}{12 e^{8}}-\frac {3 \left (A \,a^{2} c \,e^{5}-18 A a \,c^{2} d^{2} e^{3}-90 A \,c^{3} d^{4} e +3 B \,a^{2} c d \,e^{4}+90 B a \,c^{2} d^{3} e^{2}+210 B \,c^{3} d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (3 A \,a^{2} c d \,e^{5}-66 A a \,c^{2} d^{3} e^{3}-330 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+330 B a \,c^{2} d^{4} e^{2}+770 B \,c^{3} d^{6}\right ) x}{3 e^{7}}+\frac {B \,c^{3} x^{7}}{3 e}-\frac {c^{2} \left (3 A c d e -3 B a \,e^{2}-7 B c \,d^{2}\right ) x^{5}}{e^{3}}+\frac {c^{3} \left (3 A e -7 B d \right ) x^{6}}{6 e^{2}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(433\)
risch	\(\frac {B \,c^{3} x^{3}}{3 e^{5}}+\frac {c^{3} A \,x^{2}}{2 e^{5}}-\frac {5 c^{3} B d \,x^{2}}{2 e^{6}}-\frac {5 c^{3} A d x}{e^{6}}+\frac {3 c^{2} B a x}{e^{5}}+\frac {15 c^{3} B \,d^{2} x}{e^{7}}+\frac {\left (12 A a \,c^{2} d \,e^{5}+20 A \,c^{3} d^{3} e^{3}-3 B \,a^{2} c \,e^{6}-30 B a \,c^{2} d^{2} e^{4}-35 B \,c^{3} d^{4} e^{2}\right ) x^{3}-\frac {3 c e \left (A \,a^{2} e^{5}-18 A a c \,d^{2} e^{3}-35 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}+50 B a c \,d^{3} e^{2}+63 B \,c^{2} d^{5}\right ) x^{2}}{2}+\left (-A \,a^{2} c d \,e^{5}+22 A a \,c^{2} d^{3} e^{3}+47 A \,c^{3} d^{5} e -\frac {1}{3} B \,a^{3} e^{6}-3 B \,a^{2} c \,d^{2} e^{4}-65 B a \,c^{2} d^{4} e^{2}-\frac {259}{3} B \,c^{3} d^{6}\right ) x -\frac {3 A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}-75 A a \,c^{2} d^{4} e^{3}-171 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+231 B a \,c^{2} d^{5} e^{2}+319 B \,c^{3} d^{7}}{12 e}}{e^{7} \left (e x +d \right )^{4}}+\frac {3 c^{2} \ln \left (e x +d \right ) A a}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) A \,d^{2}}{e^{7}}-\frac {15 c^{2} \ln \left (e x +d \right ) B a d}{e^{6}}-\frac {35 c^{3} \ln \left (e x +d \right ) B \,d^{3}}{e^{8}}\)	\(461\)
parallelrisch	\(\frac {144 A \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{4}-9 B \,a^{2} c \,d^{3} e^{4}-3 A \,a^{2} c \,d^{2} e^{5}+75 A a \,c^{2} d^{4} e^{3}+6 A \,x^{6} c^{3} e^{7}+4 B \,x^{7} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}-4 B x \,a^{3} e^{7}-1320 B x a \,c^{2} d^{4} e^{3}-180 B \ln \left (e x +d \right ) x^{4} a \,c^{2} d \,e^{6}-375 B a \,c^{2} d^{5} e^{2}+1080 A \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{3}-2520 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}+720 A \,x^{3} c^{3} d^{3} e^{4}-875 B \,c^{3} d^{7}-720 B \ln \left (e x +d \right ) x a \,c^{2} d^{4} e^{3}-1080 B \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{3} e^{4}+216 A \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{5}-720 B \ln \left (e x +d \right ) x^{3} a \,c^{2} d^{2} e^{5}+144 A \ln \left (e x +d \right ) x^{3} a \,c^{2} d \,e^{6}+720 A \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{4}-1680 B \ln \left (e x +d \right ) x^{3} c^{3} d^{4} e^{3}+36 A \ln \left (e x +d \right ) x^{4} a \,c^{2} e^{7}+180 A \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{5}-420 B \ln \left (e x +d \right ) x^{4} c^{3} d^{3} e^{4}-36 A \,x^{5} c^{3} d \,e^{6}-14 B \,x^{6} c^{3} d \,e^{6}+180 A \ln \left (e x +d \right ) c^{3} d^{6} e +720 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-1680 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e -36 B \,x^{3} a^{2} c \,e^{7}-1680 B \,x^{3} c^{3} d^{4} e^{3}-18 A \,x^{2} a^{2} c \,e^{7}+1620 A \,x^{2} c^{3} d^{4} e^{3}-3780 B \,x^{2} c^{3} d^{5} e^{2}+1320 A x \,c^{3} d^{5} e^{2}-3080 B x \,c^{3} d^{6} e +36 B \,x^{5} a \,c^{2} e^{7}+84 B \,x^{5} c^{3} d^{2} e^{5}+375 A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}+36 A \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{3}-180 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{2}-720 B \,x^{3} a \,c^{2} d^{2} e^{5}+324 A \,x^{2} a \,c^{2} d^{2} e^{5}-54 B \,x^{2} a^{2} c d \,e^{6}-1620 B \,x^{2} a \,c^{2} d^{3} e^{4}-12 A x \,a^{2} c d \,e^{6}+264 A x a \,c^{2} d^{3} e^{4}-36 B x \,a^{2} c \,d^{2} e^{5}+144 A \,x^{3} a \,c^{2} d \,e^{6}-3 A \,a^{3} e^{7}}{12 e^{8} \left (e x +d \right )^{4}}\)	\(828\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (304) = 608\).

