### 3.2154 $$\int \frac{(a+b x+c x^2)^4}{(d+e x)^4} \, dx$$

Optimal. Leaf size=417 $\frac{x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}+\frac{2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac{c^3 x^4 (c d-b e)}{e^5}+\frac{c^4 x^5}{5 e^4}$

[Out]

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*
a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*
x^2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)/e^5 + (c^
4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(
e^9*(d + e*x)^2) - (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^9*(d + e*x)
) - (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))*Log[d + e*x])/e^9

________________________________________________________________________________________

Rubi [A]  time = 0.651569, antiderivative size = 417, 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} $\frac{x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}+\frac{2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac{c^3 x^4 (c d-b e)}{e^5}+\frac{c^4 x^5}{5 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*
a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*
x^2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)/e^5 + (c^
4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(
e^9*(d + e*x)^2) - (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^9*(d + e*x)
) - (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))*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)^4} \, dx &=\int \left (\frac{35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )}{e^8}+\frac{4 c \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)-3 b c e^2 (2 b d-a e)\right ) x}{e^7}+\frac{2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^2}{e^6}-\frac{4 c^3 (c d-b e) x^3}{e^5}+\frac{c^4 x^4}{e^4}+\frac{\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^4}+\frac{4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac{\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac{2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac{2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac{c^3 (c d-b e) x^4}{e^5}+\frac{c^4 x^5}{5 e^4}-\frac{\left (c d^2-b d e+a e^2\right )^4}{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 )^3}{e^9 (d+e x)^2}-\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^9 (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+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.22621, size = 425, normalized size = 1.02 $\frac{15 e x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (a e-2 b d)+b^4 e^4+35 c^4 d^4\right )-60 (2 c d-b e) \log (d+e x) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )+10 c^2 e^3 x^3 \left (2 c e (a e-4 b d)+3 b^2 e^2+5 c^2 d^2\right )+30 c e^2 x^2 \left (2 c^2 d e (5 b d-2 a e)+3 b c e^2 (a e-2 b d)+b^3 e^3-5 c^3 d^3\right )-\frac{30 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+\frac{30 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}-\frac{5 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^3}+15 c^3 e^4 x^4 (b e-c d)+3 c^4 e^5 x^5}{15 e^9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^
2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d +
a*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3
*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/
(d + e*x)^2 - (30*(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])/(15*e^9)

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 1265, normalized size = 3. \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)^4,x)

[Out]

