### 3.2152 $$\int \frac{(a+b x+c x^2)^4}{(d+e x)^2} \, dx$$

Optimal. Leaf size=426 $\frac{(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac{2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac{c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac{2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac{2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^7}{7 e^9}$

[Out]

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

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Rubi [A]  time = 0.721856, antiderivative size = 426, 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} $\frac{(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac{2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac{c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac{2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac{2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^7}{7 e^9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.339387, size = 780, normalized size = 1.83 $\frac{21 c^2 e^2 \left (10 a^2 e^2 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+5 a b e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )+b^2 \left (90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4+150 d^5 e x-30 d^6-9 d e^5 x^5+6 e^6 x^6\right )\right )+35 c e^3 \left (18 a^2 b e^2 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+12 a^3 e^3 \left (-d^2+d e x+e^2 x^2\right )+12 a b^2 e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+b^3 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )+35 e^4 \left (18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+12 a^3 b d e^3-3 a^4 e^4+6 a b^3 e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )+7 c^3 e \left (6 a e \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )+b \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )\right )-420 (d+e x) (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3+c^4 \left (420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6+735 d^7 e x-105 d^8-20 d e^7 x^7+15 e^8 x^8\right )}{105 e^9 (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^
6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 35*e^4*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x
^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-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)) + 35*c*e^3*(12*a^3*e^3*(-d^2 + d*e*x + e^2*x^2) + 18*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*
x^2 + e^3*x^3) + 12*a*b^2*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) + b^3*(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)) + 21*c^2*e^2*(10*a^2*e^2*(-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) + 5*a*b*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) + b^2*(-30*d^6 + 150*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*
d*e^5*x^5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-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) + b*(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)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)*Log[d
+ e*x])/(105*e^9*(d + e*x))

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Maple [B]  time = 0.055, size = 1159, normalized size = 2.7 \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)^2,x)

[Out]

4/e^2*x^3*a*b^2*c-2/e^3*x^4*a*c^3*d+1/7*c^4*x^7/e^2-6/e^7/(e*x+d)*b^2*c^2*d^6+4/e^8/(e*x+d)*b*c^3*d^7-6/e^3/(e
*x+d)*a^2*b^2*d^2-6/e^5/(e*x+d)*a^2*c^2*d^4+4/e^4/(e*x+d)*a*b^3*d^3-4/e^7/(e*x+d)*a*c^3*d^6+4/e^6/(e*x+d)*b^3*
c*d^5+12/e^4*ln(e*x+d)*a*b^3*d^2-24/e^7*ln(e*x+d)*a*c^3*d^5+20/e^6*ln(e*x+d)*b^3*c*d^4-36/e^7*ln(e*x+d)*b^2*c^
2*d^5+28/e^8*ln(e*x+d)*b*c^3*d^6+4/e^2/(e*x+d)*d*a^3*b-4/e^3/(e*x+d)*a^3*c*d^2-4/e^5*ln(e*x+d)*b^4*d^3-8/e^9*l
n(e*x+d)*c^4*d^7-1/e^5/(e*x+d)*b^4*d^4-1/e^9/(e*x+d)*c^4*d^8+1/e^2*x^4*b^3*c+2/3/e^2*x^6*b*c^3+2/e^2*x^3*a^2*c
^2+5/3/e^6*x^3*c^4*d^4+2/e^2*x^2*a*b^3-1/e^3*x^2*b^4*d-3/e^7*x^2*c^4*d^5+4/e^2*a^3*c*x+6/e^2*b^2*a^2*x+3/e^4*b
^4*d^2*x+4/5/e^2*x^5*a*c^3+6/5/e^2*x^5*b^2*c^2+3/5/e^4*x^5*c^4*d^2-1/e^5*x^4*c^4*d^3+7/e^8*c^4*d^6*x+4/e^2*ln(
e*x+d)*a^3*b-8/e^3*a*b^3*d*x+20/e^6*a*c^3*d^4*x-16/e^5*b^3*c*d^3*x+30/e^6*b^2*c^2*d^4*x-24/e^7*b*c^3*d^5*x+4/e
^4*x^3*a*c^3*d^2-8/3/e^3*x^3*b^3*c*d-3/e^3*x^4*b^2*c^2*d+3/e^4*x^4*b*c^3*d^2-8/e^5*x^2*a*c^3*d^3+6/e^4*x^2*b^3
*c*d^2-12/e^5*x^2*b^2*c^2*d^3+3/e^2*x^4*a*b*c^2-8/5/e^3*x^5*b*c^3*d-8/e^3*ln(e*x+d)*a^3*c*d-12/e^3*ln(e*x+d)*a
^2*b^2*d-24/e^5*ln(e*x+d)*a^2*c^2*d^3-6/e^3*x^2*a^2*c^2*d+6/e^2*x^2*a^2*b*c+6/e^4*x^3*b^2*c^2*d^2-16/3/e^5*x^3
*b*c^3*d^3+10/e^6*x^2*b*c^3*d^4+18/e^4*a^2*c^2*d^2*x+1/3/e^2*x^3*b^4-1/e/(e*x+d)*a^4-24/e^3*a^2*b*c*d*x+36/e^4
*a*b^2*c*d^2*x-48/e^5*a*b*c^2*d^3*x-12/e^3*x^2*a*b^2*c*d+18/e^4*x^2*a*b*c^2*d^2-8/e^3*x^3*a*b*c^2*d+36/e^4*ln(
e*x+d)*a^2*b*c*d^2-48/e^5*ln(e*x+d)*a*b^2*c*d^3+60/e^6*ln(e*x+d)*a*b*c^2*d^4+12/e^4/(e*x+d)*a^2*b*c*d^3-12/e^5
/(e*x+d)*a*b^2*c*d^4+12/e^6/(e*x+d)*a*b*c^2*d^5-1/3*c^4*d*x^6/e^3

