Integrand size = 20, antiderivative size = 320 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^4 x}{c^2}+\frac {4 a b^2 c d e^3-a b^3 e^4+8 a c^2 d e \left (c d^2-a e^2\right )-b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )-\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \] Output:
e^4*x/c^2+(4*a*b^2*c*d*e^3-a*b^3*e^4+8*a*c^2*d*e*(-a*e^2+c*d^2)-b*c*(-3*a^ 2*e^4+6*a*c*d^2*e^2+c^2*d^4)-(2*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^ 3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2*e^2+6*a*b*d*e+3*b^2*d^2))*x)/c^3/(-4*a* c+b^2)/(c*x^2+b*x+a)+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b*d)-6*a*c^2 *e^3*(a*e+2*b*d)+2*b^2*c*e^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^( 1/2))/c^3/(-4*a*c+b^2)^(3/2)+e^3*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/c^3
Time = 0.31 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {c e^4 x+\frac {-b^4 e^4 x+b^3 e^3 (-a e+4 c d x)+2 b^2 c e^2 \left (-3 c d^2 x+2 a e (d+e x)\right )-b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )-2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (-2 c^4 d^4+b^4 e^4+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (2 b d+a e)-2 b^2 c e^3 (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^3 (2 c d-b e) \log (a+x (b+c x))}{c^3} \] Input:
Integrate[(d + e*x)^4/(a + b*x + c*x^2)^2,x]
Output:
(c*e^4*x + (-(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)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(-2*c^4*d^4 + b^4*e^ 4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d + a*e) - 2*b^2*c*e^3*(b *d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(2*c*d - b*e)*Log[a + x*(b + c*x)])/c^3
Time = 0.61 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1164, 27, 1200, 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.
| \(\displaystyle \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {\int \frac {2 (d+e x)^2 \left (c d^2-2 b e d+3 a e^2-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {(d+e x)^2 \left (c d^2-e (2 b d-3 a e)-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle -\frac {2 \int \left (-\frac {(2 c d-b e) x e^3}{c}-\frac {\left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) e^2}{c^2}+\frac {c^3 d^4-2 c^2 e (b d-3 a e) d^2+a b^2 e^4-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{c^2 \left (c x^2+b x+a\right )}\right )dx}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2}-\frac {e^3 x^2 (2 c d-b e)}{2 c}\right )}{b^2-4 a c}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
Input:
Int[(d + e*x)^4/(a + b*x + c*x^2)^2,x]
Output:
-(((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*(-((e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/c^2) - (e^3*(2*c*d - b*e)*x^2)/(2*c) - ((2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh [(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)* e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*c^3)))/(b^2 - 4*a*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 1.08 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (2 e^{4} a^{2} c^{2}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-4 d \,e^{3} b^{3} c +6 d^{2} e^{2} b^{2} c^{2}-4 d^{3} e b \,c^{3}+2 d^{4} c^{4}\right ) x}{\left (4 a c -b^{2}\right ) c}+\frac {3 a^{2} b c \,e^{4}-8 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}+4 a \,b^{2} c d \,e^{3}-6 a b \,c^{2} d^{2} e^{2}+8 a \,c^{3} d^{3} e -d^{4} b \,c^{3}}{\left (4 a c -b^{2}\right ) c}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a b c \,e^{4}-8 a \,c^{2} d \,e^{3}-e^{4} b^{3}+2 b^{2} c d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 e^{4} a^{2} c -a \,b^{2} e^{4}+2 a b c d \,e^{3}-6 d^{2} e^{2} a \,c^{2}+2 b \,c^{2} d^{3} e -d^{4} c^{3}-\frac {\left (4 a b c \,e^{4}-8 a \,c^{2} d \,e^{3}-e^{4} b^{3}+2 b^{2} c d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\) | \(436\) |
| risch | \(\text {Expression too large to display}\) | \(5644\) |
Input:
int((e*x+d)^4/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
e^4*x/c^2-1/c^2*((-(2*a^2*c^2*e^4-4*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^4-4*b^3*c*d*e^3+6*b^2*c^2*d^2*e^2-4*b*c^3*d^3*e+2*c^4*d^4)/( 4*a*c-b^2)/c*x+(3*a^2*b*c*e^4-8*a^2*c^2*d*e^3-a*b^3*e^4+4*a*b^2*c*d*e^3-6* a*b*c^2*d^2*e^2+8*a*c^3*d^3*e-b*c^3*d^4)/(4*a*c-b^2)/c)/(c*x^2+b*x+a)+2/(4 *a*c-b^2)*(1/2*(4*a*b*c*e^4-8*a*c^2*d*e^3-b^3*e^4+2*b^2*c*d*e^3)/c*ln(c*x^ 2+b*x+a)+2*(3*e^4*a^2*c-a*b^2*e^4+2*a*b*c*d*e^3-6*d^2*e^2*a*c^2+2*b*c^2*d^ 3*e-d^4*c^3-1/2*(4*a*b*c*e^4-8*a*c^2*d*e^3-b^3*e^4+2*b^2*c*d*e^3)*b/c)/(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. 919 vs. \(2 (316) = 632\).
