3.1520 \(\int \frac{(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.56765, 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} \[ \frac{(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^8}+\frac{3 c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac{5 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}-\frac{3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^7}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)}-\frac{7 c^3 (d+e x)^5 (2 c d-b e)}{5 e^8}+\frac{c^4 (d+e x)^6}{3 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.274174, size = 637, normalized size = 1.61 \[ \frac{15 c^2 e^2 \left (12 a^2 e^2 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+20 a b e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+3 b^2 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )+20 c e^3 \left (27 a^2 b e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a^3 d e^3+18 a b^2 e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )-5 b^3 \left (-6 d^2 e^2 x^2-9 d^3 e x+3 d^4+2 d e^3 x^3-e^4 x^4\right )\right )+30 b e^4 \left (6 a^2 b d e^2-2 a^3 e^3+6 a b^2 e \left (-d^2+d e x+e^2 x^2\right )+b^3 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )+60 (d+e x) \log (d+e x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+6 c^3 e \left (5 a e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )-7 b \left (-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4-50 d^5 e x+10 d^6+3 d e^5 x^5-2 e^6 x^6\right )\right )+2 c^4 \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )}{60 e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^4*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x
^6 + 10*e^7*x^7) + 30*b*e^4*(6*a^2*b*d*e^2 - 2*a^3*e^3 + 6*a*b^2*e*(-d^2 + d*e*x + e^2*x^2) + b^3*(2*d^3 - 4*d
^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 20*c*e^3*(6*a^3*d*e^3 + 27*a^2*b*e^2*(-d^2 + d*e*x + e^2*x^2) + 18*a*b^2*e*
(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) - 5*b^3*(3*d^4 - 9*d^3*e*x - 6*d^2*e^2*x^2 + 2*d*e^3*x^3 - e^4*x^4
)) + 15*c^2*e^2*(12*a^2*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 20*a*b*e*(-3*d^4 + 9*d^3*e*x + 6*d^2
*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 3*b^2*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4
+ 3*e^5*x^5)) + 6*c^3*e*(5*a*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x
^5) - 7*b*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) +
60*(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)*Log[d + e*x])/(60*e
^8*(d + e*x))

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Maple [B]  time = 0.013, size = 930, normalized size = 2.4 \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)^2,x)

[Out]

-15/e^5/(e*x+d)*a*b*c^2*d^4-15/e^3*x^2*a*b*c^2*d-18/e^3*ln(e*x+d)*a^2*b*c*d+36/e^4*ln(e*x+d)*a*b^2*c*d^2-60/e^
5*ln(e*x+d)*a*b*c^2*d^3+3/e^2*ln(e*x+d)*a^2*b^2+3/e^4*ln(e*x+d)*b^4*d^2+14/e^8*ln(e*x+d)*c^4*d^6+5/e^6*x^2*c^4
*d^4+3/e^2*a*b^3*x-2/e^3*b^4*d*x-4/5/e^3*x^5*c^4*d+3/2/e^2*x^4*a*c^3-12/e^7*c^4*d^5*x+2/e^2*ln(e*x+d)*a^3*c+9/
4/e^2*x^4*b^2*c^2+3/2/e^4*x^4*c^4*d^2+7/5/e^2*x^5*b*c^3-8/3/e^5*x^3*c^4*d^3+3/e^2*x^2*a^2*c^2-24/e^3*a*b^2*c*d
*x+45/e^4*a*b*c^2*d^2*x-9/e^3/(e*x+d)*a^2*b*c*d^2+12/e^4/(e*x+d)*a*b^2*c*d^3-1/e/(e*x+d)*a^3*b+1/e^4/(e*x+d)*b
^4*d^3+2/e^8/(e*x+d)*c^4*d^7+5/3/e^2*x^3*b^3*c+1/2/e^2*x^2*b^4+1/3/e^2*c^4*x^6-24/e^5*a*c^3*d^3*x+15/e^4*b^3*c
*d^2*x+7/e^4*x^3*b*c^3*d^2-4/e^3*x^3*a*c^3*d+5/e^2*x^3*a*b*c^2-7/2/e^3*x^4*b*c^3*d+18/e^4*ln(e*x+d)*a^2*c^2*d^
2-6/e^3*ln(e*x+d)*a*b^3*d+30/e^6*ln(e*x+d)*a*c^3*d^4-20/e^5*ln(e*x+d)*b^3*c*d^3-36/e^5*b^2*c^2*d^3*x+35/e^6*b*
d^4*c^3*x+9/e^2*a^2*b*c*x-12/e^3*a^2*c^2*d*x-14/e^5*x^2*b*c^3*d^3+9/e^4*x^2*a*c^3*d^2-5/e^3*x^2*b^3*c*d+6/e^4/
(e*x+d)*a^2*c^2*d^3-3/e^3/(e*x+d)*a*b^3*d^2+45/e^6*ln(e*x+d)*b^2*c^2*d^4+27/2/e^4*x^2*b^2*c^2*d^2+6/e^2*x^2*a*
b^2*c-6/e^3*x^3*b^2*c^2*d+2/e^2/(e*x+d)*d*a^3*c+3/e^2/(e*x+d)*d*a^2*b^2-42/e^7*ln(e*x+d)*b*c^3*d^5-7/e^7/(e*x+
d)*b*c^3*d^6+6/e^6/(e*x+d)*a*c^3*d^5-5/e^5/(e*x+d)*b^3*c*d^4+9/e^6/(e*x+d)*b^2*c^2*d^5

