3.1521 \(\int \frac{(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=390 \[ \frac{x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)-2 c^3 d^2 e (35 b d-18 a e)+b^4 e^4+30 c^4 d^4\right )}{e^7}+\frac{c^2 x^3 \left (-c e (7 b d-2 a e)+3 b^2 e^2+4 c^2 d^2\right )}{e^5}-\frac{c x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )}{2 e^6}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac{c^3 x^4 (6 c d-7 b e)}{4 e^4}+\frac{2 c^4 x^5}{5 e^3} \]

[Out]

((30*c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(35*b*d - 18*a*e) - 3*b^2*c*e^3*(5*b*d - 4*a*e) + 3*c^2*e^2*(18*b^2*d^2 -
15*a*b*d*e + 2*a^2*e^2))*x)/e^7 - (c*(20*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(9*b*d - 5*a*e) - 6*c^2*d*e*(7*b*d -
3*a*e))*x^2)/(2*e^6) + (c^2*(4*c^2*d^2 + 3*b^2*e^2 - c*e*(7*b*d - 2*a*e))*x^3)/e^5 - (c^3*(6*c*d - 7*b*e)*x^4
)/(4*e^4) + (2*c^4*x^5)/(5*e^3) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(2*e^8*(d + e*x)^2) - ((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)))/(e^8*(d + e*x)) - (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))*Log[d + e*x])/e^8

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Rubi [A]  time = 0.513483, antiderivative size = 390, 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} \[ \frac{x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)-2 c^3 d^2 e (35 b d-18 a e)+b^4 e^4+30 c^4 d^4\right )}{e^7}+\frac{c^2 x^3 \left (-c e (7 b d-2 a e)+3 b^2 e^2+4 c^2 d^2\right )}{e^5}-\frac{c x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )}{2 e^6}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac{c^3 x^4 (6 c d-7 b e)}{4 e^4}+\frac{2 c^4 x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(35*b*d - 18*a*e) - 3*b^2*c*e^3*(5*b*d - 4*a*e) + 3*c^2*e^2*(18*b^2*d^2 -
15*a*b*d*e + 2*a^2*e^2))*x)/e^7 - (c*(20*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(9*b*d - 5*a*e) - 6*c^2*d*e*(7*b*d -
3*a*e))*x^2)/(2*e^6) + (c^2*(4*c^2*d^2 + 3*b^2*e^2 - c*e*(7*b*d - 2*a*e))*x^3)/e^5 - (c^3*(6*c*d - 7*b*e)*x^4
)/(4*e^4) + (2*c^4*x^5)/(5*e^3) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(2*e^8*(d + e*x)^2) - ((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)))/(e^8*(d + e*x)) - (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))*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)^3} \, dx &=\int \left (\frac{30 c^4 d^4+b^4 e^4-2 c^3 d^2 e (35 b d-18 a e)-3 b^2 c e^3 (5 b d-4 a e)+3 c^2 e^2 \left (18 b^2 d^2-15 a b d e+2 a^2 e^2\right )}{e^7}+\frac{c \left (-20 c^3 d^3+5 b^3 e^3-3 b c e^2 (9 b d-5 a e)+6 c^2 d e (7 b d-3 a e)\right ) x}{e^6}+\frac{3 c^2 \left (4 c^2 d^2+3 b^2 e^2-c e (7 b d-2 a e)\right ) x^2}{e^5}-\frac{c^3 (6 c d-7 b e) x^3}{e^4}+\frac{2 c^4 x^4}{e^3}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (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 )}{e^7 (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+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{\left (30 c^4 d^4+b^4 e^4-2 c^3 d^2 e (35 b d-18 a e)-3 b^2 c e^3 (5 b d-4 a e)+3 c^2 e^2 \left (18 b^2 d^2-15 a b d e+2 a^2 e^2\right )\right ) x}{e^7}-\frac{c \left (20 c^3 d^3-5 b^3 e^3+3 b c e^2 (9 b d-5 a e)-6 c^2 d e (7 b d-3 a e)\right ) x^2}{2 e^6}+\frac{c^2 \left (4 c^2 d^2+3 b^2 e^2-c e (7 b d-2 a e)\right ) x^3}{e^5}-\frac{c^3 (6 c d-7 b e) x^4}{4 e^4}+\frac{2 c^4 x^5}{5 e^3}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (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^8 (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+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.184208, size = 403, normalized size = 1.03 \[ \frac{20 e x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)+2 c^3 d^2 e (18 a e-35 b d)+b^4 e^4+30 c^4 d^4\right )-60 (2 c d-b e) \log (d+e x) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )+20 c^2 e^3 x^3 \left (c e (2 a e-7 b d)+3 b^2 e^2+4 c^2 d^2\right )-10 c e^2 x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )-\frac{20 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+\frac{10 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}-5 c^3 e^4 x^4 (6 c d-7 b e)+8 c^4 e^5 x^5}{20 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(30*c^4*d^4 + b^4*e^4 - 3*b^2*c*e^3*(5*b*d - 4*a*e) + 2*c^3*d^2*e*(-35*b*d + 18*a*e) + 3*c^2*e^2*(18*b^2
*d^2 - 15*a*b*d*e + 2*a^2*e^2))*x - 10*c*e^2*(20*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(9*b*d - 5*a*e) - 6*c^2*d*e*(
7*b*d - 3*a*e))*x^2 + 20*c^2*e^3*(4*c^2*d^2 + 3*b^2*e^2 + c*e*(-7*b*d + 2*a*e))*x^3 - 5*c^3*e^4*(6*c*d - 7*b*e
)*x^4 + 8*c^4*e^5*x^5 + (10*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (20*(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) - 60*(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))*Log[d + e*x])/(20*e^8
)

