Integrand size = 20, antiderivative size = 169 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {12 \left (c d^2-b d e+a e^2\right )^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:
-1/2*(e*x+d)^3*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+3*( a*e^2-b*d*e+c*d^2)*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)^2/(c*x^ 2+b*x+a)-12*(a*e^2-b*d*e+c*d^2)^2*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(- 4*a*c+b^2)^(5/2)
Leaf count is larger than twice the leaf count of optimal. \(413\) vs. \(2(169)=338\).
Time = 0.40 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.44 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {b^4 e^4 x+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )+2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^2}+\frac {b^5 e^4+2 b^3 c e^2 \left (3 c d^2-4 a e^2\right )-2 b^4 c e^3 (2 d+e x)+2 b c^2 \left (11 a^2 e^4+3 c^2 d^3 (d-4 e x)+6 a c d e^2 (d-2 e x)\right )+4 b^2 c^2 e \left (-3 c d^2 (d-e x)+a e^2 (5 d+4 e x)\right )+4 c^3 \left (3 c^2 d^4 x+6 a c d^2 e^2 x-a^2 e^3 (16 d+5 e x)\right )}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {24 \left (c d^2+e (-b d+a e)\right )^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \] Input:
Integrate[(d + e*x)^4/(a + b*x + c*x^2)^3,x]
Output:
((b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/(c ^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (b^5*e^4 + 2*b^3*c*e^2*(3*c*d^2 - 4*a*e^2) - 2*b^4*c*e^3*(2*d + e*x) + 2*b*c^2*(11*a^2*e^4 + 3*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(-3*c*d^2*(d - e*x) + a*e ^2*(5*d + 4*e*x)) + 4*c^3*(3*c^2*d^4*x + 6*a*c*d^2*e^2*x - a^2*e^3*(16*d + 5*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (24*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2
Time = 0.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1153, 1153, 1083, 219}
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.
| \(\displaystyle \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1153 |
| \(\displaystyle -\frac {3 \left (a e^2-b d e+c d^2\right ) \int \frac {(d+e x)^2}{\left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1153 |
| \(\displaystyle -\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {3 \left (a e^2-b d e+c d^2\right ) \left (\frac {4 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {3 \left (a e^2-b d e+c d^2\right ) \left (\frac {4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(d + e*x)^4/(a + b*x + c*x^2)^3,x]
Output:
-1/2*((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*(c*d^2 - b*d*e + a*e^2)*(-(((d + e*x)*(b*d - 2*a*e + (2* c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*(c*d^2 - b*d*e + a* e^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(163)=326\).
Time = 0.96 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.95
| method | result | size |
| default | \(\frac {-\frac {\left (10 e^{4} a^{2} c^{2}-8 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-6 d^{2} e^{2} b^{2} c^{2}+12 d^{3} e b \,c^{3}-6 d^{4} c^{4}\right ) x^{3}}{c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {\left (2 a^{2} b \,c^{2} e^{4}-64 a^{2} c^{3} d \,e^{3}+8 a \,b^{3} c \,e^{4}-4 a \,b^{2} c^{2} d \,e^{3}+36 a b \,c^{3} d^{2} e^{2}-b^{5} e^{4}-4 b^{4} c d \,e^{3}+18 b^{3} c^{2} d^{2} e^{2}-36 b^{2} c^{3} d^{3} e +18 b \,c^{4} d^{4}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{2}}-\frac {\left (6 a^{3} c^{2} e^{4}-10 a^{2} b^{2} c \,e^{4}+20 a^{2} b \,c^{2} d \,e^{3}+12 a^{2} c^{3} d^{2} e^{2}+a \,b^{4} e^{4}+4 a \,b^{3} c d \,e^{3}-30 a \,b^{2} c^{2} d^{2} e^{2}+20 a b \,c^{3} d^{3} e -10 a \,c^{4} d^{4}+4 b^{3} c^{2} d^{3} e -2 b^{2} c^{3} d^{4}\right ) x}{\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{2}}+\frac {10 a^{3} b c \,e^{4}-32 a^{3} c^{2} d \,e^{3}-a^{2} b^{3} e^{4}-4 a^{2} b^{2} c d \,e^{3}+36 a^{2} b \,c^{2} d^{2} e^{2}-32 a^{2} c^{3} d^{3} e -4 a \,b^{2} c^{2} d^{3} e +10 a b \,c^{3} d^{4}-b^{3} c^{2} d^{4}}{2 c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {12 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(667\) |