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Maxima [A]  time = 1.07267, size = 876, normalized size = 2.21 \begin{align*} \frac{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}}{e^{9} x + d e^{8}} + \frac{20 \, c^{4} e^{5} x^{6} - 12 \,{\left (4 \, c^{4} d e^{4} - 7 \, b c^{3} e^{5}\right )} x^{5} + 15 \,{\left (6 \, c^{4} d^{2} e^{3} - 14 \, b c^{3} d e^{4} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{5}\right )} x^{4} - 20 \,{\left (8 \, c^{4} d^{3} e^{2} - 21 \, b c^{3} d^{2} e^{3} + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{4} - 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{5}\right )} x^{3} + 30 \,{\left (10 \, c^{4} d^{4} e - 28 \, b c^{3} d^{3} e^{2} + 9 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{3} - 10 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{4} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{5}\right )} x^{2} - 60 \,{\left (12 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 12 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 15 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + 2 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} x}{60 \, e^{7}} + \frac{{\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \,{\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 )} \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)^2,x, algorithm="maxima")

[Out]

(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)/(e^9*x + d*
e^8) + 1/60*(20*c^4*e^5*x^6 - 12*(4*c^4*d*e^4 - 7*b*c^3*e^5)*x^5 + 15*(6*c^4*d^2*e^3 - 14*b*c^3*d*e^4 + 3*(3*b
^2*c^2 + 2*a*c^3)*e^5)*x^4 - 20*(8*c^4*d^3*e^2 - 21*b*c^3*d^2*e^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^4 - 5*(b^3*c +
3*a*b*c^2)*e^5)*x^3 + 30*(10*c^4*d^4*e - 28*b*c^3*d^3*e^2 + 9*(3*b^2*c^2 + 2*a*c^3)*d^2*e^3 - 10*(b^3*c + 3*a
*b*c^2)*d*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^5)*x^2 - 60*(12*c^4*d^5 - 35*b*c^3*d^4*e + 12*(3*b^2*c^2 + 2*
a*c^3)*d^3*e^2 - 15*(b^3*c + 3*a*b*c^2)*d^2*e^3 + 2*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - 3*(a*b^3 + 3*a^2*b*
c)*e^5)*x)/e^7 + (14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*
e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*log(
e*x + d)/e^8

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Fricas [B]  time = 1.49143, size = 1921, normalized size = 4.85 \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)^2,x, algorithm="fricas")

[Out]

1/60*(20*c^4*e^7*x^7 + 120*c^4*d^7 - 420*b*c^3*d^6*e - 60*a^3*b*e^7 + 180*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 300*
(b^3*c + 3*a*b*c^2)*d^4*e^3 + 60*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 180*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + 60
*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 28*(c^4*d*e^6 - 3*b*c^3*e^7)*x^6 + 3*(14*c^4*d^2*e^5 - 42*b*c^3*d*e^6 + 15*(3*b
^2*c^2 + 2*a*c^3)*e^7)*x^5 - 5*(14*c^4*d^3*e^4 - 42*b*c^3*d^2*e^5 + 15*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 20*(b^3*c
+ 3*a*b*c^2)*e^7)*x^4 + 10*(14*c^4*d^4*e^3 - 42*b*c^3*d^3*e^4 + 15*(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 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 - 30*(14*c^4*d^5*e^2 - 42*b*c^3*d^4*e^3 + 15*(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 + 3*(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 - 60*(12*c^4*d^6*e - 35*b*c^3*d^5*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 15*(b^3*c
+ 3*a*b*c^2)*d^3*e^4 + 2*(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)*x + 60*(14*c^4*
d^7 - 42*b*c^3*d^6*e + 15*(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 + 3*(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 + (14*c^4*d^6*e - 42*b*c^3
*d^5*e^2 + 15*(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 + 3*(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)*log(e*x + d))/(e^9*x + d*e^8)

