3.1519 \(\int \frac{(b+2 c x) (a+b x+c x^2)^3}{d+e x} \, dx\)

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

[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

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Rubi [A]  time = 0.581112, antiderivative size = 399, 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} \[ \frac{(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac{3 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac{5 c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{4 e^8}-\frac{3 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8}+\frac{x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{7 c^3 (d+e x)^6 (2 c d-b e)}{6 e^8}+\frac{2 c^4 (d+e x)^7}{7 e^8} \]

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 = 0.25586, size = 483, normalized size = 1.21 \[ \frac{e x \left (21 c^2 e^2 \left (20 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+3 b^2 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+35 c e^3 \left (54 a^2 b e^2 (e x-2 d)+24 a^3 e^3+24 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b^3 \left (-6 d^2 e x+12 d^3+4 d e^2 x^2-3 e^3 x^3\right )\right )+70 b^2 e^4 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+7 c^3 e \left (6 a e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )-7 b \left (20 d^3 e^2 x^2-15 d^2 e^3 x^3-30 d^4 e x+60 d^5+12 d e^4 x^4-10 e^5 x^5\right )\right )+2 c^4 \left (140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-210 d^5 e x+420 d^6-70 d e^5 x^5+60 e^6 x^6\right )\right )-420 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{420 e^8} \]

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)

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Maple [B]  time = 0.01, size = 872, normalized size = 2.2 \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),x)

[Out]

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

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Maxima [A]  time = 1.01475, size = 869, normalized size = 2.18 \begin{align*} \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{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),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

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Fricas [A]  time = 1.46347, size = 1345, normalized size = 3.37 \begin{align*} \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{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),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

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Sympy [A]  time = 1.64491, size = 626, normalized size = 1.57 \begin{align*} \frac{2 c^{4} x^{7}}{7 e} + \frac{x^{6} \left (7 b c^{3} e - 2 c^{4} d\right )}{6 e^{2}} + \frac{x^{5} \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 7 b c^{3} d e + 2 c^{4} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (15 a b c^{2} e^{3} - 6 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 9 b^{2} c^{2} d e^{2} + 7 b c^{3} d^{2} e - 2 c^{4} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 15 a b c^{2} d e^{3} + 6 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 5 b^{3} c d e^{3} + 9 b^{2} c^{2} d^{2} e^{2} - 7 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (9 a^{2} b c e^{5} - 6 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 12 a b^{2} c d e^{4} + 15 a b c^{2} d^{2} e^{3} - 6 a c^{3} d^{3} e^{2} - b^{4} d e^{4} + 5 b^{3} c d^{2} e^{3} - 9 b^{2} c^{2} d^{3} e^{2} + 7 b c^{3} d^{4} e - 2 c^{4} d^{5}\right )}{2 e^{6}} + \frac{x \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 9 a^{2} b c d e^{5} + 6 a^{2} c^{2} d^{2} e^{4} - 3 a b^{3} d e^{5} + 12 a b^{2} c d^{2} e^{4} - 15 a b c^{2} d^{3} e^{3} + 6 a c^{3} d^{4} e^{2} + b^{4} d^{2} e^{4} - 5 b^{3} c d^{3} e^{3} + 9 b^{2} c^{2} d^{4} e^{2} - 7 b c^{3} d^{5} e + 2 c^{4} d^{6}\right )}{e^{7}} + \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{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),x)

[Out]

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

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Giac [A]  time = 1.14804, size = 1002, normalized size = 2.51 \begin{align*} -{\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{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),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)