3.14.27 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=399 \[ \frac {(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac {3 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {5 c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{4 e^8}-\frac {3 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8}+\frac {x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {7 c^3 (d+e x)^6 (2 c d-b e)}{6 e^8}+\frac {2 c^4 (d+e x)^7}{7 e^8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.58, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \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 )}{3 e^8}+\frac {3 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 )}{5 e^8}-\frac {5 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 )}{4 e^8}-\frac {3 (d+e x)^2 (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 )}{2 e^8}+\frac {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^7}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {7 c^3 (d+e x)^6 (2 c d-b e)}{6 e^8}+\frac {2 c^4 (d+e x)^7}{7 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((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))*x)/e^7 - (3*(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)^2)/(2*e^8) + ((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)^3)/(3*e^8) - (5*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)^4)/(4*e^8) + (3*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^5)/(5*e^8) - (7*c^3*(2*c*d - b*e)*(d + e*x)^6)/(6*e^8
) + (2*c^4*(d + e*x)^7)/(7*e^8) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^8

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx &=\int \left (\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 )}{e^7}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)}{e^7}+\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)^2}{e^7}+\frac {5 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)^3}{e^7}+\frac {3 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)^4}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^5}{e^7}+\frac {2 c^4 (d+e x)^6}{e^7}\right ) \, dx\\ &=\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 ) x}{e^7}-\frac {3 (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)^2}{2 e^8}+\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)^3}{3 e^8}-\frac {5 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)^4}{4 e^8}+\frac {3 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)^5}{5 e^8}-\frac {7 c^3 (2 c d-b e) (d+e x)^6}{6 e^8}+\frac {2 c^4 (d+e x)^7}{7 e^8}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 483, normalized size = 1.21 \begin {gather*} \frac {e x \left (21 c^2 e^2 \left (20 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+3 b^2 \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 )+70 b^2 e^4 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+35 c e^3 \left (24 a^3 e^3+54 a^2 b e^2 (e x-2 d)+24 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )+7 c^3 e \left (6 a 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 )-7 b \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 )+2 c^4 \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 )-420 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{420 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(2*c^4*(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) + 70*b^2*e^4*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 35*c*e^3*(24*a^3
*e^3 + 54*a^2*b*e^2*(-2*d + e*x) + 24*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) - 5*b^3*(12*d^3 - 6*d^2*e*x + 4*d*
e^2*x^2 - 3*e^3*x^3)) + 21*c^2*e^2*(20*a^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b*e*(-12*d^3 + 6*d^2*e*x -
4*d*e^2*x^2 + 3*e^3*x^3) + 3*b^2*(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)) + 7*c^3*
e*(6*a*e*(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) - 7*b*(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))) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*Log
[d + e*x])/(420*e^8)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x),x]

[Out]

IntegrateAlgebraic[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x), x]

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 645, normalized size = 1.62 \begin {gather*} \frac {120 \, c^{4} e^{7} x^{7} - 70 \, {\left (2 \, c^{4} d e^{6} - 7 \, b c^{3} e^{7}\right )} x^{6} + 84 \, {\left (2 \, c^{4} d^{2} e^{5} - 7 \, b c^{3} d e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 105 \, {\left (2 \, c^{4} d^{3} e^{4} - 7 \, b c^{3} d^{2} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 140 \, {\left (2 \, c^{4} d^{4} e^{3} - 7 \, b c^{3} d^{3} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 210 \, {\left (2 \, c^{4} d^{5} e^{2} - 7 \, b c^{3} d^{4} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 420 \, {\left (2 \, c^{4} d^{6} e - 7 \, b c^{3} d^{5} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x - 420 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(120*c^4*e^7*x^7 - 70*(2*c^4*d*e^6 - 7*b*c^3*e^7)*x^6 + 84*(2*c^4*d^2*e^5 - 7*b*c^3*d*e^6 + 3*(3*b^2*c^2
+ 2*a*c^3)*e^7)*x^5 - 105*(2*c^4*d^3*e^4 - 7*b*c^3*d^2*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 5*(b^3*c + 3*a*b
*c^2)*e^7)*x^4 + 140*(2*c^4*d^4*e^3 - 7*b*c^3*d^3*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 5*(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^3 - 210*(2*c^4*d^5*e^2 - 7*b*c^3*d^4*e^3 + 3*(3*b^2*c^2 + 2*a*
c^3)*d^3*e^4 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*e^
7)*x^2 + 420*(2*c^4*d^6*e - 7*b*c^3*d^5*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2*b^2 + 2*a^3*c)*e^7)*x - 420*(2
*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*log(e*x + d))
/e^8

