Integrand size = 20, antiderivative size = 428 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx=-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 x}{e^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^2}{e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^3}{3 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^4}{4 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^5}{5 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{3 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^7}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9}+\frac {\left (c d^2-b d e+a e^2\right )^4 \log (d+e x)}{e^9} \] Output:
-4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*x/e^8+(a*e^2-b*d*e+c*d^2)^2*(14*c^2* d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^2/e^9-4/3*(-b*e+2*c*d)*(a*e^2-b* d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^3/e^9+1/4*(70*c^ 4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2 *e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^4/e^9-4/5*c*(-b*e+2*c*d)*(7* c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^5/e^9+1/3*c^2*(14*c^2*d^2+3*b^ 2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^6/e^9-4/7*c^3*(-b*e+2*c*d)*(e*x+d)^7/e^9 +1/8*c^4*(e*x+d)^8/e^9+(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)/e^9
Time = 0.27 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx=\frac {x \left (c^4 \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )+70 b e^4 \left (48 a^3 e^3+36 a^2 b e^2 (-2 d+e x)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+56 c e^3 \left (30 a^3 e^3 (-2 d+e x)+30 a^2 b e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+15 a b^2 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^3 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+84 c^2 e^2 \left (5 a^2 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 a b e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+8 c^3 e \left (7 a e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+b \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )}{840 e^8}+\frac {\left (c d^2+e (-b d+a e)\right )^4 \log (d+e x)}{e^9} \] Input:
Integrate[(a + b*x + c*x^2)^4/(d + e*x),x]
Output:
(x*(c^4*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168* d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + 70*b*e^4*(4 8*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2 *x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 56*c*e^3*(3 0*a^3*e^3*(-2*d + e*x) + 30*a^2*b*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 15*a *b^2*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^3*(60*d^4 - 30* d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 84*c^2*e^2*(5*a^2 *e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*a*b*e*(60*d^4 - 3 0*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^2*(-60*d^5 + 3 0*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 8*c^3*e*(7*a*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + b*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6))))/(840*e^8 ) + ((c*d^2 + e*(-(b*d) + a*e))^4*Log[d + e*x])/e^9
Time = 0.95 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 c (d+e x)^4 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {4 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {2 (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {4 c^3 (d+e x)^6 (2 c d-b e)}{e^8}+\frac {c^4 (d+e x)^7}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{4 e^9}+\frac {c^2 (d+e x)^6 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac {4 c (d+e x)^5 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac {4 (d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^4}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9}\) |
Input:
Int[(a + b*x + c*x^2)^4/(d + e*x),x]
Output:
(-4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*x)/e^8 + ((c*d^2 - b*d*e + a*e ^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^2)/e^9 - (4 *(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^3)/(3*e^9) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b* d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b *d*e + a^2*e^2))*(d + e*x)^4)/(4*e^9) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^ 2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^5)/(5*e^9) + (c^2*(14*c^2*d^2 + 3*b ^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^6)/(3*e^9) - (4*c^3*(2*c*d - b*e)* (d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 - b*d*e + a*e^2 )^4*Log[d + e*x])/e^9
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(874\) vs. \(2(416)=832\).
