3.1522 \(\int \frac{(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=396 \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac{c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}-\frac{c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac{c^4 x^4}{2 e^4} \]

[Out]

-((c*(40*c^3*d^3 - 5*b^3*e^3 - 2*c^2*d*e*(35*b*d - 12*a*e) + 3*b*c*e^2*(12*b*d - 5*a*e))*x)/e^7) + (c^2*(20*c^
2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c*e^2)*x^2)/(2*e^6) - (c^3*(8*c*d - 7*b*e)*x^3)/(3*e^5) + (c^4*x^4)/(2*e^
4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^3) - ((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)))/(2*e^8*(d + e*x)^2) + (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)))/(e^8*(d + e*x)) + ((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))*Log[d + e*x])/e^8

________________________________________________________________________________________

Rubi [A]  time = 0.512417, antiderivative size = 396, normalized size of antiderivative = 1., 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} \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac{c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}-\frac{c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac{c^4 x^4}{2 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

-((c*(40*c^3*d^3 - 5*b^3*e^3 - 2*c^2*d*e*(35*b*d - 12*a*e) + 3*b*c*e^2*(12*b*d - 5*a*e))*x)/e^7) + (c^2*(20*c^
2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c*e^2)*x^2)/(2*e^6) - (c^3*(8*c*d - 7*b*e)*x^3)/(3*e^5) + (c^4*x^4)/(2*e^
4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^3) - ((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)))/(2*e^8*(d + e*x)^2) + (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)))/(e^8*(d + e*x)) + ((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))*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)^4} \, dx &=\int \left (\frac{c \left (-40 c^3 d^3+5 b^3 e^3+2 c^2 d e (35 b d-12 a e)-3 b c e^2 (12 b d-5 a e)\right )}{e^7}+\frac{c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x}{e^6}-\frac{c^3 (8 c d-7 b e) x^2}{e^5}+\frac{2 c^4 x^3}{e^4}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^4}+\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 (d+e x)^3}+\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 )}{e^7 (d+e x)^2}+\frac{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 )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (40 c^3 d^3-5 b^3 e^3-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)\right ) x}{e^7}+\frac{c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2}{2 e^6}-\frac{c^3 (8 c d-7 b e) x^3}{3 e^5}+\frac{c^4 x^4}{2 e^4}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\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 )}{2 e^8 (d+e x)^2}+\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 )}{e^8 (d+e x)}+\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 ) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.170926, size = 404, normalized size = 1.02 \[ \frac{\frac{18 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{d+e x}+6 \log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )+3 c^2 e^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )-6 c e x \left (2 c^2 d e (12 a e-35 b d)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )-\frac{3 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+\frac{2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}-2 c^3 e^3 x^3 (8 c d-7 b e)+3 c^4 e^4 x^4}{6 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

(-6*c*e*(40*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(12*b*d - 5*a*e) + 2*c^2*d*e*(-35*b*d + 12*a*e))*x + 3*c^2*e^2*(20
*c^2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c*e^2)*x^2 - 2*c^3*e^3*(8*c*d - 7*b*e)*x^3 + 3*c^4*e^4*x^4 + (2*(2*c*d
- b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (3*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2
+ e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (18*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b
*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x) + 6*(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))*Log[d + e*x])/(6
*e^8)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 1023, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x)

[Out]