Time = 0.09 (sec) , antiderivative size = 746, normalized size of antiderivative = 2.38 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {4 \, B c^{3} e^{7} x^{7} - 319 \, B c^{3} d^{7} + 171 \, A c^{3} d^{6} e - 231 \, B a c^{2} d^{5} e^{2} + 75 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{3} e^{7} - 2 \, {\left (7 \, B c^{3} d e^{6} - 3 \, A c^{3} e^{7}\right )} x^{6} + 12 \, {\left (7 \, B c^{3} d^{2} e^{5} - 3 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 4 \, {\left (139 \, B c^{3} d^{3} e^{4} - 51 \, A c^{3} d^{2} e^{5} + 36 \, B a c^{2} d e^{6}\right )} x^{4} + 4 \, {\left (136 \, B c^{3} d^{4} e^{3} - 24 \, A c^{3} d^{3} e^{4} - 36 \, B a c^{2} d^{2} e^{5} + 36 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} - 6 \, {\left (74 \, B c^{3} d^{5} e^{2} - 66 \, A c^{3} d^{4} e^{3} + 126 \, B a c^{2} d^{3} e^{4} - 54 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 4 \, {\left (214 \, B c^{3} d^{6} e - 126 \, A c^{3} d^{5} e^{2} + 186 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 12 \, {\left (35 \, B c^{3} d^{7} - 15 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 4 \, {\left (35 \, B c^{3} d^{4} e^{3} - 15 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6}\right )} x^{3} + 6 \, {\left (35 \, B c^{3} d^{5} e^{2} - 15 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5}\right )} x^{2} + 4 \, {\left (35 \, B c^{3} d^{6} e - 15 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x, algorithm="fricas")
 

Output:

1/12*(4*B*c^3*e^7*x^7 - 319*B*c^3*d^7 + 171*A*c^3*d^6*e - 231*B*a*c^2*d^5* 
e^2 + 75*A*a*c^2*d^4*e^3 - 9*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 - B*a^3*d 
*e^6 - 3*A*a^3*e^7 - 2*(7*B*c^3*d*e^6 - 3*A*c^3*e^7)*x^6 + 12*(7*B*c^3*d^2 
*e^5 - 3*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 4*(139*B*c^3*d^3*e^4 - 51*A*c^ 
3*d^2*e^5 + 36*B*a*c^2*d*e^6)*x^4 + 4*(136*B*c^3*d^4*e^3 - 24*A*c^3*d^3*e^ 
4 - 36*B*a*c^2*d^2*e^5 + 36*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 - 6*(74*B*c 
^3*d^5*e^2 - 66*A*c^3*d^4*e^3 + 126*B*a*c^2*d^3*e^4 - 54*A*a*c^2*d^2*e^5 + 
 9*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^2 - 4*(214*B*c^3*d^6*e - 126*A*c^3*d^5 
*e^2 + 186*B*a*c^2*d^4*e^3 - 66*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 3*A* 
a^2*c*d*e^6 + B*a^3*e^7)*x - 12*(35*B*c^3*d^7 - 15*A*c^3*d^6*e + 15*B*a*c^ 
2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + (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 + 4*(35*B*c^3*d^4*e^3 - 15*A*c^3*d^3*e^ 
4 + 15*B*a*c^2*d^2*e^5 - 3*A*a*c^2*d*e^6)*x^3 + 6*(35*B*c^3*d^5*e^2 - 15*A 
*c^3*d^4*e^3 + 15*B*a*c^2*d^3*e^4 - 3*A*a*c^2*d^2*e^5)*x^2 + 4*(35*B*c^3*d 
^6*e - 15*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 - 3*A*a*c^2*d^3*e^4)*x)*log(e 
*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8 
)
 

Sympy [A] (verification not implemented)

Time = 32.51 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {B c^{3} x^{3}}{3 e^{5}} - \frac {c^{2} \left (- 3 A a e^{3} - 15 A c d^{2} e + 15 B a d e^{2} + 35 B c d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{2} \left (\frac {A c^{3}}{2 e^{5}} - \frac {5 B c^{3} d}{2 e^{6}}\right ) + x \left (- \frac {5 A c^{3} d}{e^{6}} + \frac {3 B a c^{2}}{e^{5}} + \frac {15 B c^{3} d^{2}}{e^{7}}\right ) + \frac {- 3 A a^{3} e^{7} - 3 A a^{2} c d^{2} e^{5} + 75 A a c^{2} d^{4} e^{3} + 171 A c^{3} d^{6} e - B a^{3} d e^{6} - 9 B a^{2} c d^{3} e^{4} - 231 B a c^{2} d^{5} e^{2} - 319 B c^{3} d^{7} + x^{3} \cdot \left (144 A a c^{2} d e^{6} + 240 A c^{3} d^{3} e^{4} - 36 B a^{2} c e^{7} - 360 B a c^{2} d^{2} e^{5} - 420 B c^{3} d^{4} e^{3}\right ) + x^{2} \left (- 18 A a^{2} c e^{7} + 324 A a c^{2} d^{2} e^{5} + 630 A c^{3} d^{4} e^{3} - 54 B a^{2} c d e^{6} - 900 B a c^{2} d^{3} e^{4} - 1134 B c^{3} d^{5} e^{2}\right ) + x \left (- 12 A a^{2} c d e^{6} + 264 A a c^{2} d^{3} e^{4} + 564 A c^{3} d^{5} e^{2} - 4 B a^{3} e^{7} - 36 B a^{2} c d^{2} e^{5} - 780 B a c^{2} d^{4} e^{3} - 1036 B c^{3} d^{6} e\right )}{12 d^{4} e^{8} + 48 d^{3} e^{9} x + 72 d^{2} e^{10} x^{2} + 48 d e^{11} x^{3} + 12 e^{12} x^{4}} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**5,x)
 

Output:

B*c**3*x**3/(3*e**5) - c**2*(-3*A*a*e**3 - 15*A*c*d**2*e + 15*B*a*d*e**2 + 
 35*B*c*d**3)*log(d + e*x)/e**8 + x**2*(A*c**3/(2*e**5) - 5*B*c**3*d/(2*e* 
*6)) + x*(-5*A*c**3*d/e**6 + 3*B*a*c**2/e**5 + 15*B*c**3*d**2/e**7) + (-3* 
A*a**3*e**7 - 3*A*a**2*c*d**2*e**5 + 75*A*a*c**2*d**4*e**3 + 171*A*c**3*d* 
*6*e - B*a**3*d*e**6 - 9*B*a**2*c*d**3*e**4 - 231*B*a*c**2*d**5*e**2 - 319 
*B*c**3*d**7 + x**3*(144*A*a*c**2*d*e**6 + 240*A*c**3*d**3*e**4 - 36*B*a** 
2*c*e**7 - 360*B*a*c**2*d**2*e**5 - 420*B*c**3*d**4*e**3) + x**2*(-18*A*a* 
*2*c*e**7 + 324*A*a*c**2*d**2*e**5 + 630*A*c**3*d**4*e**3 - 54*B*a**2*c*d* 
e**6 - 900*B*a*c**2*d**3*e**4 - 1134*B*c**3*d**5*e**2) + x*(-12*A*a**2*c*d 
*e**6 + 264*A*a*c**2*d**3*e**4 + 564*A*c**3*d**5*e**2 - 4*B*a**3*e**7 - 36 
*B*a**2*c*d**2*e**5 - 780*B*a*c**2*d**4*e**3 - 1036*B*c**3*d**6*e))/(12*d* 
*4*e**8 + 48*d**3*e**9*x + 72*d**2*e**10*x**2 + 48*d*e**11*x**3 + 12*e**12 
*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=-\frac {319 \, B c^{3} d^{7} - 171 \, A c^{3} d^{6} e + 231 \, B a c^{2} d^{5} e^{2} - 75 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{3} e^{7} + 12 \, {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 18 \, {\left (63 \, B c^{3} d^{5} e^{2} - 35 \, A c^{3} d^{4} e^{3} + 50 \, B a c^{2} d^{3} e^{4} - 18 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + A a^{2} c e^{7}\right )} x^{2} + 4 \, {\left (259 \, B c^{3} d^{6} e - 141 \, A c^{3} d^{5} e^{2} + 195 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac {2 \, B c^{3} e^{2} x^{3} - 3 \, {\left (5 \, B c^{3} d e - A c^{3} e^{2}\right )} x^{2} + 6 \, {\left (15 \, B c^{3} d^{2} - 5 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x}{6 \, e^{7}} - \frac {{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x, algorithm="maxima")
 

Output:

-1/12*(319*B*c^3*d^7 - 171*A*c^3*d^6*e + 231*B*a*c^2*d^5*e^2 - 75*A*a*c^2* 
d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + 3*A*a^3*e^ 
7 + 12*(35*B*c^3*d^4*e^3 - 20*A*c^3*d^3*e^4 + 30*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 + 18*(63*B*c^3*d^5*e^2 - 35*A*c^3*d^4*e^3 + 
 50*B*a*c^2*d^3*e^4 - 18*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + A*a^2*c*e^7)* 
x^2 + 4*(259*B*c^3*d^6*e - 141*A*c^3*d^5*e^2 + 195*B*a*c^2*d^4*e^3 - 66*A* 
a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x)/(e^12* 
x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*c^ 
3*e^2*x^3 - 3*(5*B*c^3*d*e - A*c^3*e^2)*x^2 + 6*(15*B*c^3*d^2 - 5*A*c^3*d* 
e + 3*B*a*c^2*e^2)*x)/e^7 - (35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d* 
e^2 - 3*A*a*c^2*e^3)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (304) = 608\).