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Maxima [A]  time = 1.04409, size = 1089, normalized size = 2.56 \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)^2,x, algorithm="maxima")

[Out]

-(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)/(e^10*x + d*e^9) + 1/105*(15*c^4*e^6*x^7 - 35*(c^4*d*e^5 - 2*b*c^3*e^6)*x^6 + 21*(3*c^4*d^2*e^4 - 8*b*c^
3*d*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*e^6)*x^5 - 105*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4 + (3*b^2*c^2 + 2*a*c^3)*d*e^5
- (b^3*c + 3*a*b*c^2)*e^6)*x^4 + 35*(5*c^4*d^4*e^2 - 16*b*c^3*d^3*e^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e^4 - 8*(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 - 105*(3*c^4*d^5*e - 10*b*c^3*d^4*e^2 + 4*(3
*b^2*c^2 + 2*a*c^3)*d^3*e^3 - 6*(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 - 2*(a*b^3
+ 3*a^2*b*c)*e^6)*x^2 + 105*(7*c^4*d^6 - 24*b*c^3*d^5*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 16*(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 - 8*(a*b^3 + 3*a^2*b*c)*d*e^5 + 2*(3*a^2*b^2 + 2*a^3*
c)*e^6)*x)/e^8 - 4*(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^9

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Fricas [B]  time = 1.8496, size = 2288, normalized size = 5.37 \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)^2,x, algorithm="fricas")

[Out]