6/e^4*x^2*a*b*c^2+4/3/e^8/(e*x+d)^3*b*c^3*d^7+4/e^3/(e*x+d)^2*a^3*c*d+6/e^3/(e*x+d)^2*a^2*b^2*d+12/e^5/(e*x+d)
^2*a^2*c^2*d^3-6/e^4/(e*x+d)^2*a*b^3*d^2+12/e^7/(e*x+d)^2*a*c^3*d^5-10/e^6/(e*x+d)^2*b^3*c*d^4+18/e^7/(e*x+d)^
2*b^2*c^2*d^5-14/e^8/(e*x+d)^2*b*c^3*d^6-8/e^5*x^2*a*c^3*d-12/e^5*x^2*b^2*c^2*d+20/e^6*x^2*b*c^3*d^2+12/e^4*a*
c*b^2*x+40/e^6*c^3*a*d^2*x-16/e^5*b^3*c*d*x+60/e^6*b^2*c^2*d^2*x-36/e^5/(e*x+d)*a^2*c^2*d^2+12/e^4/(e*x+d)*a*b
^3*d-60/e^7/(e*x+d)*a*c^3*d^4+40/e^6/(e*x+d)*b^3*c*d^3-90/e^7/(e*x+d)*b^2*c^2*d^4+84/e^8/(e*x+d)*b*c^3*d^5-2/e
^5/(e*x+d)^3*a^2*c^2*d^4-1/3/e/(e*x+d)^3*a^4+1/5*c^4*x^5/e^4+1/e^4*x^4*b*c^3-1/e^5*x^4*c^4*d+4/3/e^4*x^3*a*c^3
+2/e^4*x^3*b^2*c^2+10/3/e^6*x^3*c^4*d^2+2/e^4*x^2*b^3*c-10/e^7*x^2*c^4*d^3+6/e^4*c^2*a^2*x+35/e^8*c^4*d^4*x-1/
3/e^5/(e*x+d)^3*b^4*d^4-1/3/e^9/(e*x+d)^3*c^4*d^8-2/e^2/(e*x+d)^2*a^3*b+2/e^5/(e*x+d)^2*b^4*d^3+4/e^9/(e*x+d)^
2*c^4*d^7+4/e^4*ln(e*x+d)*a*b^3-4/e^5*ln(e*x+d)*b^4*d-56/e^9*ln(e*x+d)*c^4*d^5-4/e^3/(e*x+d)*a^3*c-16/3/e^5*x^
3*b*c^3*d+4/3/e^4/(e*x+d)^3*d^3*a*b^3-4/3/e^7/(e*x+d)^3*a*c^3*d^6+4/3/e^6/(e*x+d)^3*b^3*c*d^5+12/e^4*ln(e*x+d)
*a^2*b*c-24/e^5*ln(e*x+d)*a^2*c^2*d-80/e^7*ln(e*x+d)*a*c^3*d^3+40/e^6*ln(e*x+d)*b^3*c*d^2-120/e^7*ln(e*x+d)*b^
2*c^2*d^3+140/e^8*ln(e*x+d)*b*c^3*d^4-80/e^7*b*c^3*d^3*x+4/3/e^2/(e*x+d)^3*d*a^3*b-4/3/e^3/(e*x+d)^3*a^3*c*d^2
-2/e^3/(e*x+d)^3*d^2*a^2*b^2-2/e^7/(e*x+d)^3*b^2*c^2*d^6-6/e^3/(e*x+d)*a^2*b^2-6/e^5/(e*x+d)*b^4*d^2-28/e^9/(e
*x+d)*c^4*d^6+b^4*x/e^4+4/e^4/(e*x+d)^3*d^3*a^2*b*c-4/e^5/(e*x+d)^3*a*b^2*c*d^4+4/e^6/(e*x+d)^3*a*b*c^2*d^5-18
/e^4/(e*x+d)^2*a^2*b*c*d^2+24/e^5/(e*x+d)^2*a*b^2*c*d^3-30/e^6/(e*x+d)^2*a*b*c^2*d^4-48/e^5*ln(e*x+d)*a*b^2*c*
d+120/e^6*ln(e*x+d)*a*b*c^2*d^2+36/e^4/(e*x+d)*a^2*b*c*d-72/e^5/(e*x+d)*a*b^2*c*d^2+120/e^6/(e*x+d)*a*b*c^2*d^
3-48/e^5*a*b*c^2*d*x

________________________________________________________________________________________

Maxima [B]  time = 1.20639, size = 1116, normalized size = 2.68 \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)^4,x, algorithm="maxima")

[Out]

-1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 2*a^3*b*d*e^7 + a^4*e^8 + 74*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 94*(b^3*c +
3*a*b*c^2)*d^5*e^3 + 13*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 22*(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 + 6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(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 - 6*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3
*c)*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + a^3*b*e^8 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 35*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 9*(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^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9) + 1/15*(3*c^4*e^4*x^5 - 15*(c^4*d*e^3 -
b*c^3*e^4)*x^4 + 10*(5*c^4*d^2*e^2 - 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 30*(5*c^4*d^3*e - 10*b*c
^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 - (b^3*c + 3*a*b*c^2)*e^4)*x^2 + 15*(35*c^4*d^4 - 80*b*c^3*d^3*e +
20*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 16*(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^8 -
4*(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 + 1
2*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^9

________________________________________________________________________________________

Fricas [B]  time = 1.91431, size = 2708, normalized size = 6.49 \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)^4,x, algorithm="fricas")

[Out]

1/15*(3*c^4*e^8*x^8 - 365*c^4*d^8 + 1070*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 5*a^4*e^8 - 370*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 + 470*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 65*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 110*(a*b^3 + 3*a^2*b*c
)*d^3*e^5 - 10*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 3*(2*c^4*d*e^7 - 5*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 - 35*b*c^3*
d*e^7 + 10*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 3*(14*c^4*d^3*e^5 - 35*b*c^3*d^2*e^6 + 10*(3*b^2*c^2 + 2*a*c^3)*d*
e^7 - 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 15*(14*c^4*d^4*e^4 - 35*b*c^3*d^3*e^5 + 10*(3*b^2*c^2 + 2*a*c^3)*d^2*e
^6 - 10*(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 + 5*(235*c^4*d^5*e^3 - 556*b*c^3*d
^4*e^4 + 146*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 126*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d*e^7)*x^3 + 15*(67*c^4*d^6*e^2 - 136*b*c^3*d^5*e^3 + 26*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 6*(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 + 12*(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 - 15*(17*c^4*d^7*e - 74*b*c^3*d^6*e^2 + 2*a^3*b*e^8 + 34*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 54*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 18*(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 - 60*(14*c^4*d^8 - 35*b*c^3*d^7*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d^3*e^5 + (14*c^4*d^5*e^3 - 35*b*c
^3*d^4*e^4 + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 10*(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 + 3*(14*c^4*d^6*e^2 - 35*b*c^3*d^5*e^3 + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e
^4 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d*e^7)*x^2
+ 3*(14*c^4*d^7*e - 35*b*c^3*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d^2*e^6)*x)*log(e*x + d))/(e^12*x^3 + 3*d*e^11*x^2 +
3*d^2*e^10*x + d^3*e^9)

