### 3.2151 $$\int \frac{(a+b x+c x^2)^4}{d+e x} \, dx$$

Optimal. Leaf size=428 $\frac{(d+e x)^4 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{4 e^9}+\frac{c^2 (d+e x)^6 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac{4 c (d+e x)^5 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac{4 (d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac{(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac{4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^4}{e^9}-\frac{4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac{c^4 (d+e x)^8}{8 e^9}$

[Out]

(-4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*x)/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))*(d + e*x)^2)/e^9 - (4*(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)^3)/(3*e^9) + ((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)^4)/(4*e^9) - (4*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)^5)/(5*e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d
+ e*x)^6)/(3*e^9) - (4*c^3*(2*c*d - b*e)*(d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 - b*d*e +
a*e^2)^4*Log[d + e*x])/e^9

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Rubi [A]  time = 0.708334, antiderivative size = 428, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{(d+e x)^4 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{4 e^9}+\frac{c^2 (d+e x)^6 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac{4 c (d+e x)^5 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac{4 (d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac{(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac{4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^4}{e^9}-\frac{4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac{c^4 (d+e x)^8}{8 e^9}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x),x]

[Out]

(-4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*x)/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))*(d + e*x)^2)/e^9 - (4*(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)^3)/(3*e^9) + ((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)^4)/(4*e^9) - (4*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)^5)/(5*e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d
+ e*x)^6)/(3*e^9) - (4*c^3*(2*c*d - b*e)*(d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 - b*d*e +
a*e^2)^4*Log[d + e*x])/e^9

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.404188, size = 616, normalized size = 1.44 $\frac{x \left (84 c^2 e^2 \left (5 a^2 e^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+2 a b e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+b^2 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+56 c e^3 \left (30 a^2 b e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+30 a^3 e^3 (e x-2 d)+15 a b^2 e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b^3 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+70 b e^4 \left (36 a^2 b e^2 (e x-2 d)+48 a^3 e^3+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+8 c^3 e \left (7 a e \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )+b \left (140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-210 d^5 e x+420 d^6-70 d e^5 x^5+60 e^6 x^6\right )\right )+c^4 \left (-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5+420 d^6 e x-840 d^7-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac{\log (d+e x) \left (e (a e-b d)+c d^2\right )^4}{e^9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(c^4*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*
d*e^6*x^6 + 105*e^7*x^7) + 70*b*e^4*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e
^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 56*c*e^3*(30*a^3*e^3*(-2*d + e*x) + 30*a^2*b*
e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 15*a*b^2*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^3*(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)) + 84*c^2*e^2*(5*a^2*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*
e^2*x^2 + 3*e^3*x^3) + 2*a*b*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) + b^2*(-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)) + 8*c^3*e*(7*a*e*(-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) + b*(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))))/(840*e^8) + ((c*d^2 + e*(-(b*d) + a*
e))^4*Log[d + e*x])/e^9

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Maple [B]  time = 0.048, size = 1096, 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((c*x^2+b*x+a)^4/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.05015, size = 1076, normalized size = 2.51 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="maxima")

[Out]

1/840*(105*c^4*e^7*x^8 - 120*(c^4*d*e^6 - 4*b*c^3*e^7)*x^7 + 140*(c^4*d^2*e^5 - 4*b*c^3*d*e^6 + 2*(3*b^2*c^2 +
2*a*c^3)*e^7)*x^6 - 168*(c^4*d^3*e^4 - 4*b*c^3*d^2*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 4*(b^3*c + 3*a*b*c^2
)*e^7)*x^5 + 210*(c^4*d^4*e^3 - 4*b*c^3*d^3*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 4*(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^4 - 280*(c^4*d^5*e^2 - 4*b*c^3*d^4*e^3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3
*e^4 - 4*(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 - 4*(a*b^3 + 3*a^2*b*c)*e^7)*x^3 +
420*(c^4*d^6*e - 4*b*c^3*d^5*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^3*e^4 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*e^7)*x^2 - 840*(c^4*d^7
- 4*b*c^3*d^6*e - 4*a^3*b*e^7 + 2*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 4*(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 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^6)*x)/e^8 + (c^4*d^8
- 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^3 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*log
(e*x + d)/e^9

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Fricas [A]  time = 1.77465, size = 1662, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="fricas")

[Out]

1/840*(105*c^4*e^8*x^8 - 120*(c^4*d*e^7 - 4*b*c^3*e^8)*x^7 + 140*(c^4*d^2*e^6 - 4*b*c^3*d*e^7 + 2*(3*b^2*c^2 +
2*a*c^3)*e^8)*x^6 - 168*(c^4*d^3*e^5 - 4*b*c^3*d^2*e^6 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 4*(b^3*c + 3*a*b*c^2
)*e^8)*x^5 + 210*(c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 4*(b^3*c + 3*a*b*c^2)*d*e^
7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 280*(c^4*d^5*e^3 - 4*b*c^3*d^4*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3
*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 4*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 +
420*(c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 +
12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 840*(c^4*d
^7*e - 4*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 840*(c^
4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*
e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6
)*log(e*x + d))/e^9