| risch | \(\text {Expression too large to display}\) | \(1483\) |
Input:
int((e*x+d)^4/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
(-(10*a^2*c^2*e^4-8*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4- 6*b^2*c^2*d^2*e^2+12*b*c^3*d^3*e-6*c^4*d^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x ^3+1/2*(2*a^2*b*c^2*e^4-64*a^2*c^3*d*e^3+8*a*b^3*c*e^4-4*a*b^2*c^2*d*e^3+3 6*a*b*c^3*d^2*e^2-b^5*e^4-4*b^4*c*d*e^3+18*b^3*c^2*d^2*e^2-36*b^2*c^3*d^3* e+18*b*c^4*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2-(6*a^3*c^2*e^4-10*a^2*b ^2*c*e^4+20*a^2*b*c^2*d*e^3+12*a^2*c^3*d^2*e^2+a*b^4*e^4+4*a*b^3*c*d*e^3-3 0*a*b^2*c^2*d^2*e^2+20*a*b*c^3*d^3*e-10*a*c^4*d^4+4*b^3*c^2*d^3*e-2*b^2*c^ 3*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x+1/2/c^2*(10*a^3*b*c*e^4-32*a^3*c^2 *d*e^3-a^2*b^3*e^4-4*a^2*b^2*c*d*e^3+36*a^2*b*c^2*d^2*e^2-32*a^2*c^3*d^3*e -4*a*b^2*c^2*d^3*e+10*a*b*c^3*d^4-b^3*c^2*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4)) /(c*x^2+b*x+a)^2+12*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c*d ^3*e+c^2*d^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b )/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (163) = 326\).
Time = 0.11 (sec) , antiderivative size = 2631, normalized size of antiderivative = 15.57 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
[-1/2*((b^5*c^2 - 14*a*b^3*c^3 + 40*a^2*b*c^4)*d^4 + 4*(a*b^4*c^2 + 4*a^2* b^2*c^3 - 32*a^3*c^4)*d^3*e - 36*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^2 + 4*( a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*d*e^3 + (a^2*b^5 - 14*a^3*b^3*c + 40*a^4*b*c^2)*e^4 - 2*(6*(b^2*c^5 - 4*a*c^6)*d^4 - 12*(b^3*c^4 - 4*a*b*c^5 )*d^3*e + 6*(b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 12*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 - (b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)* e^4)*x^3 - (18*(b^3*c^4 - 4*a*b*c^5)*d^4 - 36*(b^4*c^3 - 4*a*b^2*c^4)*d^3* e + 18*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*e^2 - 4*(b^6*c - 3*a*b^4* c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^3 - (b^7 - 12*a*b^5*c + 30*a^2*b^3* c^2 + 8*a^3*b*c^3)*e^4)*x^2 - 12*(a^2*c^4*d^4 - 2*a^2*b*c^3*d^3*e - 2*a^3* b*c^2*d*e^3 + a^4*c^2*e^4 + (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e^2 + (c^6*d^4 - 2*b*c^5*d^3*e - 2*a*b*c^4*d*e^3 + a^2*c^4*e^4 + (b^2*c^4 + 2*a*c^5)*d^2*e ^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e - 2*a*b^2*c^3*d*e^3 + a^2*b*c^3*e ^4 + (b^3*c^3 + 2*a*b*c^4)*d^2*e^2)*x^3 + ((b^2*c^4 + 2*a*c^5)*d^4 - 2*(b^ 3*c^3 + 2*a*b*c^4)*d^3*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - 2 *(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*e^4)*x^2 + 2* (a*b*c^4*d^4 - 2*a*b^2*c^3*d^3*e - 2*a^2*b^2*c^2*d*e^3 + a^3*b*c^2*e^4 + ( a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e^2)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2* b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 2*(2*(b^4*c^3 + a*b^2*c^4 - 20*a^2*c^5)*d^4 - 4*(b^5*c^2 + a*b^3*c^3 - ...