________________________________________________________________________________________

giac [A]  time = 0.17, size = 742, normalized size = 1.86 \begin {gather*} -{\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 9 \, b^{2} c^{2} d^{5} e^{2} + 6 \, a c^{3} d^{5} e^{2} - 5 \, b^{3} c d^{4} e^{3} - 15 \, a b c^{2} d^{4} e^{3} + b^{4} d^{3} e^{4} + 12 \, a b^{2} c d^{3} e^{4} + 6 \, a^{2} c^{2} d^{3} e^{4} - 3 \, a b^{3} d^{2} e^{5} - 9 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} - a^{3} b e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (120 \, c^{4} x^{7} e^{6} - 140 \, c^{4} d x^{6} e^{5} + 168 \, c^{4} d^{2} x^{5} e^{4} - 210 \, c^{4} d^{3} x^{4} e^{3} + 280 \, c^{4} d^{4} x^{3} e^{2} - 420 \, c^{4} d^{5} x^{2} e + 840 \, c^{4} d^{6} x + 490 \, b c^{3} x^{6} e^{6} - 588 \, b c^{3} d x^{5} e^{5} + 735 \, b c^{3} d^{2} x^{4} e^{4} - 980 \, b c^{3} d^{3} x^{3} e^{3} + 1470 \, b c^{3} d^{4} x^{2} e^{2} - 2940 \, b c^{3} d^{5} x e + 756 \, b^{2} c^{2} x^{5} e^{6} + 504 \, a c^{3} x^{5} e^{6} - 945 \, b^{2} c^{2} d x^{4} e^{5} - 630 \, a c^{3} d x^{4} e^{5} + 1260 \, b^{2} c^{2} d^{2} x^{3} e^{4} + 840 \, a c^{3} d^{2} x^{3} e^{4} - 1890 \, b^{2} c^{2} d^{3} x^{2} e^{3} - 1260 \, a c^{3} d^{3} x^{2} e^{3} + 3780 \, b^{2} c^{2} d^{4} x e^{2} + 2520 \, a c^{3} d^{4} x e^{2} + 525 \, b^{3} c x^{4} e^{6} + 1575 \, a b c^{2} x^{4} e^{6} - 700 \, b^{3} c d x^{3} e^{5} - 2100 \, a b c^{2} d x^{3} e^{5} + 1050 \, b^{3} c d^{2} x^{2} e^{4} + 3150 \, a b c^{2} d^{2} x^{2} e^{4} - 2100 \, b^{3} c d^{3} x e^{3} - 6300 \, a b c^{2} d^{3} x e^{3} + 140 \, b^{4} x^{3} e^{6} + 1680 \, a b^{2} c x^{3} e^{6} + 840 \, a^{2} c^{2} x^{3} e^{6} - 210 \, b^{4} d x^{2} e^{5} - 2520 \, a b^{2} c d x^{2} e^{5} - 1260 \, a^{2} c^{2} d x^{2} e^{5} + 420 \, b^{4} d^{2} x e^{4} + 5040 \, a b^{2} c d^{2} x e^{4} + 2520 \, a^{2} c^{2} d^{2} x e^{4} + 630 \, a b^{3} x^{2} e^{6} + 1890 \, a^{2} b c x^{2} e^{6} - 1260 \, a b^{3} d x e^{5} - 3780 \, a^{2} b c d x e^{5} + 1260 \, a^{2} b^{2} x e^{6} + 840 \, a^{3} c x e^{6}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c^4*d^7 - 7*b*c^3*d^6*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e^3 + b^4