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Sympy [A]  time = 4.33576, size = 670, normalized size = 1.69 \begin{align*} \frac{c^{4} x^{6}}{3 e^{2}} - \frac{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}}{d e^{8} + e^{9} x} + \frac{x^{5} \left (7 b c^{3} e - 4 c^{4} d\right )}{5 e^{3}} + \frac{x^{4} \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 14 b c^{3} d e + 6 c^{4} d^{2}\right )}{4 e^{4}} + \frac{x^{3} \left (15 a b c^{2} e^{3} - 12 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 18 b^{2} c^{2} d e^{2} + 21 b c^{3} d^{2} e - 8 c^{4} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 30 a b c^{2} d e^{3} + 18 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 10 b^{3} c d e^{3} + 27 b^{2} c^{2} d^{2} e^{2} - 28 b c^{3} d^{3} e + 10 c^{4} d^{4}\right )}{2 e^{6}} + \frac{x \left (9 a^{2} b c e^{5} - 12 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 24 a b^{2} c d e^{4} + 45 a b c^{2} d^{2} e^{3} - 24 a c^{3} d^{3} e^{2} - 2 b^{4} d e^{4} + 15 b^{3} c d^{2} e^{3} - 36 b^{2} c^{2} d^{3} e^{2} + 35 b c^{3} d^{4} e - 12 c^{4} d^{5}\right )}{e^{7}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2} \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\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)**2,x)

[Out]

c**4*x**6/(3*e**2) - (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)/(d*e**8 + e**9*x) + x**5*
(7*b*c**3*e - 4*c**4*d)/(5*e**3) + x**4*(6*a*c**3*e**2 + 9*b**2*c**2*e**2 - 14*b*c**3*d*e + 6*c**4*d**2)/(4*e*
*4) + x**3*(15*a*b*c**2*e**3 - 12*a*c**3*d*e**2 + 5*b**3*c*e**3 - 18*b**2*c**2*d*e**2 + 21*b*c**3*d**2*e - 8*c
**4*d**3)/(3*e**5) + x**2*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 30*a*b*c**2*d*e**3 + 18*a*c**3*d**2*e**2 + b*
*4*e**4 - 10*b**3*c*d*e**3 + 27*b**2*c**2*d**2*e**2 - 28*b*c**3*d**3*e + 10*c**4*d**4)/(2*e**6) + x*(9*a**2*b*
c*e**5 - 12*a**2*c**2*d*e**4 + 3*a*b**3*e**5 - 24*a*b**2*c*d*e**4 + 45*a*b*c**2*d**2*e**3 - 24*a*c**3*d**3*e**
2 - 2*b**4*d*e**4 + 15*b**3*c*d**2*e**3 - 36*b**2*c**2*d**3*e**2 + 35*b*c**3*d**4*e - 12*c**4*d**5)/e**7 + (a*
e**2 - b*d*e + c*d**2)**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(d + e*x)/e**8

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Giac [B]  time = 1.18298, size = 1118, normalized size = 2.82 \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)^2,x, algorithm="giac")

[Out]

1/60*(20*c^4 - 84*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 45*(14*c^4*d^2*e^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*e
^4 + 2*a*c^3*e^4)*e^(-2)/(x*e + d)^2 - 100*(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e^
5 - b^3*c*e^6 - 3*a*b*c^2*e^6)*e^(-3)/(x*e + d)^3 + 30*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 90*b^2*c^2*d^2*e^
6 + 60*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a*b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c*e^8 + 6*a^2*c^2*e^8)*e^(-4)/(x
*e + d)^4 - 180*(14*c^4*d^5*e^5 - 35*b*c^3*d^4*e^6 + 30*b^2*c^2*d^3*e^7 + 20*a*c^3*d^3*e^7 - 10*b^3*c*d^2*e^8
- 30*a*b*c^2*d^2*e^8 + b^4*d*e^9 + 12*a*b^2*c*d*e^9 + 6*a^2*c^2*d*e^9 - a*b^3*e^10 - 3*a^2*b*c*e^10)*e^(-5)/(x
*e + d)^5)*(x*e + d)^6*e^(-8) - (14*c^4*d^6 - 42*b*c^3*d^5*e + 45*b^2*c^2*d^4*e^2 + 30*a*c^3*d^4*e^2 - 20*b^3*
c*d^3*e^3 - 60*a*b*c^2*d^3*e^3 + 3*b^4*d^2*e^4 + 36*a*b^2*c*d^2*e^4 + 18*a^2*c^2*d^2*e^4 - 6*a*b^3*d*e^5 - 18*
a^2*b*c*d*e^5 + 3*a^2*b^2*e^6 + 2*a^3*c*e^6)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (2*c^4*d^7*e^6/(x*e
+ d) - 7*b*c^3*d^6*e^7/(x*e + d) + 9*b^2*c^2*d^5*e^8/(x*e + d) + 6*a*c^3*d^5*e^8/(x*e + d) - 5*b^3*c*d^4*e^9/
(x*e + d) - 15*a*b*c^2*d^4*e^9/(x*e + d) + b^4*d^3*e^10/(x*e + d) + 12*a*b^2*c*d^3*e^10/(x*e + d) + 6*a^2*c^2*
d^3*e^10/(x*e + d) - 3*a*b^3*d^2*e^11/(x*e + d) - 9*a^2*b*c*d^2*e^11/(x*e + d) + 3*a^2*b^2*d*e^12/(x*e + d) +
2*a^3*c*d*e^12/(x*e + d) - a^3*b*e^13/(x*e + d))*e^(-14)