Leaf count of result is larger than twice the leaf count of optimal. 1355 vs. \(2 (162) = 324\).
Time = 14.47 (sec) , antiderivative size = 1355, normalized size of antiderivative = 8.02 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)**4/(c*x**2+b*x+a)**3,x)
Output:
-6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(x + (-384*a **3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 288*a** 2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*a* *2*b*e**4 - 72*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d** 2)**2 - 12*a*b**2*d*e**3 + 12*a*b*c*d**2*e**2 + 6*b**6*sqrt(-1/(4*a*c - b* *2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4)/(12*a**2*c*e**4 - 24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b**2*c*d**2*e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4)) + 6*sqrt(-1/(4*a *c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(x + (384*a**3*c**3*sqrt(-1 /(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 288*a**2*b**2*c**2*sqrt (-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*a**2*b*e**4 + 72*a *b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 12*a*b** 2*d*e**3 + 12*a*b*c*d**2*e**2 - 6*b**6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4 )/(12*a**2*c*e**4 - 24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b**2*c*d**2 *e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4)) + (10*a**3*b*c*e**4 - 32*a**3*c* *2*d*e**3 - a**2*b**3*e**4 - 4*a**2*b**2*c*d*e**3 + 36*a**2*b*c**2*d**2*e* *2 - 32*a**2*c**3*d**3*e - 4*a*b**2*c**2*d**3*e + 10*a*b*c**3*d**4 - b**3* c**2*d**4 + x**3*(-20*a**2*c**3*e**4 + 16*a*b**2*c**2*e**4 - 24*a*b*c**3*d *e**3 + 24*a*c**4*d**2*e**2 - 2*b**4*c*e**4 + 12*b**2*c**3*d**2*e**2 - ...
Exception generated. \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (163) = 326\).
Time = 0.38 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.84 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\frac {12 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} e x^{3} + 12 \, b^{2} c^{3} d^{2} e^{2} x^{3} + 24 \, a c^{4} d^{2} e^{2} x^{3} - 24 \, a b c^{3} d e^{3} x^{3} - 2 \, b^{4} c e^{4} x^{3} + 16 \, a b^{2} c^{2} e^{4} x^{3} - 20 \, a^{2} c^{3} e^{4} x^{3} + 18 \, b c^{4} d^{4} x^{2} - 36 \, b^{2} c^{3} d^{3} e x^{2} + 18 \, b^{3} c^{2} d^{2} e^{2} x^{2} + 36 \, a b c^{3} d^{2} e^{2} x^{2} - 4 \, b^{4} c d e^{3} x^{2} - 4 \, a b^{2} c^{2} d e^{3} x^{2} - 64 \, a^{2} c^{3} d e^{3} x^{2} - b^{5} e^{4} x^{2} + 8 \, a b^{3} c e^{4} x^{2} + 2 \, a^{2} b c^{2} e^{4} x^{2} + 4 \, b^{2} c^{3} d^{4} x + 20 \, a c^{4} d^{4} x - 8 \, b^{3} c^{2} d^{3} e x - 40 \, a b c^{3} d^{3} e x + 60 \, a b^{2} c^{2} d^{2} e^{2} x - 24 \, a^{2} c^{3} d^{2} e^{2} x - 8 \, a b^{3} c d e^{3} x - 40 \, a^{2} b c^{2} d e^{3} x - 2 \, a b^{4} e^{4} x + 20 \, a^{2} b^{2} c e^{4} x - 12 \, a^{3} c^{2} e^{4} x - b^{3} c^{2} d^{4} + 10 \, a b c^{3} d^{4} - 4 \, a b^{2} c^{2} d^{3} e - 32 \, a^{2} c^{3} d^{3} e + 36 \, a^{2} b c^{2} d^{2} e^{2} - 4 \, a^{2} b^{2} c d e^{3} - 32 \, a^{3} c^{2} d e^{3} - a^{2} b^{3} e^{4} + 10 \, a^{3} b c e^{4}}{2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
