### 3.2156 $$\int \frac{(a+b x+c x^2)^4}{(d+e x)^6} \, dx$$

Optimal. Leaf size=414 $-\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac{c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac{c^3 x^2 (3 c d-2 b e)}{e^7}+\frac{c^4 x^3}{3 e^6}$

[Out]

(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e)*x^2)/e^7 + (c^4*x^3)/(3*e^6)
- (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4)
- (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)))/(3*e^9*(d + e*x)^3) + (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)))/(e^9*(d + e*x)^2) - (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))/(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*Log[d + e*x])/e^9

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Rubi [A]  time = 0.557145, antiderivative size = 414, 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} $-\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac{c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac{c^3 x^2 (3 c d-2 b e)}{e^7}+\frac{c^4 x^3}{3 e^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^6,x]

[Out]

(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e)*x^2)/e^7 + (c^4*x^3)/(3*e^6)
- (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4)
- (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)))/(3*e^9*(d + e*x)^3) + (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)))/(e^9*(d + e*x)^2) - (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))/(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*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)^6} \, dx &=\int \left (\frac{c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right )}{e^8}-\frac{2 c^3 (3 c d-2 b e) x}{e^7}+\frac{c^4 x^2}{e^6}+\frac{\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^6}+\frac{4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^5}+\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 (d+e x)^4}+\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 )}{e^8 (d+e x)^3}+\frac{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 )}{e^8 (d+e x)^2}+\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 )}{e^8 (d+e x)}\right ) \, dx\\ &=\frac{c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac{c^3 (3 c d-2 b e) x^2}{e^7}+\frac{c^4 x^3}{3 e^6}-\frac{\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\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 )}{3 e^9 (d+e x)^3}+\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 )}{e^9 (d+e x)^2}-\frac{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 )}{e^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.233856, size = 419, normalized size = 1.01 $\frac{-\frac{15 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{d+e x}+\frac{30 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{(d+e x)^2}+15 c^2 e x \left (4 c e (a e-6 b d)+6 b^2 e^2+21 c^2 d^2\right )-\frac{10 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-60 c (2 c d-b e) \log (d+e x) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )+\frac{15 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^4}-\frac{3 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^5}+15 c^3 e^2 x^2 (2 b e-3 c d)+5 c^4 e^3 x^3}{15 e^9}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^6,x]

[Out]

(15*c^2*e*(21*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-6*b*d + a*e))*x + 15*c^3*e^2*(-3*c*d + 2*b*e)*x^2 + 5*c^4*e^3*x^3
- (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^5 + (15*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 -
(10*(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)^3 + (30*(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)))/(d + e*x)^2 - (15*(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) - 60*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7
*b*d + 3*a*e))*Log[d + e*x])/(15*e^9)

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Maple [B]  time = 0.06, size = 1341, normalized size = 3.2 \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)^6,x)

[Out]