1/105*(15*c^4*e^8*x^8 - 105*c^4*d^8 + 420*b*c^3*d^7*e + 420*a^3*b*d*e^7 - 105*a^4*e^8 - 210*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 420*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 105*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 420*(a*b^3 + 3*a^
2*b*c)*d^3*e^5 - 210*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 10*(2*c^4*d*e^7 - 7*b*c^3*e^8)*x^7 + 14*(2*c^4*d^2*e^6 -
7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 21*(2*c^4*d^3*e^5 - 7*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^
3)*d*e^7 - 5*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 35*(2*c^4*d^4*e^4 - 7*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*
e^6 - 5*(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 - 70*(2*c^4*d^5*e^3 - 7*b*c^3*d^4*
e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 5*(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 -
3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 210*(2*c^4*d^6*e^2 - 7*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 5*(
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 - 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2
+ 2*a^3*c)*e^8)*x^2 + 105*(7*c^4*d^7*e - 24*b*c^3*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 16*(b^3*c + 3*
a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 8*(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 - 420*(2*c^4*d^8 - 7*b*c^3*d^7*e - a^3*b*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + (3*a^2*b^2 + 2*
a^3*c)*d^2*e^6 + (2*c^4*d^7*e - 7*b*c^3*d^6*e^2 - a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 5*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + (3*a^2*b^2 + 2*a^3*
c)*d*e^7)*x)*log(e*x + d))/(e^10*x + d*e^9)

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Sympy [A]  time = 5.0012, size = 824, normalized size = 1.93 \begin{align*} \frac{c^{4} x^{7}}{7 e^{2}} - \frac{a^{4} e^{8} - 4 a^{3} b d e^{7} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} b^{2} d^{2} e^{6} - 12 a^{2} b c d^{3} e^{5} + 6 a^{2} c^{2} d^{4} e^{4} - 4 a b^{3} d^{3} e^{5} + 12 a b^{2} c d^{4} e^{4} - 12 a b c^{2} d^{5} e^{3} + 4 a c^{3} d^{6} e^{2} + b^{4} d^{4} e^{4} - 4 b^{3} c d^{5} e^{3} + 6 b^{2} c^{2} d^{6} e^{2} - 4 b c^{3} d^{7} e + c^{4} d^{8}}{d e^{9} + e^{10} x} + \frac{x^{6} \left (2 b c^{3} e - c^{4} d\right )}{3 e^{3}} + \frac{x^{5} \left (4 a c^{3} e^{2} + 6 b^{2} c^{2} e^{2} - 8 b c^{3} d e + 3 c^{4} d^{2}\right )}{5 e^{4}} + \frac{x^{4} \left (3 a b c^{2} e^{3} - 2 a c^{3} d e^{2} + b^{3} c e^{3} - 3 b^{2} c^{2} d e^{2} + 3 b c^{3} d^{2} e - c^{4} d^{3}\right )}{e^{5}} + \frac{x^{3} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 24 a b c^{2} d e^{3} + 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 8 b^{3} c d e^{3} + 18 b^{2} c^{2} d^{2} e^{2} - 16 b c^{3} d^{3} e + 5 c^{4} d^{4}\right )}{3 e^{6}} + \frac{x^{2} \left (6 a^{2} b c e^{5} - 6 a^{2} c^{2} d e^{4} + 2 a b^{3} e^{5} - 12 a b^{2} c d e^{4} + 18 a b c^{2} d^{2} e^{3} - 8 a c^{3} d^{3} e^{2} - b^{4} d e^{4} + 6 b^{3} c d^{2} e^{3} - 12 b^{2} c^{2} d^{3} e^{2} + 10 b c^{3} d^{4} e - 3 c^{4} d^{5}\right )}{e^{7}} + \frac{x \left (4 a^{3} c e^{6} + 6 a^{2} b^{2} e^{6} - 24 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 8 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 48 a b c^{2} d^{3} e^{3} + 20 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 16 b^{3} c d^{3} e^{3} + 30 b^{2} c^{2} d^{4} e^{2} - 24 b c^{3} d^{5} e + 7 c^{4} d^{6}\right )}{e^{8}} + \frac{4 \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^{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)**2,x)