1/e^4*ln(e*x+d)*b^4+1/2*c^4*x^4/e^4-60/e^5*ln(e*x+d)*a*b*c^2*d+9/e^3/(e*x+d)^2*a^2*b*c*d-18/e^4/(e*x+d)^2*a*b^
2*c*d^2+30/e^5/(e*x+d)^2*a*b*c^2*d^3+4/e^4/(e*x+d)^3*d^3*a*b^2*c-5/e^5/(e*x+d)^3*a*b*c^2*d^4+36/e^4/(e*x+d)*a*
b^2*c*d-90/e^5/(e*x+d)*a*b*c^2*d^2-3/e^3/(e*x+d)^3*d^2*a^2*b*c-1/3/e/(e*x+d)^3*a^3*b+1/3/e^4/(e*x+d)^3*d^3*b^4
+2/3/e^8/(e*x+d)^3*c^4*d^7-8/3*c^4/e^5*x^3*d+3*c^3/e^4*x^2*a+9/2*c^2/e^4*x^2*b^2+10*c^4/e^6*x^2*d^2+5*c/e^4*b^
3*x-40*c^4/e^7*d^3*x-1/e^2/(e*x+d)^2*a^3*c-3/2/e^2/(e*x+d)^2*a^2*b^2-3/2/e^4/(e*x+d)^2*b^4*d^2-7/e^8/(e*x+d)^2
*c^4*d^6+6/e^4*ln(e*x+d)*a^2*c^2+70/e^8*ln(e*x+d)*c^4*d^4-3/e^3/(e*x+d)*a*b^3+3/e^4/(e*x+d)*b^4*d+42/e^8/(e*x+
d)*c^4*d^5+7/3*c^3/e^4*x^3*b+70*c^3/e^6*b*d^2*x-9/e^4/(e*x+d)^2*a^2*c^2*d^2-36*c^2/e^5*b^2*d*x+60/e^6*ln(e*x+d
)*a*c^3*d^2-20/e^5*ln(e*x+d)*b^3*c*d-24*c^3/e^5*a*d*x-9/e^3/(e*x+d)*a^2*b*c+18/e^4/(e*x+d)*a^2*c^2*d+60/e^6/(e
*x+d)*a*c^3*d^3+12/e^4*ln(e*x+d)*a*b^2*c-140/e^7*ln(e*x+d)*b*c^3*d^3-5/3/e^5/(e*x+d)^3*b^3*c*d^4+90/e^6/(e*x+d
)*b^2*c^2*d^3-30/e^5/(e*x+d)*b^3*c*d^2+90/e^6*ln(e*x+d)*b^2*c^2*d^2-1/e^3/(e*x+d)^3*d^2*a*b^3+2/e^6/(e*x+d)^3*
a*c^3*d^5+2/e^4/(e*x+d)^3*d^3*a^2*c^2+1/e^2/(e*x+d)^3*d*a^2*b^2-105/e^7/(e*x+d)*b*c^3*d^4+2/3/e^2/(e*x+d)^3*d*
a^3*c+15*c^2/e^4*a*b*x-14*c^3/e^5*x^2*b*d-15/e^6/(e*x+d)^2*a*c^3*d^4+10/e^5/(e*x+d)^2*b^3*c*d^3-45/2/e^6/(e*x+
d)^2*b^2*c^2*d^4+21/e^7/(e*x+d)^2*b*c^3*d^5+3/e^6/(e*x+d)^3*b^2*c^2*d^5-7/3/e^7/(e*x+d)^3*b*c^3*d^6+3/e^3/(e*x
+d)^2*a*b^3*d

________________________________________________________________________________________

Maxima [A]  time = 1.18169, size = 905, normalized size = 2.29 \begin{align*} \frac{214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e - 2 \, a^{3} b e^{7} + 141 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 130 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 11 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 6 \,{\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} + 18 \,{\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 10 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 10 \,{\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} -{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 3 \,{\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 105 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 100 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 9 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 6 \,{\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}{6 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac{3 \, c^{4} e^{3} x^{4} - 2 \,{\left (8 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} x^{3} + 3 \,{\left (20 \, c^{4} d^{2} e - 28 \, b c^{3} d e^{2} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{2} - 6 \,{\left (40 \, c^{4} d^{3} - 70 \, b c^{3} d^{2} e + 12 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x}{6 \, e^{7}} + \frac{{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

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)^4,x, algorithm="maxima")

[Out]

1/6*(214*c^4*d^7 - 518*b*c^3*d^6*e - 2*a^3*b*e^7 + 141*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 130*(b^3*c + 3*a*b*c^2)
*d^4*e^3 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - (3*a^2*b^2 + 2*a^3*c)*d
*e^6 + 18*(14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 10*(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 - (a*b^3 + 3*a^2*b*c)*e^7)*x^2 + 3*(154*c^4*d^6*e - 378*b*c^3*d^5*e^
2 + 105*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 100*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2
*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 - (3*a^2*b^2 + 2*a^3*c)*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^
3*e^8) + 1/6*(3*c^4*e^3*x^4 - 2*(8*c^4*d*e^2 - 7*b*c^3*e^3)*x^3 + 3*(20*c^4*d^2*e - 28*b*c^3*d*e^2 + 3*(3*b^2*
c^2 + 2*a*c^3)*e^3)*x^2 - 6*(40*c^4*d^3 - 70*b*c^3*d^2*e + 12*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - 5*(b^3*c + 3*a*b*c
^2)*e^3)*x)/e^7 + (70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*
e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*log(e*x + d)/e^8