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Sympy [A]  time = 1.83222, size = 796, normalized size = 1.86 \begin{align*} \frac{c^{4} x^{8}}{8 e} + \frac{x^{7} \left (4 b c^{3} e - c^{4} d\right )}{7 e^{2}} + \frac{x^{6} \left (4 a c^{3} e^{2} + 6 b^{2} c^{2} e^{2} - 4 b c^{3} d e + c^{4} d^{2}\right )}{6 e^{3}} + \frac{x^{5} \left (12 a b c^{2} e^{3} - 4 a c^{3} d e^{2} + 4 b^{3} c e^{3} - 6 b^{2} c^{2} d e^{2} + 4 b c^{3} d^{2} e - c^{4} d^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 12 a b c^{2} d e^{3} + 4 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + c^{4} d^{4}\right )}{4 e^{5}} + \frac{x^{3} \left (12 a^{2} b c e^{5} - 6 a^{2} c^{2} d e^{4} + 4 a b^{3} e^{5} - 12 a b^{2} c d e^{4} + 12 a b c^{2} d^{2} e^{3} - 4 a c^{3} d^{3} e^{2} - b^{4} d e^{4} + 4 b^{3} c d^{2} e^{3} - 6 b^{2} c^{2} d^{3} e^{2} + 4 b c^{3} d^{4} e - c^{4} d^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (4 a^{3} c e^{6} + 6 a^{2} b^{2} e^{6} - 12 a^{2} b c d e^{5} + 6 a^{2} c^{2} d^{2} e^{4} - 4 a b^{3} d e^{5} + 12 a b^{2} c d^{2} e^{4} - 12 a b c^{2} d^{3} e^{3} + 4 a c^{3} d^{4} e^{2} + b^{4} d^{2} e^{4} - 4 b^{3} c d^{3} e^{3} + 6 b^{2} c^{2} d^{4} e^{2} - 4 b c^{3} d^{5} e + c^{4} d^{6}\right )}{2 e^{7}} + \frac{x \left (4 a^{3} b e^{7} - 4 a^{3} c d e^{6} - 6 a^{2} b^{2} d e^{6} + 12 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 4 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 12 a b c^{2} d^{4} e^{3} - 4 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 4 b^{3} c d^{4} e^{3} - 6 b^{2} c^{2} d^{5} e^{2} + 4 b c^{3} d^{6} e - c^{4} d^{7}\right )}{e^{8}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{4} \log{\left (d + e x \right )}}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d),x)

[Out]

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

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Giac [B]  time = 1.16693, size = 1276, normalized size = 2.98 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d),x, algorithm="giac")

[Out]

(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^
4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*
a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*e^(-9)*log(abs(x*e + d)) + 1/840*(105*c^4*x^8*e^7 - 120*c^4*d*x^7*e^6
+ 140*c^4*d^2*x^6*e^5 - 168*c^4*d^3*x^5*e^4 + 210*c^4*d^4*x^4*e^3 - 280*c^4*d^5*x^3*e^2 + 420*c^4*d^6*x^2*e -
840*c^4*d^7*x + 480*b*c^3*x^7*e^7 - 560*b*c^3*d*x^6*e^6 + 672*b*c^3*d^2*x^5*e^5 - 840*b*c^3*d^3*x^4*e^4 + 112
0*b*c^3*d^4*x^3*e^3 - 1680*b*c^3*d^5*x^2*e^2 + 3360*b*c^3*d^6*x*e + 840*b^2*c^2*x^6*e^7 + 560*a*c^3*x^6*e^7 -
1008*b^2*c^2*d*x^5*e^6 - 672*a*c^3*d*x^5*e^6 + 1260*b^2*c^2*d^2*x^4*e^5 + 840*a*c^3*d^2*x^4*e^5 - 1680*b^2*c^2
*d^3*x^3*e^4 - 1120*a*c^3*d^3*x^3*e^4 + 2520*b^2*c^2*d^4*x^2*e^3 + 1680*a*c^3*d^4*x^2*e^3 - 5040*b^2*c^2*d^5*x
*e^2 - 3360*a*c^3*d^5*x*e^2 + 672*b^3*c*x^5*e^7 + 2016*a*b*c^2*x^5*e^7 - 840*b^3*c*d*x^4*e^6 - 2520*a*b*c^2*d*
x^4*e^6 + 1120*b^3*c*d^2*x^3*e^5 + 3360*a*b*c^2*d^2*x^3*e^5 - 1680*b^3*c*d^3*x^2*e^4 - 5040*a*b*c^2*d^3*x^2*e^
4 + 3360*b^3*c*d^4*x*e^3 + 10080*a*b*c^2*d^4*x*e^3 + 210*b^4*x^4*e^7 + 2520*a*b^2*c*x^4*e^7 + 1260*a^2*c^2*x^4
*e^7 - 280*b^4*d*x^3*e^6 - 3360*a*b^2*c*d*x^3*e^6 - 1680*a^2*c^2*d*x^3*e^6 + 420*b^4*d^2*x^2*e^5 + 5040*a*b^2*
c*d^2*x^2*e^5 + 2520*a^2*c^2*d^2*x^2*e^5 - 840*b^4*d^3*x*e^4 - 10080*a*b^2*c*d^3*x*e^4 - 5040*a^2*c^2*d^3*x*e^
4 + 1120*a*b^3*x^3*e^7 + 3360*a^2*b*c*x^3*e^7 - 1680*a*b^3*d*x^2*e^6 - 5040*a^2*b*c*d*x^2*e^6 + 3360*a*b^3*d^2
*x*e^5 + 10080*a^2*b*c*d^2*x*e^5 + 2520*a^2*b^2*x^2*e^7 + 1680*a^3*c*x^2*e^7 - 5040*a^2*b^2*d*x*e^6 - 3360*a^3
*c*d*x*e^6 + 3360*a^3*b*x*e^7)*e^(-8)