*d^3*e^4 + 12*a*b^2*c*d^3*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e^6 + 2*
a^3*c*d*e^6 - a^3*b*e^7)*e^(-8)*log(abs(x*e + d)) + 1/420*(120*c^4*x^7*e^6 - 140*c^4*d*x^6*e^5 + 168*c^4*d^2*x
^5*e^4 - 210*c^4*d^3*x^4*e^3 + 280*c^4*d^4*x^3*e^2 - 420*c^4*d^5*x^2*e + 840*c^4*d^6*x + 490*b*c^3*x^6*e^6 - 5
88*b*c^3*d*x^5*e^5 + 735*b*c^3*d^2*x^4*e^4 - 980*b*c^3*d^3*x^3*e^3 + 1470*b*c^3*d^4*x^2*e^2 - 2940*b*c^3*d^5*x
*e + 756*b^2*c^2*x^5*e^6 + 504*a*c^3*x^5*e^6 - 945*b^2*c^2*d*x^4*e^5 - 630*a*c^3*d*x^4*e^5 + 1260*b^2*c^2*d^2*
x^3*e^4 + 840*a*c^3*d^2*x^3*e^4 - 1890*b^2*c^2*d^3*x^2*e^3 - 1260*a*c^3*d^3*x^2*e^3 + 3780*b^2*c^2*d^4*x*e^2 +
2520*a*c^3*d^4*x*e^2 + 525*b^3*c*x^4*e^6 + 1575*a*b*c^2*x^4*e^6 - 700*b^3*c*d*x^3*e^5 - 2100*a*b*c^2*d*x^3*e^
5 + 1050*b^3*c*d^2*x^2*e^4 + 3150*a*b*c^2*d^2*x^2*e^4 - 2100*b^3*c*d^3*x*e^3 - 6300*a*b*c^2*d^3*x*e^3 + 140*b^
4*x^3*e^6 + 1680*a*b^2*c*x^3*e^6 + 840*a^2*c^2*x^3*e^6 - 210*b^4*d*x^2*e^5 - 2520*a*b^2*c*d*x^2*e^5 - 1260*a^2
*c^2*d*x^2*e^5 + 420*b^4*d^2*x*e^4 + 5040*a*b^2*c*d^2*x*e^4 + 2520*a^2*c^2*d^2*x*e^4 + 630*a*b^3*x^2*e^6 + 189
0*a^2*b*c*x^2*e^6 - 1260*a*b^3*d*x*e^5 - 3780*a^2*b*c*d*x*e^5 + 1260*a^2*b^2*x*e^6 + 840*a^3*c*x*e^6)*e^(-7)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 872, normalized size = 2.19 \begin {gather*} \frac {2 c^{4} x^{7}}{7 e}+\frac {7 b \,c^{3} x^{6}}{6 e}-\frac {c^{4} d \,x^{6}}{3 e^{2}}+\frac {6 a \,c^{3} x^{5}}{5 e}+\frac {9 b^{2} c^{2} x^{5}}{5 e}-\frac {7 b \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {2 c^{4} d^{2} x^{5}}{5 e^{3}}+\frac {15 a b \,c^{2} x^{4}}{4 e}-\frac {3 a \,c^{3} d \,x^{4}}{2 e^{2}}+\frac {5 b^{3} c \,x^{4}}{4 e}-\frac {9 b^{2} c^{2} d \,x^{4}}{4 e^{2}}+\frac {7 b \,c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {c^{4} d^{3} x^{4}}{2 e^{4}}+\frac {2 a^{2} c^{2} x^{3}}{e}+\frac {4 a \,b^{2} c \,x^{3}}{e}-\frac {5 a b \,c^{2} d \,x^{3}}{e^{2}}+\frac {2 a \,c^{3} d^{2} x^{3}}{e^{3}}+\frac {b^{4} x^{3}}{3 e}-\frac {5 b^{3} c d \,x^{3}}{3 e^{2}}+\frac {3 b^{2} c^{2} d^{2} x^{3}}{e^{3}}-\frac {7 b \,c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {2 c^{4} d^{4} x^{3}}{3 e^{5}}+\frac {9 a^{2} b c \,x^{2}}{2 e}-\frac {3 a^{2} c^{2} d \,x^{2}}{e^{2}}+\frac {3 a \,b^{3} x^{2}}{2 e}-\frac {6 a \,b^{2} c d \,x^{2}}{e^{2}}+\frac {15 a b \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {3 a \,c^{3} d^{3} x^{2}}{e^{4}}-\frac {b^{4} d \,x^{2}}{2 e^{2}}+\frac {5 b^{3} c \,d^{2} x^{2}}{2 e^{3}}-\frac {9 b^{2} c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {7 b \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {c^{4} d^{5} x^{2}}{e^{6}}+\frac {a^{3} b \ln \left (e x +d \right )}{e}-\frac {2 a^{3} c d \ln \left (e x +d \right )}{e^{2}}+\frac {2 a^{3} c x}{e}-\frac {3 a^{2} b^{2} d \ln \left (e x +d \right )}{e^{2}}+\frac {3 a^{2} b^{2} x}{e}+\frac {9 a^{2} b c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {9 a^{2} b c d x}{e^{2}}-\frac {6 a^{2} c^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {6 a^{2} c^{2} d^{2} x}{e^{3}}+\frac {3 a \,b^{3} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 a \,b^{3} d x}{e^{2}}-\frac {12 a \,b^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {12 a \,b^{2} c \,d^{2} x}{e^{3}}+\frac {15 a b \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {15 a b \,c^{2} d^{3} x}{e^{4}}-\frac {6 a \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {6 a \,c^{3} d^{4} x}{e^{5}}-\frac {b^{4} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b^{4} d^{2} x}{e^{3}}+\frac {5 b^{3} c \,d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {5 b^{3} c \,d^{3} x}{e^{4}}-\frac {9 b^{2} c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {9 b^{2} c^{2} d^{4} x}{e^{5}}+\frac {7 b \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {7 b \,c^{3} d^{5} x}{e^{6}}-\frac {2 c^{4} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {2 c^{4} d^{6} x}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d),x)