________________________________________________________________________________________

Fricas [B]  time = 1.54624, size = 2175, normalized size = 5.49 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x, algorithm="fricas")

[Out]

1/6*(3*c^4*e^7*x^7 + 214*c^4*d^7 - 518*b*c^3*d^6*e - 2*a^3*b*e^7 + 141*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 130*(b^
3*c + 3*a*b*c^2)*d^4*e^3 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - (3*a^2*
b^2 + 2*a^3*c)*d*e^6 - 7*(c^4*d*e^6 - 2*b*c^3*e^7)*x^6 + 3*(7*c^4*d^2*e^5 - 14*b*c^3*d*e^6 + 3*(3*b^2*c^2 + 2*
a*c^3)*e^7)*x^5 - 15*(7*c^4*d^3*e^4 - 14*b*c^3*d^2*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 2*(b^3*c + 3*a*b*c^2)
*e^7)*x^4 - (556*c^4*d^4*e^3 - 1022*b*c^3*d^3*e^4 + 189*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 90*(b^3*c + 3*a*b*c^2)
*d*e^6)*x^3 - 3*(136*c^4*d^5*e^2 - 182*b*c^3*d^4*e^3 + 9*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 + 30*(b^3*c + 3*a*b*c^2
)*d^2*e^5 - 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 + 6*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 + 3*(74*c^4*d^6*e - 238*b*
c^3*d^5*e^2 + 81*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 90*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d^2*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 - (3*a^2*b^2 + 2*a^3*c)*e^7)*x + 6*(70*c^4*d^7 - 140*b*c^3*d^6*e +
30*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 20*(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 + (
70*c^4*d^4*e^3 - 140*b*c^3*d^3*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 20*(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 + 3*(70*c^4*d^5*e^2 - 140*b*c^3*d^4*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 -
20*(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)*x^2 + 3*(70*c^4*d^6*e - 140*b*c^3*d^5*e
^2 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 20*(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)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

________________________________________________________________________________________

Sympy [B]  time = 91.6159, size = 814, normalized size = 2.06 \begin{align*} \frac{c^{4} x^{4}}{2 e^{4}} - \frac{2 a^{3} b e^{7} + 2 a^{3} c d e^{6} + 3 a^{2} b^{2} d e^{6} + 18 a^{2} b c d^{2} e^{5} - 66 a^{2} c^{2} d^{3} e^{4} + 6 a b^{3} d^{2} e^{5} - 132 a b^{2} c d^{3} e^{4} + 390 a b c^{2} d^{4} e^{3} - 282 a c^{3} d^{5} e^{2} - 11 b^{4} d^{3} e^{4} + 130 b^{3} c d^{4} e^{3} - 423 b^{2} c^{2} d^{5} e^{2} + 518 b c^{3} d^{6} e - 214 c^{4} d^{7} + x^{2} \left (54 a^{2} b c e^{7} - 108 a^{2} c^{2} d e^{6} + 18 a b^{3} e^{7} - 216 a b^{2} c d e^{6} + 540 a b c^{2} d^{2} e^{5} - 360 a c^{3} d^{3} e^{4} - 18 b^{4} d e^{6} + 180 b^{3} c d^{2} e^{5} - 540 b^{2} c^{2} d^{3} e^{4} + 630 b c^{3} d^{4} e^{3} - 252 c^{4} d^{5} e^{2}\right ) + x \left (6 a^{3} c e^{7} + 9 a^{2} b^{2} e^{7} + 54 a^{2} b c d e^{6} - 162 a^{2} c^{2} d^{2} e^{5} + 18 a b^{3} d e^{6} - 324 a b^{2} c d^{2} e^{5} + 900 a b c^{2} d^{3} e^{4} - 630 a c^{3} d^{4} e^{3} - 27 b^{4} d^{2} e^{5} + 300 b^{3} c d^{3} e^{4} - 945 b^{2} c^{2} d^{4} e^{3} + 1134 b c^{3} d^{5} e^{2} - 462 c^{4} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} + \frac{x^{3} \left (7 b c^{3} e - 8 c^{4} d\right )}{3 e^{5}} + \frac{x^{2} \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 28 b c^{3} d e + 20 c^{4} d^{2}\right )}{2 e^{6}} + \frac{x \left (15 a b c^{2} e^{3} - 24 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 36 b^{2} c^{2} d e^{2} + 70 b c^{3} d^{2} e - 40 c^{4} d^{3}\right )}{e^{7}} + \frac{\left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right ) \log{\left (d + e x \right )}}{e^{8}} \end{align*}