________________________________________________________________________________________

Sympy [B]  time = 111.556, size = 933, normalized size = 2.24 \begin{align*} \frac{c^{4} x^{5}}{5 e^{4}} - \frac{a^{4} e^{8} + 2 a^{3} b d e^{7} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} b^{2} d^{2} e^{6} - 66 a^{2} b c d^{3} e^{5} + 78 a^{2} c^{2} d^{4} e^{4} - 22 a b^{3} d^{3} e^{5} + 156 a b^{2} c d^{4} e^{4} - 282 a b c^{2} d^{5} e^{3} + 148 a c^{3} d^{6} e^{2} + 13 b^{4} d^{4} e^{4} - 94 b^{3} c d^{5} e^{3} + 222 b^{2} c^{2} d^{6} e^{2} - 214 b c^{3} d^{7} e + 73 c^{4} d^{8} + x^{2} \left (12 a^{3} c e^{8} + 18 a^{2} b^{2} e^{8} - 108 a^{2} b c d e^{7} + 108 a^{2} c^{2} d^{2} e^{6} - 36 a b^{3} d e^{7} + 216 a b^{2} c d^{2} e^{6} - 360 a b c^{2} d^{3} e^{5} + 180 a c^{3} d^{4} e^{4} + 18 b^{4} d^{2} e^{6} - 120 b^{3} c d^{3} e^{5} + 270 b^{2} c^{2} d^{4} e^{4} - 252 b c^{3} d^{5} e^{3} + 84 c^{4} d^{6} e^{2}\right ) + x \left (6 a^{3} b e^{8} + 12 a^{3} c d e^{7} + 18 a^{2} b^{2} d e^{7} - 162 a^{2} b c d^{2} e^{6} + 180 a^{2} c^{2} d^{3} e^{5} - 54 a b^{3} d^{2} e^{6} + 360 a b^{2} c d^{3} e^{5} - 630 a b c^{2} d^{4} e^{4} + 324 a c^{3} d^{5} e^{3} + 30 b^{4} d^{3} e^{5} - 210 b^{3} c d^{4} e^{4} + 486 b^{2} c^{2} d^{5} e^{3} - 462 b c^{3} d^{6} e^{2} + 156 c^{4} d^{7} e\right )}{3 d^{3} e^{9} + 9 d^{2} e^{10} x + 9 d e^{11} x^{2} + 3 e^{12} x^{3}} + \frac{x^{4} \left (b c^{3} e - c^{4} d\right )}{e^{5}} + \frac{x^{3} \left (4 a c^{3} e^{2} + 6 b^{2} c^{2} e^{2} - 16 b c^{3} d e + 10 c^{4} d^{2}\right )}{3 e^{6}} + \frac{x^{2} \left (6 a b c^{2} e^{3} - 8 a c^{3} d e^{2} + 2 b^{3} c e^{3} - 12 b^{2} c^{2} d e^{2} + 20 b c^{3} d^{2} e - 10 c^{4} d^{3}\right )}{e^{7}} + \frac{x \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 48 a b c^{2} d e^{3} + 40 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 16 b^{3} c d e^{3} + 60 b^{2} c^{2} d^{2} e^{2} - 80 b c^{3} d^{3} e + 35 c^{4} d^{4}\right )}{e^{8}} + \frac{4 \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^{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)**4,x)

[Out]