4*c/e^6*ln(e*x+d)*b^3-56*c^4/e^9*ln(e*x+d)*d^3-1/5/e^5/(e*x+d)^5*d^4*b^4-1/5/e^9/(e*x+d)^5*c^4*d^8-6/e^5/(e*x+
d)*c^2*a^2-70/e^9/(e*x+d)*c^4*d^4+6*c^2/e^6*b^2*x+21*c^4/e^8*d^2*x-1/e^2/(e*x+d)^4*a^3*b+1/e^5/(e*x+d)^4*b^4*d
^3+2/e^9/(e*x+d)^4*c^4*d^7-4/3/e^3/(e*x+d)^3*a^3*c-2/e^3/(e*x+d)^3*a^2*b^2-2/e^5/(e*x+d)^3*b^4*d^2-28/3/e^9/(e
*x+d)^3*c^4*d^6-2/e^4/(e*x+d)^2*a*b^3+2/e^5/(e*x+d)^2*b^4*d+28/e^9/(e*x+d)^2*c^4*d^5+2*c^3/e^6*x^2*b-3*c^4/e^7
*x^2*d+4*c^3/e^6*a*x-30/e^7/(e*x+d)^3*b^2*c^2*d^4+28/e^8/(e*x+d)^3*b*c^3*d^5-6/e^4/(e*x+d)^2*a^2*b*c+12/e^5/(e
*x+d)^2*a^2*c^2*d+40/e^7/(e*x+d)^2*a*c^3*d^3-20/e^6/(e*x+d)^2*b^3*c*d^2+60/e^7/(e*x+d)^2*b^2*c^2*d^3-24*c^3/e^
7*b*d*x+2/e^3/(e*x+d)^4*a^3*c*d+3/e^3/(e*x+d)^4*a^2*b^2*d+6/e^5/(e*x+d)^4*a^2*c^2*d^3-3/e^4/(e*x+d)^4*a*b^3*d^
2+6/e^7/(e*x+d)^4*a*c^3*d^5-5/e^6/(e*x+d)^4*b^3*c*d^4+9/e^7/(e*x+d)^4*b^2*c^2*d^5-7/e^8/(e*x+d)^4*b*c^3*d^6-12
/e^5/(e*x+d)^3*a^2*c^2*d^2+4/e^4/(e*x+d)^3*a*b^3*d-20/e^7/(e*x+d)^3*a*c^3*d^4+40/3/e^6/(e*x+d)^3*b^3*c*d^3+12*
c^2/e^6*ln(e*x+d)*a*b-24*c^3/e^7*ln(e*x+d)*a*d-1/5/e/(e*x+d)^5*a^4-1/e^5/(e*x+d)*b^4-70/e^8/(e*x+d)^2*b*c^3*d^
4-12/e^5/(e*x+d)*a*b^2*c-60/e^7/(e*x+d)*c^3*a*d^2+20/e^6/(e*x+d)*b^3*c*d-90/e^7/(e*x+d)*b^2*c^2*d^2+140/e^8/(e
*x+d)*b*c^3*d^3+1/3*c^4*x^3/e^6-36*c^2/e^7*ln(e*x+d)*b^2*d+84*c^3/e^8*ln(e*x+d)*b*d^2+4/5/e^2/(e*x+d)^5*d*a^3*
b-4/5/e^3/(e*x+d)^5*a^3*c*d^2-6/5/e^3/(e*x+d)^5*d^2*a^2*b^2-6/5/e^5/(e*x+d)^5*a^2*c^2*d^4+4/5/e^4/(e*x+d)^5*d^
3*a*b^3-4/5/e^7/(e*x+d)^5*a*c^3*d^6+4/5/e^6/(e*x+d)^5*d^5*b^3*c-6/5/e^7/(e*x+d)^5*b^2*c^2*d^6+4/5/e^8/(e*x+d)^
5*b*c^3*d^7+12/e^4/(e*x+d)^3*a^2*b*c*d-24/e^5/(e*x+d)^3*a*b^2*c*d^2+40/e^6/(e*x+d)^3*a*b*c^2*d^3+24/e^5/(e*x+d
)^2*a*b^2*c*d-60/e^6/(e*x+d)^2*a*b*c^2*d^2+12/5/e^4/(e*x+d)^5*d^3*a^2*b*c+12/5/e^6/(e*x+d)^5*d^5*a*b*c^2+60/e^
6/(e*x+d)*a*b*c^2*d-12/5/e^5/(e*x+d)^5*d^4*a*b^2*c-9/e^4/(e*x+d)^4*a^2*b*c*d^2+12/e^5/(e*x+d)^4*a*b^2*c*d^3-15
/e^6/(e*x+d)^4*a*b*c^2*d^4

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Maxima [B]  time = 1.13759, size = 1148, normalized size = 2.77 \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)^6,x, algorithm="maxima")

[Out]

-1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 3*a^3*b*d*e^7 + 3*a^4*e^8 + 261*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 137*(b
^3*c + 3*a*b*c^2)*d^5*e^3 + 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 + 15*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 20*(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 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 50*(
3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 30*(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 + (a*b^3
+ 3*a^2*b*c)*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 195*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(b^3
*c + 3*a*b*c^2)*d^3*e^5 + 3*(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 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3
- 125*(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 + 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)/(e^14*x^5 + 5*d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x +
d^5*e^9) + 1/3*(c^4*e^2*x^3 - 3*(3*c^4*d*e - 2*b*c^3*e^2)*x^2 + 3*(21*c^4*d^2 - 24*b*c^3*d*e + 2*(3*b^2*c^2 +
2*a*c^3)*e^2)*x)/e^8 - 4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e
^3)*log(e*x + d)/e^9

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Fricas [B]  time = 1.77927, size = 2645, normalized size = 6.39 \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)^6,x, algorithm="fricas")

[Out]