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)**4,x)

[Out]

c**4*x**4/(2*e**4) - (2*a**3*b*e**7 + 2*a**3*c*d*e**6 + 3*a**2*b**2*d*e**6 + 18*a**2*b*c*d**2*e**5 - 66*a**2*c
**2*d**3*e**4 + 6*a*b**3*d**2*e**5 - 132*a*b**2*c*d**3*e**4 + 390*a*b*c**2*d**4*e**3 - 282*a*c**3*d**5*e**2 -
11*b**4*d**3*e**4 + 130*b**3*c*d**4*e**3 - 423*b**2*c**2*d**5*e**2 + 518*b*c**3*d**6*e - 214*c**4*d**7 + x**2*
(54*a**2*b*c*e**7 - 108*a**2*c**2*d*e**6 + 18*a*b**3*e**7 - 216*a*b**2*c*d*e**6 + 540*a*b*c**2*d**2*e**5 - 360
*a*c**3*d**3*e**4 - 18*b**4*d*e**6 + 180*b**3*c*d**2*e**5 - 540*b**2*c**2*d**3*e**4 + 630*b*c**3*d**4*e**3 - 2
52*c**4*d**5*e**2) + x*(6*a**3*c*e**7 + 9*a**2*b**2*e**7 + 54*a**2*b*c*d*e**6 - 162*a**2*c**2*d**2*e**5 + 18*a
*b**3*d*e**6 - 324*a*b**2*c*d**2*e**5 + 900*a*b*c**2*d**3*e**4 - 630*a*c**3*d**4*e**3 - 27*b**4*d**2*e**5 + 30
0*b**3*c*d**3*e**4 - 945*b**2*c**2*d**4*e**3 + 1134*b*c**3*d**5*e**2 - 462*c**4*d**6*e))/(6*d**3*e**8 + 18*d**
2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3) + x**3*(7*b*c**3*e - 8*c**4*d)/(3*e**5) + x**2*(6*a*c**3*e**2 + 9*b
**2*c**2*e**2 - 28*b*c**3*d*e + 20*c**4*d**2)/(2*e**6) + x*(15*a*b*c**2*e**3 - 24*a*c**3*d*e**2 + 5*b**3*c*e**
3 - 36*b**2*c**2*d*e**2 + 70*b*c**3*d**2*e - 40*c**4*d**3)/e**7 + (6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 60*a*
b*c**2*d*e**3 + 60*a*c**3*d**2*e**2 + b**4*e**4 - 20*b**3*c*d*e**3 + 90*b**2*c**2*d**2*e**2 - 140*b*c**3*d**3*
e + 70*c**4*d**4)*log(d + e*x)/e**8