[Out]

-15/e^4*x*a*b*c^2*d^3+12/e^3*x*a*b^2*c*d^2-9/e^2*x*a^2*b*c*d+15/2/e^3*x^2*a*b*c^2*d^2-6/e^2*x^2*a*b^2*c*d+15/e
^5*ln(e*x+d)*a*b*c^2*d^4-12/e^4*ln(e*x+d)*a*b^2*c*d^3+9/e^3*ln(e*x+d)*a^2*b*c*d^2-5/e^2*x^3*a*b*c^2*d+2/e^7*x*
c^4*d^6+1/e*ln(e*x+d)*a^3*b-1/e^4*ln(e*x+d)*b^4*d^3-2/e^8*ln(e*x+d)*c^4*d^7+7/6/e*x^6*b*c^3-1/3/e^2*x^6*c^4*d+
1/e^3*x*b^4*d^2+2/5/e^3*x^5*c^4*d^2+6/5/e*x^5*a*c^3+9/5/e*x^5*b^2*c^2-1/2/e^2*x^2*b^4*d+3/2/e*x^2*a*b^3+2/3/e^
5*x^3*c^4*d^4+2/e*x^3*a^2*c^2-1/2/e^4*x^4*c^4*d^3+5/4/e*x^4*b^3*c+2/e*x*a^3*c+3/e*x*a^2*b^2-1/e^6*x^2*c^4*d^5-
6/e^6*ln(e*x+d)*a*c^3*d^5+5/e^5*ln(e*x+d)*b^3*c*d^4-9/e^6*ln(e*x+d)*b^2*c^2*d^5+7/e^7*ln(e*x+d)*b*c^3*d^6-3/e^
2*x^2*a^2*c^2*d+2/e^3*x^3*a*c^3*d^2+3/e^3*ln(e*x+d)*a*b^3*d^2-3/e^4*x^2*a*c^3*d^3+5/2/e^3*x^2*b^3*c*d^2+6/e^3*
x*a^2*c^2*d^2+7/2/e^5*x^2*b*c^3*d^4-3/e^2*x*a*b^3*d+3/e^3*x^3*b^2*c^2*d^2-7/3/e^4*x^3*b*c^3*d^3+9/2/e*x^2*a^2*
b*c+6/e^5*x*a*c^3*d^4-5/e^4*x*b^3*c*d^3-2/e^2*ln(e*x+d)*a^3*c*d-3/e^2*ln(e*x+d)*a^2*b^2*d-6/e^4*ln(e*x+d)*a^2*
c^2*d^3-9/2/e^4*x^2*b^2*c^2*d^3+4/e*x^3*a*b^2*c-9/4/e^2*x^4*b^2*c^2*d+7/4/e^3*x^4*b*c^3*d^2+15/4/e*x^4*a*b*c^2
-3/2/e^2*x^4*a*c^3*d-7/5/e^2*x^5*b*c^3*d-5/3/e^2*x^3*b^3*c*d-7/e^6*x*b*c^3*d^5+9/e^5*x*b^2*c^2*d^4+2/7/e*c^4*x
^7+1/3/e*x^3*b^4

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 644, normalized size = 1.61 \begin {gather*} \frac {120 \, c^{4} e^{6} x^{7} - 70 \, {\left (2 \, c^{4} d e^{5} - 7 \, b c^{3} e^{6}\right )} x^{6} + 84 \, {\left (2 \, c^{4} d^{2} e^{4} - 7 \, b c^{3} d e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{6}\right )} x^{5} - 105 \, {\left (2 \, c^{4} d^{3} e^{3} - 7 \, b c^{3} d^{2} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{5} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{6}\right )} x^{4} + 140 \, {\left (2 \, c^{4} d^{4} e^{2} - 7 \, b c^{3} d^{3} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{4} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{6}\right )} x^{3} - 210 \, {\left (2 \, c^{4} d^{5} e - 7 \, b c^{3} d^{4} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{5} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{6}\right )} x^{2} + 420 \, {\left (2 \, c^{4} d^{6} - 7 \, b c^{3} d^{5} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(120*c^4*e^6*x^7 - 70*(2*c^4*d*e^5 - 7*b*c^3*e^6)*x^6 + 84*(2*c^4*d^2*e^4 - 7*b*c^3*d*e^5 + 3*(3*b^2*c^2
+ 2*a*c^3)*e^6)*x^5 - 105*(2*c^4*d^3*e^3 - 7*b*c^3*d^2*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^5 - 5*(b^3*c + 3*a*b
*c^2)*e^6)*x^4 + 140*(2*c^4*d^4*e^2 - 7*b*c^3*d^3*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^4 - 5*(b^3*c + 3*a*b*c^2
)*d*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^6)*x^3 - 210*(2*c^4*d^5*e - 7*b*c^3*d^4*e^2 + 3*(3*b^2*c^2 + 2*a*c^
3)*d^3*e^3 - 5*(b^3*c + 3*a*b*c^2)*d^2*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^5 - 3*(a*b^3 + 3*a^2*b*c)*e^6)
*x^2 + 420*(2*c^4*d^6 - 7*b*c^3*d^5*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^3*e^3 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*x)/e^7 - (2*c^4*
d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*log(e*x + d)/e^8