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Maple [B]  time = 0.017, size = 978, normalized size = 2.5 \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)^3,x)

[Out]

2/5*c^4*x^5/e^3+1/e^3*b^4*x-45/e^4*a*b*c^2*d*x-9/2/e^3/(e*x+d)^2*d^2*a^2*b*c+6/e^4/(e*x+d)^2*a*b^2*c*d^3-15/2/
e^5/(e*x+d)^2*a*b*c^2*d^4-36/e^4*ln(e*x+d)*a*b^2*c*d+90/e^5*ln(e*x+d)*a*b*c^2*d^2+18/e^3/(e*x+d)*a^2*b*c*d+60/
e^5/(e*x+d)*a*b*c^2*d^3-36/e^4/(e*x+d)*a*b^2*c*d^2-3/e^2/(e*x+d)*a^2*b^2-3/e^4/(e*x+d)*b^4*d^2-14/e^8/(e*x+d)*
c^4*d^6+2/e^3*x^3*a*c^3+3/e^3*x^3*b^2*c^2+4/e^5*x^3*c^4*d^2+5/2/e^3*x^2*b^3*c-10/e^6*x^2*c^4*d^3+6/e^3*a^2*c^2
*x+30/e^7*c^4*d^4*x-1/2/e/(e*x+d)^2*a^3*b+1/2/e^4/(e*x+d)^2*b^4*d^3+7/4/e^3*x^4*b*c^3-3/2/e^4*x^4*c^4*d+1/e^8/
(e*x+d)^2*c^4*d^7+3/e^3*ln(e*x+d)*a*b^3-3/e^4*ln(e*x+d)*b^4*d-42/e^8*ln(e*x+d)*c^4*d^5-2/e^2/(e*x+d)*a^3*c-3/2
/e^3/(e*x+d)^2*d^2*a*b^3+3/e^6/(e*x+d)^2*a*c^3*d^5-5/2/e^5/(e*x+d)^2*b^3*c*d^4-7/e^4*x^3*b*c^3*d+15/2/e^3*x^2*
a*b*c^2-60/e^6*ln(e*x+d)*a*c^3*d^3+30/e^5*ln(e*x+d)*b^3*c*d^2+3/e^4/(e*x+d)^2*a^2*c^2*d^3+3/2/e^2/(e*x+d)^2*d*
a^2*b^2+36/e^5*a*c^3*d^2*x-15/e^4*b^3*c*d*x+54/e^5*b^2*c^2*d^2*x-27/2/e^4*x^2*b^2*c^2*d+21/e^5*x^2*b*c^3*d^2+1
2/e^3*a*b^2*c*x-9/e^4*x^2*a*c^3*d+20/e^5/(e*x+d)*b^3*c*d^3-45/e^6/(e*x+d)*b^2*c^2*d^4+42/e^7/(e*x+d)*b*c^3*d^5
-90/e^6*ln(e*x+d)*b^2*c^2*d^3+105/e^7*ln(e*x+d)*b*c^3*d^4-18/e^4/(e*x+d)*a^2*c^2*d^2+6/e^3/(e*x+d)*a*b^3*d-30/
e^6/(e*x+d)*a*c^3*d^4+9/2/e^6/(e*x+d)^2*b^2*c^2*d^5-7/2/e^7/(e*x+d)^2*b*c^3*d^6+9/e^3*ln(e*x+d)*a^2*b*c-18/e^4
*ln(e*x+d)*a^2*c^2*d-70/e^6*b*d^3*c^3*x+1/e^2/(e*x+d)^2*d*a^3*c