Time = 0.13 (sec) , antiderivative size = 1857, normalized size of antiderivative = 5.80 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^3 + (b^5*c - 8*a*b^3*c^2 + 16 *a^2*b*c^3)*e^4*x^2 - (b^3*c^3 - 4*a*b*c^4)*d^4 + 8*(a*b^2*c^3 - 4*a^2*c^4 )*d^3*e - 6*(a*b^3*c^2 - 4*a^2*b*c^3)*d^2*e^2 + 4*(a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d*e^3 - (a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e^4 + (2*a*c^4* d^4 - 4*a*b*c^3*d^3*e + 12*a^2*c^3*d^2*e^2 + 2*(a*b^3*c - 6*a^2*b*c^2)*d*e ^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^4 + (2*c^5*d^4 - 4*b*c^4*d^3*e + 12*a*c^4*d^2*e^2 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^3 - (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^4)*x^2 + (2*b*c^4*d^4 - 4*b^2*c^3*d^3*e + 12*a*b*c^3*d^2*e^2 + 2*(b^4*c - 6*a*b^2*c^2)*d*e^3 - (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^4)*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*(b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c ^3 - 4*a*b*c^4)*d^3*e + 6*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^2 - 4* (b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d*e^3 + (b^6 - 9*a*b^4*c + 26*a^2*b^2 *c^2 - 24*a^3*c^3)*e^4)*x + (2*(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d*e^ 3 - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4 + (2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4)*x^2 + (2*(b ^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^3 - (b^6 - 8*a*b^4*c + 16*a^2*b^2*c ^2)*e^4)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a ^2*b*c^5)*x), ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^3 + (b^5*c - ...
Leaf count of result is larger than twice the leaf count of optimal. 1924 vs. \(2 (323) = 646\).
Time = 8.11 (sec) , antiderivative size = 1924, normalized size of antiderivative = 6.01 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**4/(c*x**2+b*x+a)**2,x)
Output:
(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6 *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 - 2* b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a* *2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**2*b*c*e**4 - 16*a**2* c**4*(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e** 4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 32*a**2*c**2*d*e**3 + 2*a*b**3 *e**4 + 8*a*b**2*c**3*(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3) *(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3 *(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a*b**2*c*d* e**3 - 12*a*b*c**2*d**2*e**2 - b**4*c**2*(-e**3*(b*e - 2*c*d)/c**3 - sqrt( -(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6 ))) + 4*b**2*c**2*d**3*e - 2*b*c**3*d**4)/(12*a**2*c**2*e**4 - 12*a*b**2*c *e**4 + 24*a*b*c**2*d*e**3 - 24*a*c**3*d**2*e**2 + 2*b**4*e**4 - 4*b**3*c* d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4)) + (-e**3*(b*e - 2*c*d)/c**3 + sqr t(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*a*b**2*c*e**4 + 12*a*b*c**2...
