### 3.2158 $$\int \frac{(a+b x+c x^2)^4}{(d+e x)^8} \, dx$$

Optimal. Leaf size=424 $-\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac{2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac{4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac{c^4 x}{e^8}$

[Out]

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

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Rubi [A]  time = 0.481811, antiderivative size = 424, 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} $-\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac{2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac{4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac{c^4 x}{e^8}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

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

Mathematica [A]  time = 0.548716, size = 748, normalized size = 1.76 $-\frac{6 c^2 e^2 \left (a^2 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 a b e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+15 b^2 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )\right )+c e^3 \left (9 a^2 b e^2 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 a^3 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+12 a b^2 e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+10 b^3 \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )+e^4 \left (6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+10 a^3 b e^3 (d+7 e x)+15 a^4 e^4+3 a b^3 e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+c^3 e \left (60 a e \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )-b d \left (20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+7203 d^5 e x+1089 d^6+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 c^3 (d+e x)^7 (2 c d-b e) \log (d+e x)+c^4 \left (24843 d^6 e^2 x^2+35525 d^5 e^3 x^3+28175 d^4 e^4 x^4+11025 d^3 e^5 x^5+735 d^2 e^6 x^6+9261 d^7 e x+1443 d^8-735 d e^7 x^7-105 e^8 x^8\right )}{105 e^9 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

-(c^4*(1443*d^8 + 9261*d^7*e*x + 24843*d^6*e^2*x^2 + 35525*d^5*e^3*x^3 + 28175*d^4*e^4*x^4 + 11025*d^3*e^5*x^5
+ 735*d^2*e^6*x^6 - 735*d*e^7*x^7 - 105*e^8*x^8) + e^4*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2
*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d^3*e*x
+ 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + c*e^3*(4*a^3*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 9*a^2*b*e^2*
(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 12*a*b^2*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 +
35*e^4*x^4) + 10*b^3*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 6*c^2*
e^2*(a^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*a*b*e*(d^5 + 7*d^4*e*x + 21*d^
3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 15*b^2*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^
3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + c^3*e*(60*a*e*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3
*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - b*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2 + 306
25*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*c^3*(2*c*d - b*e)*(d + e*x)^7*Log[
d + e*x])/(105*e^9*(d + e*x)^7)

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Maple [B]  time = 0.054, size = 1374, 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)^8,x)

[Out]