Time = 1.12 (sec) , antiderivative size = 875, normalized size of antiderivative = 2.04
| method | result | size |
| norman | \(\frac {\left (4 a^{3} b \,e^{7}-4 d \,e^{6} c \,a^{3}-6 a^{2} b^{2} d \,e^{6}+12 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+4 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+12 a b \,c^{2} d^{4} e^{3}-4 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+4 b^{3} c \,d^{4} e^{3}-6 b^{2} c^{2} d^{5} e^{2}+4 b \,c^{3} d^{6} e -d^{7} c^{4}\right ) x}{e^{8}}+\frac {c^{4} x^{8}}{8 e}+\frac {\left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-12 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}-4 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}-12 a b \,c^{2} d^{3} e^{3}+4 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}-4 b^{3} c \,d^{3} e^{3}+6 b^{2} c^{2} d^{4} e^{2}-4 b \,c^{3} d^{5} e +d^{6} c^{4}\right ) x^{2}}{2 e^{7}}+\frac {\left (12 a^{2} b c \,e^{5}-6 a^{2} c^{2} d \,e^{4}+4 a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+12 a b \,c^{2} d^{2} e^{3}-4 a \,c^{3} d^{3} e^{2}-b^{4} d \,e^{4}+4 b^{3} c \,d^{2} e^{3}-6 b^{2} c^{2} d^{3} e^{2}+4 b \,c^{3} d^{4} e -c^{4} d^{5}\right ) x^{3}}{3 e^{6}}+\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-12 a b \,c^{2} d \,e^{3}+4 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-4 d \,e^{3} b^{3} c +6 d^{2} e^{2} b^{2} c^{2}-4 d^{3} e b \,c^{3}+d^{4} c^{4}\right ) x^{4}}{4 e^{5}}+\frac {c^{2} \left (4 a c \,e^{2}+6 b^{2} e^{2}-4 b c d e +c^{2} d^{2}\right ) x^{6}}{6 e^{3}}+\frac {c^{3} \left (4 b e -c d \right ) x^{7}}{7 e^{2}}+\frac {c \left (12 a b c \,e^{3}-4 d \,e^{2} a \,c^{2}+4 b^{3} e^{3}-6 d \,e^{2} b^{2} c +4 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) x^{5}}{5 e^{4}}+\frac {\left (a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(875\) |
| default | \(\text {Expression too large to display}\) | \(1004\) |
| risch | \(\text {Expression too large to display}\) | \(1096\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1099\) |
Input:
int((c*x^2+b*x+a)^4/(e*x+d),x,method=_RETURNVERBOSE)
Output:
(4*a^3*b*e^7-4*a^3*c*d*e^6-6*a^2*b^2*d*e^6+12*a^2*b*c*d^2*e^5-6*a^2*c^2*d^ 3*e^4+4*a*b^3*d^2*e^5-12*a*b^2*c*d^3*e^4+12*a*b*c^2*d^4*e^3-4*a*c^3*d^5*e^ 2-b^4*d^3*e^4+4*b^3*c*d^4*e^3-6*b^2*c^2*d^5*e^2+4*b*c^3*d^6*e-c^4*d^7)/e^8 *x+1/8*c^4/e*x^8+1/2/e^7*(4*a^3*c*e^6+6*a^2*b^2*e^6-12*a^2*b*c*d*e^5+6*a^2 *c^2*d^2*e^4-4*a*b^3*d*e^5+12*a*b^2*c*d^2*e^4-12*a*b*c^2*d^3*e^3+4*a*c^3*d ^4*e^2+b^4*d^2*e^4-4*b^3*c*d^3*e^3+6*b^2*c^2*d^4*e^2-4*b*c^3*d^5*e+c^4*d^6 )*x^2+1/3/e^6*(12*a^2*b*c*e^5-6*a^2*c^2*d*e^4+4*a*b^3*e^5-12*a*b^2*c*d*e^4 +12*a*b*c^2*d^2*e^3-4*a*c^3*d^3*e^2-b^4*d*e^4+4*b^3*c*d^2*e^3-6*b^2*c^2*d^ 3*e^2+4*b*c^3*d^4*e-c^4*d^5)*x^3+1/4/e^5*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-12* a*b*c^2*d*e^3+4*a*c^3*d^2*e^2+b^4*e^4-4*b^3*c*d*e^3+6*b^2*c^2*d^2*e^2-4*b* c^3*d^3*e+c^4*d^4)*x^4+1/6*c^2/e^3*(4*a*c*e^2+6*b^2*e^2-4*b*c*d*e+c^2*d^2) *x^6+1/7*c^3/e^2*(4*b*e-c*d)*x^7+1/5/e^4*c*(12*a*b*c*e^3-4*a*c^2*d*e^2+4*b ^3*e^3-6*b^2*c*d*e^2+4*b*c^2*d^2*e-c^3*d^3)*x^5+(a^4*e^8-4*a^3*b*d*e^7+4*a ^3*c*d^2*e^6+6*a^2*b^2*d^2*e^6-12*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4-4*a*b^ 3*d^3*e^5+12*a*b^2*c*d^4*e^4-12*a*b*c^2*d^5*e^3+4*a*c^3*d^6*e^2+b^4*d^4*e^ 4-4*b^3*c*d^5*e^3+6*b^2*c^2*d^6*e^2-4*b*c^3*d^7*e+c^4*d^8)/e^9*ln(e*x+d)
Time = 0.08 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="fricas")
Output:
1/840*(105*c^4*e^8*x^8 - 120*(c^4*d*e^7 - 4*b*c^3*e^8)*x^7 + 140*(c^4*d^2* e^6 - 4*b*c^3*d*e^7 + 2*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 168*(c^4*d^3*e^5 - 4*b*c^3*d^2*e^6 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 4*(b^3*c + 3*a*b*c^2)* e^8)*x^5 + 210*(c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d^ 2*e^6 - 4*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)* x^4 - 280*(c^4*d^5*e^3 - 4*b*c^3*d^4*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 4*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 420*(c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 2*(3 *b^2*c^2 + 2*a*c^3)*d^4*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 + 12*a* b^2*c + 6*a^2*c^2)*d^2*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 840*(c^4*d^7*e - 4*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 2*(3* b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4 + 12*a*b ^2*c + 6*a^2*c^2)*d^3*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 840*(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^ 8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3* a^2*b^2 + 2*a^3*c)*d^2*e^6)*log(e*x + d))/e^9
Time = 0.95 (sec) , antiderivative size = 808, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate((c*x**2+b*x+a)**4/(e*x+d),x)
Output:
c**4*x**8/(8*e) + x**7*(4*b*c**3/(7*e) - c**4*d/(7*e**2)) + x**6*(2*a*c**3 /(3*e) + b**2*c**2/e - 2*b*c**3*d/(3*e**2) + c**4*d**2/(6*e**3)) + x**5*(1 2*a*b*c**2/(5*e) - 4*a*c**3*d/(5*e**2) + 4*b**3*c/(5*e) - 6*b**2*c**2*d/(5 *e**2) + 4*b*c**3*d**2/(5*e**3) - c**4*d**3/(5*e**4)) + x**4*(3*a**2*c**2/ (2*e) + 3*a*b**2*c/e - 3*a*b*c**2*d/e**2 + a*c**3*d**2/e**3 + b**4/(4*e) - b**3*c*d/e**2 + 3*b**2*c**2*d**2/(2*e**3) - b*c**3*d**3/e**4 + c**4*d**4/ (4*e**5)) + x**3*(4*a**2*b*c/e - 2*a**2*c**2*d/e**2 + 4*a*b**3/(3*e) - 4*a *b**2*c*d/e**2 + 4*a*b*c**2*d**2/e**3 - 4*a*c**3*d**3/(3*e**4) - b**4*d/(3 *e**2) + 4*b**3*c*d**2/(3*e**3) - 2*b**2*c**2*d**3/e**4 + 4*b*c**3*d**4/(3 *e**5) - c**4*d**5/(3*e**6)) + x**2*(2*a**3*c/e + 3*a**2*b**2/e - 6*a**2*b *c*d/e**2 + 3*a**2*c**2*d**2/e**3 - 2*a*b**3*d/e**2 + 6*a*b**2*c*d**2/e**3 - 6*a*b*c**2*d**3/e**4 + 2*a*c**3*d**4/e**5 + b**4*d**2/(2*e**3) - 2*b**3 *c*d**3/e**4 + 3*b**2*c**2*d**4/e**5 - 2*b*c**3*d**5/e**6 + c**4*d**6/(2*e **7)) + x*(4*a**3*b/e - 4*a**3*c*d/e**2 - 6*a**2*b**2*d/e**2 + 12*a**2*b*c *d**2/e**3 - 6*a**2*c**2*d**3/e**4 + 4*a*b**3*d**2/e**3 - 12*a*b**2*c*d**3 /e**4 + 12*a*b*c**2*d**4/e**5 - 4*a*c**3*d**5/e**6 - b**4*d**3/e**4 + 4*b* *3*c*d**4/e**5 - 6*b**2*c**2*d**5/e**6 + 4*b*c**3*d**6/e**7 - c**4*d**7/e* *8) + (a*e**2 - b*d*e + c*d**2)**4*log(d + e*x)/e**9
Time = 0.05 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="maxima")
Output:
1/840*(105*c^4*e^7*x^8 - 120*(c^4*d*e^6 - 4*b*c^3*e^7)*x^7 + 140*(c^4*d^2* e^5 - 4*b*c^3*d*e^6 + 2*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^6 - 168*(c^4*d^3*e^4 - 4*b*c^3*d^2*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 4*(b^3*c + 3*a*b*c^2)* e^7)*x^5 + 210*(c^4*d^4*e^3 - 4*b*c^3*d^3*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^ 2*e^5 - 4*(b^3*c + 3*a*b*c^2)*d*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)* x^4 - 280*(c^4*d^5*e^2 - 4*b*c^3*d^4*e^3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^2*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 4*(a*b^3 + 3*a^2*b*c)*e^7)*x^3 + 420*(c^4*d^6*e - 4*b*c^3*d^5*e^2 + 2*(3*b ^2*c^2 + 2*a*c^3)*d^4*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^3*e^4 + (b^4 + 12*a*b^ 2*c + 6*a^2*c^2)*d^2*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d*e^6 + 2*(3*a^2*b^2 + 2* a^3*c)*e^7)*x^2 - 840*(c^4*d^7 - 4*b*c^3*d^6*e - 4*a^3*b*e^7 + 2*(3*b^2*c^ 2 + 2*a*c^3)*d^5*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + 2*(3*a^2*b^2 + 2*a^3 *c)*d*e^6)*x)/e^8 + (c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2 *(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12 *a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b ^2 + 2*a^3*c)*d^2*e^6)*log(e*x + d)/e^9
Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (416) = 832\).