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Maxima [A]  time = 1.08434, size = 886, normalized size = 2.27 \begin{align*} -\frac{26 \, c^{4} d^{7} - 77 \, b c^{3} d^{6} e + a^{3} b e^{7} + 27 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 35 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 5 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 9 \,{\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} + 2 \,{\left (14 \, c^{4} d^{6} e - 42 \, b c^{3} d^{5} e^{2} + 15 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 20 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 3 \,{\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}{2 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac{8 \, c^{4} e^{4} x^{5} - 5 \,{\left (6 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{4} + 20 \,{\left (4 \, c^{4} d^{2} e^{2} - 7 \, b c^{3} d e^{3} +{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{3} - 10 \,{\left (20 \, c^{4} d^{3} e - 42 \, b c^{3} d^{2} e^{2} + 9 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} - 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{2} + 20 \,{\left (30 \, c^{4} d^{4} - 70 \, b c^{3} d^{3} e + 18 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 15 \,{\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 )} x}{20 \, e^{7}} - \frac{3 \,{\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} -{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\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)^3,x, algorithm="maxima")

[Out]

-1/2*(26*c^4*d^7 - 77*b*c^3*d^6*e + a^3*b*e^7 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 35*(b^3*c + 3*a*b*c^2)*d^4*
e^3 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 9*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6 +
2*(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)/(e^10*x^2 +
2*d*e^9*x + d^2*e^8) + 1/20*(8*c^4*e^4*x^5 - 5*(6*c^4*d*e^3 - 7*b*c^3*e^4)*x^4 + 20*(4*c^4*d^2*e^2 - 7*b*c^3*
d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 10*(20*c^4*d^3*e - 42*b*c^3*d^2*e^2 + 9*(3*b^2*c^2 + 2*a*c^3)*d*e^3 -
5*(b^3*c + 3*a*b*c^2)*e^4)*x^2 + 20*(30*c^4*d^4 - 70*b*c^3*d^3*e + 18*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 15*(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)*x)/e^7 - 3*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^
2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*
a^2*b*c)*e^5)*log(e*x + d)/e^8

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Fricas [B]  time = 1.57594, size = 2090, normalized size = 5.36 \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)^3,x, algorithm="fricas")

[Out]

1/20*(8*c^4*e^7*x^7 - 260*c^4*d^7 + 770*b*c^3*d^6*e - 10*a^3*b*e^7 - 270*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 + 350*(
b^3*c + 3*a*b*c^2)*d^4*e^3 - 50*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 90*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - 10*(
3*a^2*b^2 + 2*a^3*c)*d*e^6 - 7*(2*c^4*d*e^6 - 5*b*c^3*e^7)*x^6 + 2*(14*c^4*d^2*e^5 - 35*b*c^3*d*e^6 + 10*(3*b^
2*c^2 + 2*a*c^3)*e^7)*x^5 - 5*(14*c^4*d^3*e^4 - 35*b*c^3*d^2*e^5 + 10*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 10*(b^3*c
+ 3*a*b*c^2)*e^7)*x^4 + 20*(14*c^4*d^4*e^3 - 35*b*c^3*d^3*e^4 + 10*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 10*(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 + 10*(100*c^4*d^5*e^2 - 238*b*c^3*d^4*e^3 + 63*(3*
b^2*c^2 + 2*a*c^3)*d^3*e^4 - 55*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6)*x^2 + 20
*(16*c^4*d^6*e - 28*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 - 2*(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 - 60*(14*c^4*d^7
- 35*b*c^3*d^6*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d^2*e^5 + (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 + 2*(14*c^4*d^6*e - 35*b*c^3*d^5*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^
2*e^8)