________________________________________________________________________________________

Giac [A]  time = 1.18968, size = 915, normalized size = 2.31 \begin{align*}{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (3 \, c^{4} x^{4} e^{12} - 16 \, c^{4} d x^{3} e^{11} + 60 \, c^{4} d^{2} x^{2} e^{10} - 240 \, c^{4} d^{3} x e^{9} + 14 \, b c^{3} x^{3} e^{12} - 84 \, b c^{3} d x^{2} e^{11} + 420 \, b c^{3} d^{2} x e^{10} + 27 \, b^{2} c^{2} x^{2} e^{12} + 18 \, a c^{3} x^{2} e^{12} - 216 \, b^{2} c^{2} d x e^{11} - 144 \, a c^{3} d x e^{11} + 30 \, b^{3} c x e^{12} + 90 \, a b c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac{{\left (214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e + 423 \, b^{2} c^{2} d^{5} e^{2} + 282 \, a c^{3} d^{5} e^{2} - 130 \, b^{3} c d^{4} e^{3} - 390 \, a b c^{2} d^{4} e^{3} + 11 \, b^{4} d^{3} e^{4} + 132 \, a b^{2} c d^{3} e^{4} + 66 \, a^{2} c^{2} d^{3} e^{4} - 6 \, a b^{3} d^{2} e^{5} - 18 \, a^{2} b c d^{2} e^{5} - 3 \, a^{2} b^{2} d e^{6} - 2 \, a^{3} c d e^{6} - 2 \, a^{3} b e^{7} + 18 \,{\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 30 \, b^{2} c^{2} d^{3} e^{4} + 20 \, a c^{3} d^{3} e^{4} - 10 \, b^{3} c d^{2} e^{5} - 30 \, a b c^{2} d^{2} e^{5} + b^{4} d e^{6} + 12 \, a b^{2} c d e^{6} + 6 \, a^{2} c^{2} d e^{6} - a b^{3} e^{7} - 3 \, a^{2} b c e^{7}\right )} x^{2} + 3 \,{\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 315 \, b^{2} c^{2} d^{4} e^{3} + 210 \, a c^{3} d^{4} e^{3} - 100 \, b^{3} c d^{3} e^{4} - 300 \, a b c^{2} d^{3} e^{4} + 9 \, b^{4} d^{2} e^{5} + 108 \, a b^{2} c d^{2} e^{5} + 54 \, a^{2} c^{2} d^{2} e^{5} - 6 \, a b^{3} d e^{6} - 18 \, a^{2} b c d e^{6} - 3 \, a^{2} b^{2} e^{7} - 2 \, a^{3} c e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

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)^4,x, algorithm="giac")

[Out]

(70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 + b^
4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*e^(-8)*log(abs(x*e + d)) + 1/6*(3*c^4*x^4*e^12 - 16*c^4*d*x^3*e^11 + 6
0*c^4*d^2*x^2*e^10 - 240*c^4*d^3*x*e^9 + 14*b*c^3*x^3*e^12 - 84*b*c^3*d*x^2*e^11 + 420*b*c^3*d^2*x*e^10 + 27*b
^2*c^2*x^2*e^12 + 18*a*c^3*x^2*e^12 - 216*b^2*c^2*d*x*e^11 - 144*a*c^3*d*x*e^11 + 30*b^3*c*x*e^12 + 90*a*b*c^2
*x*e^12)*e^(-16) + 1/6*(214*c^4*d^7 - 518*b*c^3*d^6*e + 423*b^2*c^2*d^5*e^2 + 282*a*c^3*d^5*e^2 - 130*b^3*c*d^
4*e^3 - 390*a*b*c^2*d^4*e^3 + 11*b^4*d^3*e^4 + 132*a*b^2*c*d^3*e^4 + 66*a^2*c^2*d^3*e^4 - 6*a*b^3*d^2*e^5 - 18
*a^2*b*c*d^2*e^5 - 3*a^2*b^2*d*e^6 - 2*a^3*c*d*e^6 - 2*a^3*b*e^7 + 18*(14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 30*
b^2*c^2*d^3*e^4 + 20*a*c^3*d^3*e^4 - 10*b^3*c*d^2*e^5 - 30*a*b*c^2*d^2*e^5 + b^4*d*e^6 + 12*a*b^2*c*d*e^6 + 6*
a^2*c^2*d*e^6 - a*b^3*e^7 - 3*a^2*b*c*e^7)*x^2 + 3*(154*c^4*d^6*e - 378*b*c^3*d^5*e^2 + 315*b^2*c^2*d^4*e^3 +
210*a*c^3*d^4*e^3 - 100*b^3*c*d^3*e^4 - 300*a*b*c^2*d^3*e^4 + 9*b^4*d^2*e^5 + 108*a*b^2*c*d^2*e^5 + 54*a^2*c^2
*d^2*e^5 - 6*a*b^3*d*e^6 - 18*a^2*b*c*d*e^6 - 3*a^2*b^2*e^7 - 2*a^3*c*e^7)*x)*e^(-8)/(x*e + d)^3