Time = 0.36 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="giac")
Output:
1/840*(105*c^4*e^7*x^8 - 120*c^4*d*e^6*x^7 + 480*b*c^3*e^7*x^7 + 140*c^4*d ^2*e^5*x^6 - 560*b*c^3*d*e^6*x^6 + 840*b^2*c^2*e^7*x^6 + 560*a*c^3*e^7*x^6 - 168*c^4*d^3*e^4*x^5 + 672*b*c^3*d^2*e^5*x^5 - 1008*b^2*c^2*d*e^6*x^5 - 672*a*c^3*d*e^6*x^5 + 672*b^3*c*e^7*x^5 + 2016*a*b*c^2*e^7*x^5 + 210*c^4*d ^4*e^3*x^4 - 840*b*c^3*d^3*e^4*x^4 + 1260*b^2*c^2*d^2*e^5*x^4 + 840*a*c^3* d^2*e^5*x^4 - 840*b^3*c*d*e^6*x^4 - 2520*a*b*c^2*d*e^6*x^4 + 210*b^4*e^7*x ^4 + 2520*a*b^2*c*e^7*x^4 + 1260*a^2*c^2*e^7*x^4 - 280*c^4*d^5*e^2*x^3 + 1 120*b*c^3*d^4*e^3*x^3 - 1680*b^2*c^2*d^3*e^4*x^3 - 1120*a*c^3*d^3*e^4*x^3 + 1120*b^3*c*d^2*e^5*x^3 + 3360*a*b*c^2*d^2*e^5*x^3 - 280*b^4*d*e^6*x^3 - 3360*a*b^2*c*d*e^6*x^3 - 1680*a^2*c^2*d*e^6*x^3 + 1120*a*b^3*e^7*x^3 + 336 0*a^2*b*c*e^7*x^3 + 420*c^4*d^6*e*x^2 - 1680*b*c^3*d^5*e^2*x^2 + 2520*b^2* c^2*d^4*e^3*x^2 + 1680*a*c^3*d^4*e^3*x^2 - 1680*b^3*c*d^3*e^4*x^2 - 5040*a *b*c^2*d^3*e^4*x^2 + 420*b^4*d^2*e^5*x^2 + 5040*a*b^2*c*d^2*e^5*x^2 + 2520 *a^2*c^2*d^2*e^5*x^2 - 1680*a*b^3*d*e^6*x^2 - 5040*a^2*b*c*d*e^6*x^2 + 252 0*a^2*b^2*e^7*x^2 + 1680*a^3*c*e^7*x^2 - 840*c^4*d^7*x + 3360*b*c^3*d^6*e* x - 5040*b^2*c^2*d^5*e^2*x - 3360*a*c^3*d^5*e^2*x + 3360*b^3*c*d^4*e^3*x + 10080*a*b*c^2*d^4*e^3*x - 840*b^4*d^3*e^4*x - 10080*a*b^2*c*d^3*e^4*x - 5 040*a^2*c^2*d^3*e^4*x + 3360*a*b^3*d^2*e^5*x + 10080*a^2*b*c*d^2*e^5*x - 5 040*a^2*b^2*d*e^6*x - 3360*a^3*c*d*e^6*x + 3360*a^3*b*e^7*x)/e^8 + (c^4*d^ 8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e...