1/15*(5*c^4*e^8*x^8 - 743*c^4*d^8 + 1377*b*c^3*d^7*e - 3*a^3*b*d*e^7 - 3*a^4*e^8 - 261*(3*b^2*c^2 + 2*a*c^3)*d
^6*e^2 + 137*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 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 - 10*(2*c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + 10*(14*c^4*d^2*e^6 - 21*b*c^3*d*e
^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 25*(47*c^4*d^3*e^5 - 60*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7)
*x^5 + 5*(335*c^4*d^4*e^4 - 240*b*c^3*d^3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 60*(b^3*c + 3*a*b*c^2)*d*e^
7 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 10*(85*c^4*d^5*e^3 - 390*b*c^3*d^4*e^4 + 120*(3*b^2*c^2 + 2*a*
c^3)*d^3*e^5 - 90*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(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 - 10*(365*c^4*d^6*e^2 - 810*b*c^3*d^5*e^3 + 180*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(b^3*c + 3*a*b*c
^2)*d^3*e^5 + 3*(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 - 5*(575*c^4*d^7*e - 1125*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 225*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 125*(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 + 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 - 60*(14*c^4*d^8 - 21*b*c^3*d^7*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - (b^3*c + 3*a*b*c^2)*
d^5*e^3 + (14*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - (b^3*c + 3*a*b*c^2)*e^8)*x^5 +
5*(14*c^4*d^4*e^4 - 21*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - (b^3*c + 3*a*b*c^2)*d*e^7)*x^4 + 10*(
14*c^4*d^5*e^3 - 21*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - (b^3*c + 3*a*b*c^2)*d^2*e^6)*x^3 + 10*(1
4*c^4*d^6*e^2 - 21*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - (b^3*c + 3*a*b*c^2)*d^3*e^5)*x^2 + 5*(14*
c^4*d^7*e - 21*b*c^3*d^6*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - (b^3*c + 3*a*b*c^2)*d^4*e^4)*x)*log(e*x + d))
/(e^14*x^5 + 5*d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x + d^5*e^9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.12174, size = 1135, normalized size = 2.74 \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)^6,x, algorithm="giac")

[Out]

-4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*e^(-9)*log(abs(
x*e + d)) + 1/3*(c^4*x^3*e^12 - 9*c^4*d*x^2*e^11 + 63*c^4*d^2*x*e^10 + 6*b*c^3*x^2*e^12 - 72*b*c^3*d*x*e^11 +
18*b^2*c^2*x*e^12 + 12*a*c^3*x*e^12)*e^(-18) - 1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 783*b^2*c^2*d^6*e^2 + 52
2*a*c^3*d^6*e^2 - 137*b^3*c*d^5*e^3 - 411*a*b*c^2*d^5*e^3 + 3*b^4*d^4*e^4 + 36*a*b^2*c*d^4*e^4 + 18*a^2*c^2*d^
4*e^4 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 3*a^2*b^2*d^2*e^6 + 2*a^3*c*d^2*e^6 + 3*a^3*b*d*e^7 + 15*(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)*x^4 + 3*a^4*e^8 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 150*b^2*c^2
*d^3*e^5 + 100*a*c^3*d^3*e^5 - 30*b^3*c*d^2*e^6 - 90*a*b*c^2*d^2*e^6 + b^4*d*e^7 + 12*a*b^2*c*d*e^7 + 6*a^2*c^
2*d*e^7 + a*b^3*e^8 + 3*a^2*b*c*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 585*b^2*c^2*d^4*e^4 + 390
*a*c^3*d^4*e^4 - 110*b^3*c*d^3*e^5 - 330*a*b*c^2*d^3*e^5 + 3*b^4*d^2*e^6 + 36*a*b^2*c*d^2*e^6 + 18*a^2*c^2*d^2
*e^6 + 3*a*b^3*d*e^7 + 9*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^2 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*
e^2 + 693*b^2*c^2*d^5*e^3 + 462*a*c^3*d^5*e^3 - 125*b^3*c*d^4*e^4 - 375*a*b*c^2*d^4*e^4 + 3*b^4*d^3*e^5 + 36*a
*b^2*c*d^3*e^5 + 18*a^2*c^2*d^3*e^5 + 3*a*b^3*d^2*e^6 + 9*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 +
3*a^3*b*e^8)*x)*e^(-9)/(x*e + d)^5