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 697, normalized size = 1.75 \begin {gather*} x^3\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{3\,e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{3\,e}\right )-x^4\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{4\,e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{2\,e}-\frac {3\,a\,b\,\left (b^2+3\,a\,c\right )}{2\,e}\right )+x^6\,\left (\frac {7\,b\,c^3}{6\,e}-\frac {c^4\,d}{3\,e^2}\right )+x\,\left (\frac {2\,c\,a^3+3\,a^2\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{e}-\frac {3\,a\,b\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )+x^5\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{5\,e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{5\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6-9\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4-3\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-15\,a\,b\,c^2\,d^4\,e^3+6\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-5\,b^3\,c\,d^4\,e^3+9\,b^2\,c^2\,d^5\,e^2-7\,b\,c^3\,d^6\,e+2\,c^4\,d^7\right )}{e^8}+\frac {2\,c^4\,x^7}{7\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x),x)

[Out]

x^3*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(3*e) + (d*((d*((6*a*c^3 + 9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2)
)/e))/e - (5*b*c*(3*a*c + b^2))/e))/(3*e)) - x^4*((d*((6*a*c^3 + 9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^
2))/e))/(4*e) - (5*b*c*(3*a*c + b^2))/(4*e)) - x^2*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((6*a*c^3 +
9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2))/e))/e - (5*b*c*(3*a*c + b^2))/e))/e))/(2*e) - (3*a*b*(3*a*c +
b^2))/(2*e)) + x^6*((7*b*c^3)/(6*e) - (c^4*d)/(3*e^2)) + x*((2*a^3*c + 3*a^2*b^2)/e + (d*((d*((b^4 + 6*a^2*c^
2 + 12*a*b^2*c)/e + (d*((d*((6*a*c^3 + 9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2))/e))/e - (5*b*c*(3*a*c
+ b^2))/e))/e))/e - (3*a*b*(3*a*c + b^2))/e))/e) + x^5*((6*a*c^3 + 9*b^2*c^2)/(5*e) - (d*((7*b*c^3)/e - (2*c^4
*d)/e^2))/(5*e)) - (log(d + e*x)*(2*c^4*d^7 - a^3*b*e^7 + b^4*d^3*e^4 - 3*a*b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 + 6*
a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 + 6*a^2*c^2*d^3*e^4 + 9*b^2*c^2*d^5*e^2 + 2*a^3*c*d*e^6 - 7*b*c^3*d^6*e - 15*a
*b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 - 9*a^2*b*c*d^2*e^5))/e^8 + (2*c^4*x^7)/(7*e)