c**4*x**5/(5*e**4) - (a**4*e**8 + 2*a**3*b*d*e**7 + 4*a**3*c*d**2*e**6 + 6*a**2*b**2*d**2*e**6 - 66*a**2*b*c*d
**3*e**5 + 78*a**2*c**2*d**4*e**4 - 22*a*b**3*d**3*e**5 + 156*a*b**2*c*d**4*e**4 - 282*a*b*c**2*d**5*e**3 + 14
8*a*c**3*d**6*e**2 + 13*b**4*d**4*e**4 - 94*b**3*c*d**5*e**3 + 222*b**2*c**2*d**6*e**2 - 214*b*c**3*d**7*e + 7
3*c**4*d**8 + x**2*(12*a**3*c*e**8 + 18*a**2*b**2*e**8 - 108*a**2*b*c*d*e**7 + 108*a**2*c**2*d**2*e**6 - 36*a*
b**3*d*e**7 + 216*a*b**2*c*d**2*e**6 - 360*a*b*c**2*d**3*e**5 + 180*a*c**3*d**4*e**4 + 18*b**4*d**2*e**6 - 120
*b**3*c*d**3*e**5 + 270*b**2*c**2*d**4*e**4 - 252*b*c**3*d**5*e**3 + 84*c**4*d**6*e**2) + x*(6*a**3*b*e**8 + 1
2*a**3*c*d*e**7 + 18*a**2*b**2*d*e**7 - 162*a**2*b*c*d**2*e**6 + 180*a**2*c**2*d**3*e**5 - 54*a*b**3*d**2*e**6
+ 360*a*b**2*c*d**3*e**5 - 630*a*b*c**2*d**4*e**4 + 324*a*c**3*d**5*e**3 + 30*b**4*d**3*e**5 - 210*b**3*c*d**
4*e**4 + 486*b**2*c**2*d**5*e**3 - 462*b*c**3*d**6*e**2 + 156*c**4*d**7*e))/(3*d**3*e**9 + 9*d**2*e**10*x + 9*
d*e**11*x**2 + 3*e**12*x**3) + x**4*(b*c**3*e - c**4*d)/e**5 + x**3*(4*a*c**3*e**2 + 6*b**2*c**2*e**2 - 16*b*c
**3*d*e + 10*c**4*d**2)/(3*e**6) + x**2*(6*a*b*c**2*e**3 - 8*a*c**3*d*e**2 + 2*b**3*c*e**3 - 12*b**2*c**2*d*e*
*2 + 20*b*c**3*d**2*e - 10*c**4*d**3)/e**7 + x*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 48*a*b*c**2*d*e**3 + 40*
a*c**3*d**2*e**2 + b**4*e**4 - 16*b**3*c*d*e**3 + 60*b**2*c**2*d**2*e**2 - 80*b*c**3*d**3*e + 35*c**4*d**4)/e*
*8 + 4*(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**9

________________________________________________________________________________________

Giac [B]  time = 1.10175, size = 1168, normalized size = 2.8 \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)^4,x, algorithm="giac")

[Out]

-4*(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^(-9)*log(abs(x*e + d)) + 1/1
5*(3*c^4*x^5*e^16 - 15*c^4*d*x^4*e^15 + 50*c^4*d^2*x^3*e^14 - 150*c^4*d^3*x^2*e^13 + 525*c^4*d^4*x*e^12 + 15*b
*c^3*x^4*e^16 - 80*b*c^3*d*x^3*e^15 + 300*b*c^3*d^2*x^2*e^14 - 1200*b*c^3*d^3*x*e^13 + 30*b^2*c^2*x^3*e^16 + 2
0*a*c^3*x^3*e^16 - 180*b^2*c^2*d*x^2*e^15 - 120*a*c^3*d*x^2*e^15 + 900*b^2*c^2*d^2*x*e^14 + 600*a*c^3*d^2*x*e^
14 + 30*b^3*c*x^2*e^16 + 90*a*b*c^2*x^2*e^16 - 240*b^3*c*d*x*e^15 - 720*a*b*c^2*d*x*e^15 + 15*b^4*x*e^16 + 180
*a*b^2*c*x*e^16 + 90*a^2*c^2*x*e^16)*e^(-20) - 1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 222*b^2*c^2*d^6*e^2 + 148*a
*c^3*d^6*e^2 - 94*b^3*c*d^5*e^3 - 282*a*b*c^2*d^5*e^3 + 13*b^4*d^4*e^4 + 156*a*b^2*c*d^4*e^4 + 78*a^2*c^2*d^4*
e^4 - 22*a*b^3*d^3*e^5 - 66*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 + 2*a^3*b*d*e^7 + a^4*e^8 +
6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 45*b^2*c^2*d^4*e^4 + 30*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 - 60*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 - 6*a*b^3*d*e^7 - 18*a^2*b*c*d*e^7 + 3*a^2*b^
2*e^8 + 2*a^3*c*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + 81*b^2*c^2*d^5*e^3 + 54*a*c^3*d^5*e^3 - 35*b^3
*c*d^4*e^4 - 105*a*b*c^2*d^4*e^4 + 5*b^4*d^3*e^5 + 60*a*b^2*c*d^3*e^5 + 30*a^2*c^2*d^3*e^5 - 9*a*b^3*d^2*e^6 -
27*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 + a^3*b*e^8)*x)*e^(-9)/(x*e + d)^3