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Sympy [A]  time = 19.3531, size = 721, normalized size = 1.85 \begin{align*} \frac{2 c^{4} x^{5}}{5 e^{3}} - \frac{a^{3} b e^{7} + 2 a^{3} c d e^{6} + 3 a^{2} b^{2} d e^{6} - 27 a^{2} b c d^{2} e^{5} + 30 a^{2} c^{2} d^{3} e^{4} - 9 a b^{3} d^{2} e^{5} + 60 a b^{2} c d^{3} e^{4} - 105 a b c^{2} d^{4} e^{3} + 54 a c^{3} d^{5} e^{2} + 5 b^{4} d^{3} e^{4} - 35 b^{3} c d^{4} e^{3} + 81 b^{2} c^{2} d^{5} e^{2} - 77 b c^{3} d^{6} e + 26 c^{4} d^{7} + x \left (4 a^{3} c e^{7} + 6 a^{2} b^{2} e^{7} - 36 a^{2} b c d e^{6} + 36 a^{2} c^{2} d^{2} e^{5} - 12 a b^{3} d e^{6} + 72 a b^{2} c d^{2} e^{5} - 120 a b c^{2} d^{3} e^{4} + 60 a c^{3} d^{4} e^{3} + 6 b^{4} d^{2} e^{5} - 40 b^{3} c d^{3} e^{4} + 90 b^{2} c^{2} d^{4} e^{3} - 84 b c^{3} d^{5} e^{2} + 28 c^{4} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac{x^{4} \left (7 b c^{3} e - 6 c^{4} d\right )}{4 e^{4}} + \frac{x^{3} \left (2 a c^{3} e^{2} + 3 b^{2} c^{2} e^{2} - 7 b c^{3} d e + 4 c^{4} d^{2}\right )}{e^{5}} + \frac{x^{2} \left (15 a b c^{2} e^{3} - 18 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 27 b^{2} c^{2} d e^{2} + 42 b c^{3} d^{2} e - 20 c^{4} d^{3}\right )}{2 e^{6}} + \frac{x \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 45 a b c^{2} d e^{3} + 36 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 15 b^{3} c d e^{3} + 54 b^{2} c^{2} d^{2} e^{2} - 70 b c^{3} d^{3} e + 30 c^{4} d^{4}\right )}{e^{7}} + \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \left (3 a c e^{2} + b^{2} e^{2} - 7 b c d e + 7 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)**3,x)

[Out]

2*c**4*x**5/(5*e**3) - (a**3*b*e**7 + 2*a**3*c*d*e**6 + 3*a**2*b**2*d*e**6 - 27*a**2*b*c*d**2*e**5 + 30*a**2*c
**2*d**3*e**4 - 9*a*b**3*d**2*e**5 + 60*a*b**2*c*d**3*e**4 - 105*a*b*c**2*d**4*e**3 + 54*a*c**3*d**5*e**2 + 5*
b**4*d**3*e**4 - 35*b**3*c*d**4*e**3 + 81*b**2*c**2*d**5*e**2 - 77*b*c**3*d**6*e + 26*c**4*d**7 + x*(4*a**3*c*
e**7 + 6*a**2*b**2*e**7 - 36*a**2*b*c*d*e**6 + 36*a**2*c**2*d**2*e**5 - 12*a*b**3*d*e**6 + 72*a*b**2*c*d**2*e*
*5 - 120*a*b*c**2*d**3*e**4 + 60*a*c**3*d**4*e**3 + 6*b**4*d**2*e**5 - 40*b**3*c*d**3*e**4 + 90*b**2*c**2*d**4
*e**3 - 84*b*c**3*d**5*e**2 + 28*c**4*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2) + x**4*(7*b*c**3*e -
6*c**4*d)/(4*e**4) + x**3*(2*a*c**3*e**2 + 3*b**2*c**2*e**2 - 7*b*c**3*d*e + 4*c**4*d**2)/e**5 + x**2*(15*a*b*
c**2*e**3 - 18*a*c**3*d*e**2 + 5*b**3*c*e**3 - 27*b**2*c**2*d*e**2 + 42*b*c**3*d**2*e - 20*c**4*d**3)/(2*e**6)
+ x*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 45*a*b*c**2*d*e**3 + 36*a*c**3*d**2*e**2 + b**4*e**4 - 15*b**3*c*d
*e**3 + 54*b**2*c**2*d**2*e**2 - 70*b*c**3*d**3*e + 30*c**4*d**4)/e**7 + 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d
**2)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)*log(d + e*x)/e**8