Exception generated. \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^2,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
Time = 0.35 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} - \frac {2 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 12 \, a c^{3} d^{2} e^{2} + 2 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - b^{4} e^{4} + 6 \, a b^{2} c e^{4} - 6 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac {\frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} x}{c} + \frac {b c^{3} d^{4} - 8 \, a c^{3} d^{3} e + 6 \, a b c^{2} d^{2} e^{2} - 4 \, a b^{2} c d e^{3} + 8 \, a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 3 \, a^{2} b c e^{4}}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \] Input:
integrate((e*x+d)^4/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
e^4*x/c^2 - 2*(2*c^4*d^4 - 4*b*c^3*d^3*e + 12*a*c^3*d^2*e^2 + 2*b^3*c*d*e^ 3 - 12*a*b*c^2*d*e^3 - b^4*e^4 + 6*a*b^2*c*e^4 - 6*a^2*c^2*e^4)*arctan((2* c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + (2 *c*d*e^3 - b*e^4)*log(c*x^2 + b*x + a)/c^3 - ((2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*a*c^3*d^2*e^2 - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3 + b^4*e^4 - 4*a*b^2*c*e^4 + 2*a^2*c^2*e^4)*x/c + (b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*a*b^2*c*d*e^3 + 8*a^2*c^2*d*e^3 + a*b^3*e^4 - 3*a ^2*b*c*e^4)/c)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)
Time = 5.86 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {-3\,a^2\,b\,c\,e^4+8\,a^2\,c^2\,d\,e^3+a\,b^3\,e^4-4\,a\,b^2\,c\,d\,e^3+6\,a\,b\,c^2\,d^2\,e^2-8\,a\,c^3\,d^3\,e+b\,c^3\,d^4}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,a^2\,c^2\,e^4-4\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-4\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,b\,c^3\,e^4+256\,d\,a^3\,c^4\,e^3+96\,a^2\,b^3\,c^2\,e^4-192\,d\,a^2\,b^2\,c^3\,e^3-24\,a\,b^5\,c\,e^4+48\,d\,a\,b^4\,c^2\,e^3+2\,b^7\,e^4-4\,d\,b^6\,c\,e^3\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {e^4\,x}{c^2}-\frac {2\,\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (6\,a^2\,c^2\,e^4-6\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-2\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:
int((d + e*x)^4/(a + b*x + c*x^2)^2,x)
Output:
((a*b^3*e^4 + b*c^3*d^4 + 8*a^2*c^2*d*e^3 - 3*a^2*b*c*e^4 - 8*a*c^3*d^3*e - 4*a*b^2*c*d*e^3 + 6*a*b*c^2*d^2*e^2)/(c*(4*a*c - b^2)) + (x*(b^4*e^4 + 2 *c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2* c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(c*(4*a*c - b^2 )))/(a*c^2 + c^3*x^2 + b*c^2*x) + (log(a + b*x + c*x^2)*(2*b^7*e^4 - 128*a ^3*b*c^3*e^4 + 256*a^3*c^4*d*e^3 + 96*a^2*b^3*c^2*e^4 - 24*a*b^5*c*e^4 - 4 *b^6*c*d*e^3 + 48*a*b^4*c^2*d*e^3 - 192*a^2*b^2*c^3*d*e^3))/(2*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) + (e^4*x)/c^2 - (2*atan((2*c* x)/(4*a*c - b^2)^(1/2) - (b^3*c^2 - 4*a*b*c^3)/(c^2*(4*a*c - b^2)^(3/2)))* (b^4*e^4 - 2*c^4*d^4 + 6*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 - 6*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 2*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(c^3*(4*a*c - b^2)^(3/2 ))
Time = 0.21 (sec) , antiderivative size = 2019, normalized size of antiderivative = 6.31 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^4/(c*x^2+b*x+a)^2,x)
Output:
( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**2 *e**4 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* *3*c*e**4 - 24*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 - 12*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 + 24*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 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c *x)/sqrt(4*a*c - b**2))*a**2*b*c**3*e**4*x**2 - 2*sqrt(4*a*c - b**2)*atan( (b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*e**4 + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*d*e**3 + 12*sqrt(4*a*c - b**2)*atan ((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*e**4*x - 24*sqrt(4*a*c - b**2)*a tan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**2*d*e**3*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**2*e**4*x**2 - 8*sqrt (4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**3*d**3*e + 2 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**3*d**2 *e**2*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b** 2*c**3*d*e**3*x**2 + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* *2))*a*b*c**4*d**4 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b **2))*a*b*c**4*d**2*e**2*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt (4*a*c - b**2))*b**6*e**4*x + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 *a*c - b**2))*b**5*c*d*e**3*x - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s...