-3/e^4/(e*x+d)^4*a^2*b*c+6/e^5/(e*x+d)^4*a^2*c^2*d+2/3/e^5/(e*x+d)^6*b^4*d^3+4/3/e^9/(e*x+d)^6*c^4*d^7-4/5/e^3
/(e*x+d)^5*a^3*c-6/5/e^3/(e*x+d)^5*a^2*b^2-6/5/e^5/(e*x+d)^5*b^4*d^2-28/5/e^9/(e*x+d)^5*c^4*d^6-4*c^3/e^7/(e*x
+d)*a-6*c^2/e^7/(e*x+d)*b^2-28*c^4/e^9/(e*x+d)*d^2+14/e^9/(e*x+d)^4*c^4*d^5-2/e^5/(e*x+d)^3*c^2*a^2-70/3/e^9/(
e*x+d)^3*c^4*d^4-2*c/e^6/(e*x+d)^2*b^3+28*c^4/e^9/(e*x+d)^2*d^3+4*c^3/e^8*ln(e*x+d)*b-8*c^4/e^9*ln(e*x+d)*d-1/
e^4/(e*x+d)^4*a*b^3+1/e^5/(e*x+d)^4*b^4*d-1/7/e^5/(e*x+d)^7*d^4*b^4-1/7/e^9/(e*x+d)^7*c^4*d^8-2/3/e^2/(e*x+d)^
6*a^3*b+36/5/e^4/(e*x+d)^5*a^2*b*c*d-72/5/e^5/(e*x+d)^5*a*b^2*c*d^2-35/e^8/(e*x+d)^4*b*c^3*d^4-4/e^5/(e*x+d)^3
*a*b^2*c-20/e^7/(e*x+d)^3*c^3*a*d^2+28*c^3/e^8/(e*x+d)*b*d+4/3/e^3/(e*x+d)^6*a^3*c*d+2/e^3/(e*x+d)^6*a^2*b^2*d
+4/e^5/(e*x+d)^6*a^2*c^2*d^3-2/e^4/(e*x+d)^6*a*b^3*d^2+4/e^7/(e*x+d)^6*a*c^3*d^5-10/3/e^6/(e*x+d)^6*b^3*c*d^4-
14/3/e^8/(e*x+d)^6*b*c^3*d^6-36/5/e^5/(e*x+d)^5*a^2*c^2*d^2+20/e^7/(e*x+d)^4*a*c^3*d^3-10/e^6/(e*x+d)^4*b^3*c*
d^2+4/7/e^2/(e*x+d)^7*d*a^3*b-4/7/e^3/(e*x+d)^7*a^3*c*d^2-6/7/e^3/(e*x+d)^7*d^2*a^2*b^2-6/7/e^5/(e*x+d)^7*a^2*
c^2*d^4+4/7/e^4/(e*x+d)^7*d^3*a*b^3+12/5/e^4/(e*x+d)^5*a*b^3*d-12/e^7/(e*x+d)^5*a*c^3*d^4+8/e^6/(e*x+d)^5*b^3*
c*d^3-18/e^7/(e*x+d)^5*b^2*c^2*d^4+84/5/e^8/(e*x+d)^5*b*c^3*d^5+20/e^6/(e*x+d)^3*a*b*c^2*d-6/e^4/(e*x+d)^6*a^2
*b*c*d^2+8/e^5/(e*x+d)^6*a*b^2*c*d^3-10/e^6/(e*x+d)^6*a*b*c^2*d^4-30/e^6/(e*x+d)^4*a*b*c^2*d^2-1/7/e/(e*x+d)^7
*a^4-1/3/e^5/(e*x+d)^3*b^4-4/7/e^7/(e*x+d)^7*a*c^3*d^6+4/7/e^6/(e*x+d)^7*d^5*b^3*c-6/7/e^7/(e*x+d)^7*d^6*b^2*c
^2+4/7/e^8/(e*x+d)^7*d^7*b*c^3+20/3/e^6/(e*x+d)^3*b^3*c*d-30/e^7/(e*x+d)^3*b^2*c^2*d^2+140/3/e^8/(e*x+d)^3*b*c
^3*d^3-6*c^2/e^6/(e*x+d)^2*a*b+12*c^3/e^7/(e*x+d)^2*a*d+18*c^2/e^7/(e*x+d)^2*b^2*d-42*c^3/e^8/(e*x+d)^2*b*d^2+
30/e^7/(e*x+d)^4*b^2*c^2*d^3+24/e^6/(e*x+d)^5*a*b*c^2*d^3+6/e^7/(e*x+d)^6*b^2*c^2*d^5+12/e^5/(e*x+d)^4*a*b^2*c
*d+12/7/e^4/(e*x+d)^7*d^3*a^2*b*c-12/7/e^5/(e*x+d)^7*d^4*a*b^2*c+12/7/e^6/(e*x+d)^7*d^5*a*b*c^2+c^4*x/e^8

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Maxima [B]  time = 1.12256, size = 1177, normalized size = 2.78 \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)^8,x, algorithm="maxima")

[Out]

-1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 10*a^3*b*d*e^7 + 15*a^4*e^8 + 30*(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 + 3*(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 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 210*(70*c^4
*d^3*e^5 - 63*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 + 35*(910*c^4*d^4*e
^4 - 770*b*c^3*d^3*e^5 + 30*(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 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30*(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 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 21*(1218*c^4*d
^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(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 + 3*(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 + 7*(1338*c^4*d^7*e
- 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(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 + 3*(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)/(e^16*
x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e^11*x^2 + 7*d^6*e^10*x + d^
7*e^9) + c^4*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(e*x + d)/e^9