________________________________________________________________________________________

sympy [A]  time = 1.36, size = 641, normalized size = 1.61 \begin {gather*} \frac {2 c^{4} x^{7}}{7 e} + x^{6} \left (\frac {7 b c^{3}}{6 e} - \frac {c^{4} d}{3 e^{2}}\right ) + x^{5} \left (\frac {6 a c^{3}}{5 e} + \frac {9 b^{2} c^{2}}{5 e} - \frac {7 b c^{3} d}{5 e^{2}} + \frac {2 c^{4} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {15 a b c^{2}}{4 e} - \frac {3 a c^{3} d}{2 e^{2}} + \frac {5 b^{3} c}{4 e} - \frac {9 b^{2} c^{2} d}{4 e^{2}} + \frac {7 b c^{3} d^{2}}{4 e^{3}} - \frac {c^{4} d^{3}}{2 e^{4}}\right ) + x^{3} \left (\frac {2 a^{2} c^{2}}{e} + \frac {4 a b^{2} c}{e} - \frac {5 a b c^{2} d}{e^{2}} + \frac {2 a c^{3} d^{2}}{e^{3}} + \frac {b^{4}}{3 e} - \frac {5 b^{3} c d}{3 e^{2}} + \frac {3 b^{2} c^{2} d^{2}}{e^{3}} - \frac {7 b c^{3} d^{3}}{3 e^{4}} + \frac {2 c^{4} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {9 a^{2} b c}{2 e} - \frac {3 a^{2} c^{2} d}{e^{2}} + \frac {3 a b^{3}}{2 e} - \frac {6 a b^{2} c d}{e^{2}} + \frac {15 a b c^{2} d^{2}}{2 e^{3}} - \frac {3 a c^{3} d^{3}}{e^{4}} - \frac {b^{4} d}{2 e^{2}} + \frac {5 b^{3} c d^{2}}{2 e^{3}} - \frac {9 b^{2} c^{2} d^{3}}{2 e^{4}} + \frac {7 b c^{3} d^{4}}{2 e^{5}} - \frac {c^{4} d^{5}}{e^{6}}\right ) + x \left (\frac {2 a^{3} c}{e} + \frac {3 a^{2} b^{2}}{e} - \frac {9 a^{2} b c d}{e^{2}} + \frac {6 a^{2} c^{2} d^{2}}{e^{3}} - \frac {3 a b^{3} d}{e^{2}} + \frac {12 a b^{2} c d^{2}}{e^{3}} - \frac {15 a b c^{2} d^{3}}{e^{4}} + \frac {6 a c^{3} d^{4}}{e^{5}} + \frac {b^{4} d^{2}}{e^{3}} - \frac {5 b^{3} c d^{3}}{e^{4}} + \frac {9 b^{2} c^{2} d^{4}}{e^{5}} - \frac {7 b c^{3} d^{5}}{e^{6}} + \frac {2 c^{4} d^{6}}{e^{7}}\right ) + \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d),x)

[Out]

2*c**4*x**7/(7*e) + x**6*(7*b*c**3/(6*e) - c**4*d/(3*e**2)) + x**5*(6*a*c**3/(5*e) + 9*b**2*c**2/(5*e) - 7*b*c
**3*d/(5*e**2) + 2*c**4*d**2/(5*e**3)) + x**4*(15*a*b*c**2/(4*e) - 3*a*c**3*d/(2*e**2) + 5*b**3*c/(4*e) - 9*b*
*2*c**2*d/(4*e**2) + 7*b*c**3*d**2/(4*e**3) - c**4*d**3/(2*e**4)) + x**3*(2*a**2*c**2/e + 4*a*b**2*c/e - 5*a*b
*c**2*d/e**2 + 2*a*c**3*d**2/e**3 + b**4/(3*e) - 5*b**3*c*d/(3*e**2) + 3*b**2*c**2*d**2/e**3 - 7*b*c**3*d**3/(
3*e**4) + 2*c**4*d**4/(3*e**5)) + x**2*(9*a**2*b*c/(2*e) - 3*a**2*c**2*d/e**2 + 3*a*b**3/(2*e) - 6*a*b**2*c*d/
e**2 + 15*a*b*c**2*d**2/(2*e**3) - 3*a*c**3*d**3/e**4 - b**4*d/(2*e**2) + 5*b**3*c*d**2/(2*e**3) - 9*b**2*c**2
*d**3/(2*e**4) + 7*b*c**3*d**4/(2*e**5) - c**4*d**5/e**6) + x*(2*a**3*c/e + 3*a**2*b**2/e - 9*a**2*b*c*d/e**2
+ 6*a**2*c**2*d**2/e**3 - 3*a*b**3*d/e**2 + 12*a*b**2*c*d**2/e**3 - 15*a*b*c**2*d**3/e**4 + 6*a*c**3*d**4/e**5
+ b**4*d**2/e**3 - 5*b**3*c*d**3/e**4 + 9*b**2*c**2*d**4/e**5 - 7*b*c**3*d**5/e**6 + 2*c**4*d**6/e**7) + (b*e
- 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**8

________________________________________________________________________________________