12*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^ 2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c ^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^5*d^4*x^3 - 24*b*c^4*d^3*e*x^3 + 12*b^ 2*c^3*d^2*e^2*x^3 + 24*a*c^4*d^2*e^2*x^3 - 24*a*b*c^3*d*e^3*x^3 - 2*b^4*c* e^4*x^3 + 16*a*b^2*c^2*e^4*x^3 - 20*a^2*c^3*e^4*x^3 + 18*b*c^4*d^4*x^2 - 3 6*b^2*c^3*d^3*e*x^2 + 18*b^3*c^2*d^2*e^2*x^2 + 36*a*b*c^3*d^2*e^2*x^2 - 4* b^4*c*d*e^3*x^2 - 4*a*b^2*c^2*d*e^3*x^2 - 64*a^2*c^3*d*e^3*x^2 - b^5*e^4*x ^2 + 8*a*b^3*c*e^4*x^2 + 2*a^2*b*c^2*e^4*x^2 + 4*b^2*c^3*d^4*x + 20*a*c^4* d^4*x - 8*b^3*c^2*d^3*e*x - 40*a*b*c^3*d^3*e*x + 60*a*b^2*c^2*d^2*e^2*x - 24*a^2*c^3*d^2*e^2*x - 8*a*b^3*c*d*e^3*x - 40*a^2*b*c^2*d*e^3*x - 2*a*b^4* e^4*x + 20*a^2*b^2*c*e^4*x - 12*a^3*c^2*e^4*x - b^3*c^2*d^4 + 10*a*b*c^3*d ^4 - 4*a*b^2*c^2*d^3*e - 32*a^2*c^3*d^3*e + 36*a^2*b*c^2*d^2*e^2 - 4*a^2*b ^2*c*d*e^3 - 32*a^3*c^2*d*e^3 - a^2*b^3*e^4 + 10*a^3*b*c*e^4)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)^2)
Time = 0.44 (sec) , antiderivative size = 798, normalized size of antiderivative = 4.72 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\frac {12\,\mathrm {atan}\left (\frac {\left (\frac {6\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {12\,c\,x\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,a^2\,e^4-12\,a\,b\,d\,e^3+12\,a\,c\,d^2\,e^2+6\,b^2\,d^2\,e^2-12\,b\,c\,d^3\,e+6\,c^2\,d^4}\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {-10\,a^3\,b\,c\,e^4+32\,a^3\,c^2\,d\,e^3+a^2\,b^3\,e^4+4\,a^2\,b^2\,c\,d\,e^3-36\,a^2\,b\,c^2\,d^2\,e^2+32\,a^2\,c^3\,d^3\,e+4\,a\,b^2\,c^2\,d^3\,e-10\,a\,b\,c^3\,d^4+b^3\,c^2\,d^4}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (-2\,a^2\,b\,c^2\,e^4+64\,a^2\,c^3\,d\,e^3-8\,a\,b^3\,c\,e^4+4\,a\,b^2\,c^2\,d\,e^3-36\,a\,b\,c^3\,d^2\,e^2+b^5\,e^4+4\,b^4\,c\,d\,e^3-18\,b^3\,c^2\,d^2\,e^2+36\,b^2\,c^3\,d^3\,e-18\,b\,c^4\,d^4\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^3\,\left (10\,a^2\,c^2\,e^4-8\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-6\,b^2\,c^2\,d^2\,e^2+12\,b\,c^3\,d^3\,e-6\,c^4\,d^4\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (6\,a^3\,c^2\,e^4-10\,a^2\,b^2\,c\,e^4+20\,a^2\,b\,c^2\,d\,e^3+12\,a^2\,c^3\,d^2\,e^2+a\,b^4\,e^4+4\,a\,b^3\,c\,d\,e^3-30\,a\,b^2\,c^2\,d^2\,e^2+20\,a\,b\,c^3\,d^3\,e-10\,a\,c^4\,d^4+4\,b^3\,c^2\,d^3\,e-2\,b^2\,c^3\,d^4\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \] Input:
int((d + e*x)^4/(a + b*x + c*x^2)^3,x)
Output:
(12*atan((((6*(b^5 + 16*a^2*b*c^2 - 8*a*b^3*c)*(a*e^2 + c*d^2 - b*d*e)^2)/ ((4*a*c - b^2)^(5/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (12*c*x*(a*e^2 + c* d^2 - b*d*e)^2)/(4*a*c - b^2)^(5/2))*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(6*a^ 2*e^4 + 6*c^2*d^4 + 6*b^2*d^2*e^2 - 12*a*b*d*e^3 - 12*b*c*d^3*e + 12*a*c*d ^2*e^2))*(a*e^2 + c*d^2 - b*d*e)^2)/(4*a*c - b^2)^(5/2) - ((a^2*b^3*e^4 + b^3*c^2*d^4 + 32*a^2*c^3*d^3*e + 32*a^3*c^2*d*e^3 - 10*a*b*c^3*d^4 - 10*a^ 3*b*c*e^4 + 4*a*b^2*c^2*d^3*e + 4*a^2*b^2*c*d*e^3 - 36*a^2*b*c^2*d^2*e^2)/ (2*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(b^5*e^4 - 18*b*c^4*d^4 - 2* a^2*b*c^2*e^4 + 64*a^2*c^3*d*e^3 + 36*b^2*c^3*d^3*e - 18*b^3*c^2*d^2*e^2 - 8*a*b^3*c*e^4 + 4*b^4*c*d*e^3 - 36*a*b*c^3*d^2*e^2 + 4*a*b^2*c^2*d*e^3))/ (2*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^3*(b^4*e^4 - 6*c^4*d^4 + 10*a^ 2*c^2*e^4 - 12*a*c^3*d^2*e^2 - 6*b^2*c^2*d^2*e^2 - 8*a*b^2*c*e^4 + 12*b*c^ 3*d^3*e + 12*a*b*c^2*d*e^3))/(c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(a*b^ 4*e^4 - 10*a*c^4*d^4 + 6*a^3*c^2*e^4 - 2*b^2*c^3*d^4 - 10*a^2*b^2*c*e^4 + 4*b^3*c^2*d^3*e + 12*a^2*c^3*d^2*e^2 + 20*a*b*c^3*d^3*e + 4*a*b^3*c*d*e^3 + 20*a^2*b*c^2*d*e^3 - 30*a*b^2*c^2*d^2*e^2))/(c^2*(b^4 + 16*a^2*c^2 - 8*a *b^2*c)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3)
Time = 0.22 (sec) , antiderivative size = 2767, normalized size of antiderivative = 16.37 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^4/(c*x^2+b*x+a)^3,x)
Output:
(24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b*c*e**4 - 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c*d *e**3 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b* *2*c*e**4*x + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a **3*b*c**2*d**2*e**2 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**2*e**4*x**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqr t(4*a*c - b**2))*a**2*b**3*c*d**2*e**2 - 96*sqrt(4*a*c - b**2)*atan((b + 2 *c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c*d*e**3*x + 24*sqrt(4*a*c - b**2)*ata n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c*e**4*x**2 - 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*d**3*e + 96*sqrt (4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*d**2*e* *2*x - 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b** 2*c**2*d*e**3*x**2 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b **2))*a**2*b**2*c**2*e**4*x**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq rt(4*a*c - b**2))*a**2*b*c**3*d**4 + 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x )/sqrt(4*a*c - b**2))*a**2*b*c**3*d**2*e**2*x**2 + 24*sqrt(4*a*c - b**2)*a tan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**3*e**4*x**4 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*d**2*e**2*x - 48*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*d*e**3*x**2 - 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**...