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Fricas [B]  time = 1.80653, size = 2275, 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)^8,x, algorithm="fricas")

[Out]

1/105*(105*c^4*e^8*x^8 + 735*c^4*d*e^7*x^7 - 1443*c^4*d^8 + 1089*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 15*a^4*e^8 - 3
0*(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 - 3*
(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 - 105*(7*c^4*d^2*e^6 - 28*b*c^3*d*e^7 + 2*(3*b^2
*c^2 + 2*a*c^3)*e^8)*x^6 - 105*(105*c^4*d^3*e^5 - 126*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 2*(b^3*c
+ 3*a*b*c^2)*e^8)*x^5 - 35*(805*c^4*d^4*e^4 - 770*b*c^3*d^3*e^5 + 30*(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 - 35*(1015*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30
*(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 + 3*(a*
b^3 + 3*a^2*b*c)*e^8)*x^3 - 21*(1183*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(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 + 3*(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 - 7*(1323*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(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 + 3*(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 - b*c^3*d^7*e + (2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 7*(2*c^4*d
^2*e^6 - b*c^3*d*e^7)*x^6 + 21*(2*c^4*d^3*e^5 - b*c^3*d^2*e^6)*x^5 + 35*(2*c^4*d^4*e^4 - b*c^3*d^3*e^5)*x^4 +
35*(2*c^4*d^5*e^3 - b*c^3*d^4*e^4)*x^3 + 21*(2*c^4*d^6*e^2 - b*c^3*d^5*e^3)*x^2 + 7*(2*c^4*d^7*e - b*c^3*d^6*e
^2)*x)*log(e*x + d))/(e^16*x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e
^11*x^2 + 7*d^6*e^10*x + d^7*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)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.15361, size = 1125, normalized size = 2.65 \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)^8,x, algorithm="giac")

[Out]

c^4*x*e^(-8) - 4*(2*c^4*d - b*c^3*e)*e^(-9)*log(abs(x*e + d)) - 1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 90*b^
2*c^2*d^6*e^2 + 60*a*c^3*d^6*e^2 + 10*b^3*c*d^5*e^3 + 30*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 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + 3*b^2*c^2*e^8
+ 2*a*c^3*e^8)*x^6 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 210*(70*c^4*d^3*e^5 - 63*b*c^3*d^2*e^6 + 9*b^2*c^2
*d*e^7 + 6*a*c^3*d*e^7 + b^3*c*e^8 + 3*a*b*c^2*e^8)*x^5 + 10*a^3*b*d*e^7 + 35*(910*c^4*d^4*e^4 - 770*b*c^3*d^3
*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^3*d^2*e^6 + 10*b^3*c*d*e^7 + 30*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 + 15*a^4*e^8 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 90*b^2*c^2*d^3*e^5 + 60*a*c^3*d^3
*e^5 + 10*b^3*c*d^2*e^6 + 30*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 + 3*a*b^3*e^8 +
9*a^2*b*c*e^8)*x^3 + 21*(1218*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 90*b^2*c^2*d^4*e^4 + 60*a*c^3*d^4*e^4 + 10*b^3
*c*d^3*e^5 + 30*a*b*c^2*d^3*e^5 + b^4*d^2*e^6 + 12*a*b^2*c*d^2*e^6 + 6*a^2*c^2*d^2*e^6 + 3*a*b^3*d*e^7 + 9*a^2
*b*c*d*e^7 + 6*a^2*b^2*e^8 + 4*a^3*c*e^8)*x^2 + 7*(1338*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 90*b^2*c^2*d^5*e^3 +
60*a*c^3*d^5*e^3 + 10*b^3*c*d^4*e^4 + 30*a*b*c^2*d^4*e^4 + b^4*d^3*e^5 + 12*a*b^2*c*d^3*e^5 + 6*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 + 6*a^2*b^2*d*e^7 + 4*a^3*c*d*e^7 + 10*a^3*b*e^8)*x)*e^(-9)/(x*e + d)^
7