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Giac [A]  time = 1.20149, size = 937, normalized size = 2.4 \begin{align*} -3 \,{\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 30 \, b^{2} c^{2} d^{3} e^{2} + 20 \, a c^{3} d^{3} e^{2} - 10 \, b^{3} c d^{2} e^{3} - 30 \, a b c^{2} d^{2} e^{3} + b^{4} d e^{4} + 12 \, a b^{2} c d e^{4} + 6 \, a^{2} c^{2} d e^{4} - a b^{3} e^{5} - 3 \, a^{2} b c e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{20} \,{\left (8 \, c^{4} x^{5} e^{12} - 30 \, c^{4} d x^{4} e^{11} + 80 \, c^{4} d^{2} x^{3} e^{10} - 200 \, c^{4} d^{3} x^{2} e^{9} + 600 \, c^{4} d^{4} x e^{8} + 35 \, b c^{3} x^{4} e^{12} - 140 \, b c^{3} d x^{3} e^{11} + 420 \, b c^{3} d^{2} x^{2} e^{10} - 1400 \, b c^{3} d^{3} x e^{9} + 60 \, b^{2} c^{2} x^{3} e^{12} + 40 \, a c^{3} x^{3} e^{12} - 270 \, b^{2} c^{2} d x^{2} e^{11} - 180 \, a c^{3} d x^{2} e^{11} + 1080 \, b^{2} c^{2} d^{2} x e^{10} + 720 \, a c^{3} d^{2} x e^{10} + 50 \, b^{3} c x^{2} e^{12} + 150 \, a b c^{2} x^{2} e^{12} - 300 \, b^{3} c d x e^{11} - 900 \, a b c^{2} d x e^{11} + 20 \, b^{4} x e^{12} + 240 \, a b^{2} c x e^{12} + 120 \, a^{2} c^{2} x e^{12}\right )} e^{\left (-15\right )} - \frac{{\left (26 \, c^{4} d^{7} - 77 \, b c^{3} d^{6} e + 81 \, b^{2} c^{2} d^{5} e^{2} + 54 \, a c^{3} d^{5} e^{2} - 35 \, b^{3} c d^{4} e^{3} - 105 \, a b c^{2} d^{4} e^{3} + 5 \, b^{4} d^{3} e^{4} + 60 \, a b^{2} c d^{3} e^{4} + 30 \, a^{2} c^{2} d^{3} e^{4} - 9 \, a b^{3} d^{2} e^{5} - 27 \, 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} + 2 \,{\left (14 \, c^{4} d^{6} e - 42 \, b c^{3} d^{5} e^{2} + 45 \, b^{2} c^{2} d^{4} e^{3} + 30 \, a c^{3} d^{4} e^{3} - 20 \, b^{3} c d^{3} e^{4} - 60 \, a b c^{2} d^{3} e^{4} + 3 \, b^{4} d^{2} e^{5} + 36 \, a b^{2} c d^{2} e^{5} + 18 \, 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 )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-3*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*e^(-8)*log(abs(x*e + d)) + 1/2
0*(8*c^4*x^5*e^12 - 30*c^4*d*x^4*e^11 + 80*c^4*d^2*x^3*e^10 - 200*c^4*d^3*x^2*e^9 + 600*c^4*d^4*x*e^8 + 35*b*c
^3*x^4*e^12 - 140*b*c^3*d*x^3*e^11 + 420*b*c^3*d^2*x^2*e^10 - 1400*b*c^3*d^3*x*e^9 + 60*b^2*c^2*x^3*e^12 + 40*
a*c^3*x^3*e^12 - 270*b^2*c^2*d*x^2*e^11 - 180*a*c^3*d*x^2*e^11 + 1080*b^2*c^2*d^2*x*e^10 + 720*a*c^3*d^2*x*e^1
0 + 50*b^3*c*x^2*e^12 + 150*a*b*c^2*x^2*e^12 - 300*b^3*c*d*x*e^11 - 900*a*b*c^2*d*x*e^11 + 20*b^4*x*e^12 + 240
*a*b^2*c*x*e^12 + 120*a^2*c^2*x*e^12)*e^(-15) - 1/2*(26*c^4*d^7 - 77*b*c^3*d^6*e + 81*b^2*c^2*d^5*e^2 + 54*a*c
^3*d^5*e^2 - 35*b^3*c*d^4*e^3 - 105*a*b*c^2*d^4*e^3 + 5*b^4*d^3*e^4 + 60*a*b^2*c*d^3*e^4 + 30*a^2*c^2*d^3*e^4
- 9*a*b^3*d^2*e^5 - 27*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 + 2*(14*c^4*d^6*e - 42*b*
c^3*d^5*e^2 + 45*b^2*c^2*d^4*e^3 + 30*a*c^3*d^4*e^3 - 20*b^3*c*d^3*e^4 - 60*a*b*c^2*d^3*e^4 + 3*b^4*d^2*e^5 +
36*a*b^2*c*d^2*e^5 + 18*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)^2