Time = 5.33 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
int((a + b*x + c*x^2)^4/(d + e*x),x)
Output:
x^4*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(4*e) + (d*((d*((4*a*c^3 + 6*b^2*c^2)/ e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*b*c*(3*a*c + b^2))/e))/(4*e )) - x^5*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e) )/(5*e) - (4*b*c*(3*a*c + b^2))/(5*e)) - x^3*((d*((b^4 + 6*a^2*c^2 + 12*a* b^2*c)/e + (d*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2 ))/e))/e - (4*b*c*(3*a*c + b^2))/e))/e))/(3*e) - (4*a*b*(3*a*c + b^2))/(3* e)) + x^7*((4*b*c^3)/(7*e) - (c^4*d)/(7*e^2)) - x*((d*((4*a^3*c + 6*a^2*b^ 2)/e + (d*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((4*a*c^3 + 6*b^2 *c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*b*c*(3*a*c + b^2))/e) )/e))/e - (4*a*b*(3*a*c + b^2))/e))/e))/e - (4*a^3*b)/e) + x^2*((4*a^3*c + 6*a^2*b^2)/(2*e) + (d*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((4* a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*b*c*(3*a *c + b^2))/e))/e))/e - (4*a*b*(3*a*c + b^2))/e))/(2*e)) + x^6*((4*a*c^3 + 6*b^2*c^2)/(6*e) - (d*((4*b*c^3)/e - (c^4*d)/e^2))/(6*e)) + (log(d + e*x)* (a^4*e^8 + c^4*d^8 + b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a ^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 6 *b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12 *a*b^2*c*d^4*e^4 - 12*a^2*b*c*d^3*e^5))/e^9 + (c^4*x^8)/(8*e)
Time = 0.25 (sec) , antiderivative size = 1098, normalized size of antiderivative = 2.57 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^4/(e*x+d),x)
Output:
(840*log(d + e*x)*a**4*e**8 - 3360*log(d + e*x)*a**3*b*d*e**7 + 3360*log(d + e*x)*a**3*c*d**2*e**6 + 5040*log(d + e*x)*a**2*b**2*d**2*e**6 - 10080*l og(d + e*x)*a**2*b*c*d**3*e**5 + 5040*log(d + e*x)*a**2*c**2*d**4*e**4 - 3 360*log(d + e*x)*a*b**3*d**3*e**5 + 10080*log(d + e*x)*a*b**2*c*d**4*e**4 - 10080*log(d + e*x)*a*b*c**2*d**5*e**3 + 3360*log(d + e*x)*a*c**3*d**6*e* *2 + 840*log(d + e*x)*b**4*d**4*e**4 - 3360*log(d + e*x)*b**3*c*d**5*e**3 + 5040*log(d + e*x)*b**2*c**2*d**6*e**2 - 3360*log(d + e*x)*b*c**3*d**7*e + 840*log(d + e*x)*c**4*d**8 + 3360*a**3*b*e**8*x - 3360*a**3*c*d*e**7*x + 1680*a**3*c*e**8*x**2 - 5040*a**2*b**2*d*e**7*x + 2520*a**2*b**2*e**8*x** 2 + 10080*a**2*b*c*d**2*e**6*x - 5040*a**2*b*c*d*e**7*x**2 + 3360*a**2*b*c *e**8*x**3 - 5040*a**2*c**2*d**3*e**5*x + 2520*a**2*c**2*d**2*e**6*x**2 - 1680*a**2*c**2*d*e**7*x**3 + 1260*a**2*c**2*e**8*x**4 + 3360*a*b**3*d**2*e **6*x - 1680*a*b**3*d*e**7*x**2 + 1120*a*b**3*e**8*x**3 - 10080*a*b**2*c*d **3*e**5*x + 5040*a*b**2*c*d**2*e**6*x**2 - 3360*a*b**2*c*d*e**7*x**3 + 25 20*a*b**2*c*e**8*x**4 + 10080*a*b*c**2*d**4*e**4*x - 5040*a*b*c**2*d**3*e* *5*x**2 + 3360*a*b*c**2*d**2*e**6*x**3 - 2520*a*b*c**2*d*e**7*x**4 + 2016* a*b*c**2*e**8*x**5 - 3360*a*c**3*d**5*e**3*x + 1680*a*c**3*d**4*e**4*x**2 - 1120*a*c**3*d**3*e**5*x**3 + 840*a*c**3*d**2*e**6*x**4 - 672*a*c**3*d*e* *7*x**5 + 560*a*c**3*e**8*x**6 - 840*b**4*d**3*e**5*x + 420*b**4*d**2*e**6 *x**2 - 280*b**4*d*e**7*x**3 + 210*b**4*e**8*x**4 + 3360*b**3*c*d**4*e*...