[Out]

c**4*x**7/(7*e**2) - (a**4*e**8 - 4*a**3*b*d*e**7 + 4*a**3*c*d**2*e**6 + 6*a**2*b**2*d**2*e**6 - 12*a**2*b*c*d
**3*e**5 + 6*a**2*c**2*d**4*e**4 - 4*a*b**3*d**3*e**5 + 12*a*b**2*c*d**4*e**4 - 12*a*b*c**2*d**5*e**3 + 4*a*c*
*3*d**6*e**2 + b**4*d**4*e**4 - 4*b**3*c*d**5*e**3 + 6*b**2*c**2*d**6*e**2 - 4*b*c**3*d**7*e + c**4*d**8)/(d*e
**9 + e**10*x) + x**6*(2*b*c**3*e - c**4*d)/(3*e**3) + x**5*(4*a*c**3*e**2 + 6*b**2*c**2*e**2 - 8*b*c**3*d*e +
3*c**4*d**2)/(5*e**4) + x**4*(3*a*b*c**2*e**3 - 2*a*c**3*d*e**2 + b**3*c*e**3 - 3*b**2*c**2*d*e**2 + 3*b*c**3
*d**2*e - c**4*d**3)/e**5 + x**3*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 24*a*b*c**2*d*e**3 + 12*a*c**3*d**2*e*
*2 + b**4*e**4 - 8*b**3*c*d*e**3 + 18*b**2*c**2*d**2*e**2 - 16*b*c**3*d**3*e + 5*c**4*d**4)/(3*e**6) + x**2*(6
*a**2*b*c*e**5 - 6*a**2*c**2*d*e**4 + 2*a*b**3*e**5 - 12*a*b**2*c*d*e**4 + 18*a*b*c**2*d**2*e**3 - 8*a*c**3*d*
*3*e**2 - b**4*d*e**4 + 6*b**3*c*d**2*e**3 - 12*b**2*c**2*d**3*e**2 + 10*b*c**3*d**4*e - 3*c**4*d**5)/e**7 + x
*(4*a**3*c*e**6 + 6*a**2*b**2*e**6 - 24*a**2*b*c*d*e**5 + 18*a**2*c**2*d**2*e**4 - 8*a*b**3*d*e**5 + 36*a*b**2
*c*d**2*e**4 - 48*a*b*c**2*d**3*e**3 + 20*a*c**3*d**4*e**2 + 3*b**4*d**2*e**4 - 16*b**3*c*d**3*e**3 + 30*b**2*
c**2*d**4*e**2 - 24*b*c**3*d**5*e + 7*c**4*d**6)/e**8 + 4*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e
*x)/e**9

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Giac [B]  time = 1.09843, size = 1368, normalized size = 3.21 \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)^2,x, algorithm="giac")

[Out]

1/105*(15*c^4 - 70*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 42*(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 - 105*(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 + 35*(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 - 210*(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 + 210*(14*c^4*d^6*e^6 - 42*b*c^3*d^5*e^7 + 45*b^2*c^2*d^4*e^8 + 30*a*c^3*d^4*e^8 - 20*b^3*c*d^3*e^9
- 60*a*b*c^2*d^3*e^9 + 3*b^4*d^2*e^10 + 36*a*b^2*c*d^2*e^10 + 18*a^2*c^2*d^2*e^10 - 6*a*b^3*d*e^11 - 18*a^2*b
*c*d*e^11 + 3*a^2*b^2*e^12 + 2*a^3*c*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-9) + 4*(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^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^4*d^8*e^7/(x*e + d) - 4*b*c^3*d^7*e^8/(x*e + d) + 6*b^2*c^2*d
^6*e^9/(x*e + d) + 4*a*c^3*d^6*e^9/(x*e + d) - 4*b^3*c*d^5*e^10/(x*e + d) - 12*a*b*c^2*d^5*e^10/(x*e + d) + b^
4*d^4*e^11/(x*e + d) + 12*a*b^2*c*d^4*e^11/(x*e + d) + 6*a^2*c^2*d^4*e^11/(x*e + d) - 4*a*b^3*d^3*e^12/(x*e +
d) - 12*a^2*b*c*d^3*e^12/(x*e + d) + 6*a^2*b^2*d^2*e^13/(x*e + d) + 4*a^3*c*d^2*e^13/(x*e + d) - 4*a^3*b*d*e^1
4/(x*e + d) + a^4*e^15/(x*e + d))*e^(-16)