Time = 0.15 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/6*(2*B*c^3 - 3*(7*B*c^3*d*e - A*c^3*e^2)/((e*x + d)*e) + 18*(7*B*c^3*d^2 
*e^2 - 2*A*c^3*d*e^3 + B*a*c^2*e^4)/((e*x + d)^2*e^2))*(e*x + d)^3/e^8 + ( 
35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*log(abs( 
e*x + d)/((e*x + d)^2*abs(e)))/e^8 - 1/12*(420*B*c^3*d^4*e^36/(e*x + d) - 
126*B*c^3*d^5*e^36/(e*x + d)^2 + 28*B*c^3*d^6*e^36/(e*x + d)^3 - 3*B*c^3*d 
^7*e^36/(e*x + d)^4 - 240*A*c^3*d^3*e^37/(e*x + d) + 90*A*c^3*d^4*e^37/(e* 
x + d)^2 - 24*A*c^3*d^5*e^37/(e*x + d)^3 + 3*A*c^3*d^6*e^37/(e*x + d)^4 + 
360*B*a*c^2*d^2*e^38/(e*x + d) - 180*B*a*c^2*d^3*e^38/(e*x + d)^2 + 60*B*a 
*c^2*d^4*e^38/(e*x + d)^3 - 9*B*a*c^2*d^5*e^38/(e*x + d)^4 - 144*A*a*c^2*d 
*e^39/(e*x + d) + 108*A*a*c^2*d^2*e^39/(e*x + d)^2 - 48*A*a*c^2*d^3*e^39/( 
e*x + d)^3 + 9*A*a*c^2*d^4*e^39/(e*x + d)^4 + 36*B*a^2*c*e^40/(e*x + d) - 
54*B*a^2*c*d*e^40/(e*x + d)^2 + 36*B*a^2*c*d^2*e^40/(e*x + d)^3 - 9*B*a^2* 
c*d^3*e^40/(e*x + d)^4 + 18*A*a^2*c*e^41/(e*x + d)^2 - 24*A*a^2*c*d*e^41/( 
e*x + d)^3 + 9*A*a^2*c*d^2*e^41/(e*x + d)^4 + 4*B*a^3*e^42/(e*x + d)^3 - 3 
*B*a^3*d*e^42/(e*x + d)^4 + 3*A*a^3*e^43/(e*x + d)^4)/e^44
 

Mupad [B] (verification not implemented)

Time = 6.14 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx=x^2\,\left (\frac {A\,c^3}{2\,e^5}-\frac {5\,B\,c^3\,d}{2\,e^6}\right )-x\,\left (\frac {5\,d\,\left (\frac {A\,c^3}{e^5}-\frac {5\,B\,c^3\,d}{e^6}\right )}{e}-\frac {3\,B\,a\,c^2}{e^5}+\frac {10\,B\,c^3\,d^2}{e^7}\right )-\frac {\frac {B\,a^3\,d\,e^6+3\,A\,a^3\,e^7+9\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+231\,B\,a\,c^2\,d^5\,e^2-75\,A\,a\,c^2\,d^4\,e^3+319\,B\,c^3\,d^7-171\,A\,c^3\,d^6\,e}{12\,e}+x^2\,\left (\frac {9\,B\,a^2\,c\,d\,e^5}{2}+\frac {3\,A\,a^2\,c\,e^6}{2}+75\,B\,a\,c^2\,d^3\,e^3-27\,A\,a\,c^2\,d^2\,e^4+\frac {189\,B\,c^3\,d^5\,e}{2}-\frac {105\,A\,c^3\,d^4\,e^2}{2}\right )+x^3\,\left (3\,B\,a^2\,c\,e^6+30\,B\,a\,c^2\,d^2\,e^4-12\,A\,a\,c^2\,d\,e^5+35\,B\,c^3\,d^4\,e^2-20\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{3}+3\,B\,a^2\,c\,d^2\,e^4+A\,a^2\,c\,d\,e^5+65\,B\,a\,c^2\,d^4\,e^2-22\,A\,a\,c^2\,d^3\,e^3+\frac {259\,B\,c^3\,d^6}{3}-47\,A\,c^3\,d^5\,e\right )}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-\frac {\ln \left (d+e\,x\right )\,\left (35\,B\,c^3\,d^3-15\,A\,c^3\,d^2\,e+15\,B\,a\,c^2\,d\,e^2-3\,A\,a\,c^2\,e^3\right )}{e^8}+\frac {B\,c^3\,x^3}{3\,e^5} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^5,x)
 

Output:

x^2*((A*c^3)/(2*e^5) - (5*B*c^3*d)/(2*e^6)) - x*((5*d*((A*c^3)/e^5 - (5*B* 
c^3*d)/e^6))/e - (3*B*a*c^2)/e^5 + (10*B*c^3*d^2)/e^7) - ((3*A*a^3*e^7 + 3 
19*B*c^3*d^7 + B*a^3*d*e^6 - 171*A*c^3*d^6*e - 75*A*a*c^2*d^4*e^3 + 3*A*a^ 
2*c*d^2*e^5 + 231*B*a*c^2*d^5*e^2 + 9*B*a^2*c*d^3*e^4)/(12*e) + x^2*((3*A* 
a^2*c*e^6)/2 + (189*B*c^3*d^5*e)/2 - (105*A*c^3*d^4*e^2)/2 - 27*A*a*c^2*d^ 
2*e^4 + 75*B*a*c^2*d^3*e^3 + (9*B*a^2*c*d*e^5)/2) + x^3*(3*B*a^2*c*e^6 - 2 
0*A*c^3*d^3*e^3 + 35*B*c^3*d^4*e^2 + 30*B*a*c^2*d^2*e^4 - 12*A*a*c^2*d*e^5 
) + x*((B*a^3*e^6)/3 + (259*B*c^3*d^6)/3 - 47*A*c^3*d^5*e - 22*A*a*c^2*d^3 
*e^3 + 65*B*a*c^2*d^4*e^2 + 3*B*a^2*c*d^2*e^4 + A*a^2*c*d*e^5))/(d^4*e^7 + 
 e^11*x^4 + 4*d^3*e^8*x + 4*d*e^10*x^3 + 6*d^2*e^9*x^2) - (log(d + e*x)*(3 
5*B*c^3*d^3 - 3*A*a*c^2*e^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2))/e^8 + (B 
*c^3*x^3)/(3*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.69 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^5,x)
 

Output:

(36*log(d + e*x)*a**2*c**2*d**5*e**3 + 144*log(d + e*x)*a**2*c**2*d**4*e** 
4*x + 216*log(d + e*x)*a**2*c**2*d**3*e**5*x**2 + 144*log(d + e*x)*a**2*c* 
*2*d**2*e**6*x**3 + 36*log(d + e*x)*a**2*c**2*d*e**7*x**4 - 180*log(d + e* 
x)*a*b*c**2*d**6*e**2 - 720*log(d + e*x)*a*b*c**2*d**5*e**3*x - 1080*log(d 
 + e*x)*a*b*c**2*d**4*e**4*x**2 - 720*log(d + e*x)*a*b*c**2*d**3*e**5*x**3 
 - 180*log(d + e*x)*a*b*c**2*d**2*e**6*x**4 + 180*log(d + e*x)*a*c**3*d**7 
*e + 720*log(d + e*x)*a*c**3*d**6*e**2*x + 1080*log(d + e*x)*a*c**3*d**5*e 
**3*x**2 + 720*log(d + e*x)*a*c**3*d**4*e**4*x**3 + 180*log(d + e*x)*a*c** 
3*d**3*e**5*x**4 - 420*log(d + e*x)*b*c**3*d**8 - 1680*log(d + e*x)*b*c**3 
*d**7*e*x - 2520*log(d + e*x)*b*c**3*d**6*e**2*x**2 - 1680*log(d + e*x)*b* 
c**3*d**5*e**3*x**3 - 420*log(d + e*x)*b*c**3*d**4*e**4*x**4 - 3*a**4*d*e* 
*7 - a**3*b*d**2*e**6 - 4*a**3*b*d*e**7*x - 3*a**3*c*d**3*e**5 - 12*a**3*c 
*d**2*e**6*x - 18*a**3*c*d*e**7*x**2 + 9*a**2*b*c*e**8*x**4 + 39*a**2*c**2 
*d**5*e**3 + 120*a**2*c**2*d**4*e**4*x + 108*a**2*c**2*d**3*e**5*x**2 - 36 
*a**2*c**2*d*e**7*x**4 - 195*a*b*c**2*d**6*e**2 - 600*a*b*c**2*d**5*e**3*x 
 - 540*a*b*c**2*d**4*e**4*x**2 + 180*a*b*c**2*d**2*e**6*x**4 + 36*a*b*c**2 
*d*e**7*x**5 + 195*a*c**3*d**7*e + 600*a*c**3*d**6*e**2*x + 540*a*c**3*d** 
5*e**3*x**2 - 180*a*c**3*d**3*e**5*x**4 - 36*a*c**3*d**2*e**6*x**5 + 6*a*c 
**3*d*e**7*x**6 - 455*b*c**3*d**8 - 1400*b*c**3*d**7*e*x - 1260*b*c**3*d** 
6*e**2*x**2 + 420*b*c**3*d**4*e**4*x**4 + 84*b*c**3*d**3*e**5*x**5 - 14...