3.2159 \(\int \frac{(a+b x+c x^2)^4}{(d+e x)^9} \, dx\)

Optimal. Leaf size=435 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{4 e^9 (d+e x)^4}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^4}{8 e^9 (d+e x)^8}+\frac{4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac{c^4 \log (d+e x)}{e^9} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.462165, antiderivative size = 435, 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} \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{4 e^9 (d+e x)^4}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^4}{8 e^9 (d+e x)^8}+\frac{4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac{c^4 \log (d+e x)}{e^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^9,x]

[Out]

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

Mathematica [A]  time = 0.352392, size = 740, normalized size = 1.7 \[ \frac{-6 c^2 e^2 \left (3 a^2 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+10 a b e \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+15 b^2 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )\right )-4 c e^3 \left (9 a^2 b e^2 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+5 a^3 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+9 a b^2 e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 b^3 \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )-3 e^4 \left (10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+20 a^3 b e^3 (d+8 e x)+35 a^4 e^4+4 a b^3 e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )-60 c^3 e \left (a e \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )+7 b \left (28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+8 d^6 e x+d^7+28 d e^6 x^6+8 e^7 x^7\right )\right )+c^4 d \left (57624 d^5 e^2 x^2+107408 d^4 e^3 x^3+122500 d^3 e^4 x^4+86240 d^2 e^5 x^5+17424 d^6 e x+2283 d^7+35280 d e^6 x^6+6720 e^7 x^7\right )+840 c^4 (d+e x)^8 \log (d+e x)}{840 e^9 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^9,x]

[Out]

(c^4*d*(2283*d^7 + 17424*d^6*e*x + 57624*d^5*e^2*x^2 + 107408*d^4*e^3*x^3 + 122500*d^3*e^4*x^4 + 86240*d^2*e^5
*x^5 + 35280*d*e^6*x^6 + 6720*e^7*x^7) - 3*e^4*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 +
8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^
2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) - 4*c*e^3*(5*a^3*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 9*a^2*b*e^2*(d^3 +
8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 9*a*b^2*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*
x^4) + 5*b^3*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) - 6*c^2*e^2*(3*a
^2*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 10*a*b*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2
*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 15*b^2*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3
+ 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) - 60*c^3*e*(a*e*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3
*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6) + 7*b*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 +
70*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)) + 840*c^4*(d + e*x)^8*Log[d + e*x])/(840*e^9*(d +
e*x)^8)

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Maple [B]  time = 0.053, size = 1382, 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)^9,x)

[Out]

-4*c^3/e^8/(e*x+d)*b+8*c^4/e^9/(e*x+d)*d-45/2/e^7/(e*x+d)^4*b^2*c^2*d^2+35/e^8/(e*x+d)^4*b*c^3*d^3-4*c^2/e^6/(
e*x+d)^3*a*b+8*c^3/e^7/(e*x+d)^3*a*d+12*c^2/e^7/(e*x+d)^3*b^2*d-28*c^3/e^8/(e*x+d)^3*b*d^2+14*c^3/e^8/(e*x+d)^
2*b*d+1/2/e^2/(e*x+d)^8*a^3*b*d-15/e^7/(e*x+d)^4*c^3*a*d^2+5/e^6/(e*x+d)^4*b^3*c*d+48/5/e^5/(e*x+d)^5*d*a*b^2*
c-1/4*b^4/e^5/(e*x+d)^4+36/7/e^7/(e*x+d)^7*d^5*b^2*c^2-4/e^8/(e*x+d)^7*b*c^3*d^6-3/e^5/(e*x+d)^4*a*c*b^2+1/2/e
^8/(e*x+d)^8*b*c^3*d^7-6/e^5/(e*x+d)^6*a^2*c^2*d^2+2/e^4/(e*x+d)^6*a*d*b^3-10/e^7/(e*x+d)^6*a*c^3*d^4+20/3/e^6
/(e*x+d)^6*d^3*b^3*c-15/e^7/(e*x+d)^6*d^4*b^2*c^2+14/e^8/(e*x+d)^6*b*c^3*d^5-12/5/e^4/(e*x+d)^5*a^2*b*c+24/5/e
^5/(e*x+d)^5*d*a^2*c^2-1/2/e^3/(e*x+d)^8*a^3*c*d^2-3/4/e^3/(e*x+d)^8*a^2*b^2*d^2-3/4/e^5/(e*x+d)^8*a^2*c^2*d^4
+1/2/e^4/(e*x+d)^8*d^3*a*b^3-1/2/e^7/(e*x+d)^8*a*c^3*d^6+1/2/e^6/(e*x+d)^8*d^5*b^3*c-3/4/e^7/(e*x+d)^8*d^6*b^2
*c^2+8/7/e^3/(e*x+d)^7*a^3*c*d+12/7/e^3/(e*x+d)^7*d*a^2*b^2+24/7/e^5/(e*x+d)^7*a^2*c^2*d^3-12/e^5/(e*x+d)^6*a*
b^2*c*d^2+20/e^6/(e*x+d)^6*d^3*a*b*c^2-1/8/e/(e*x+d)^8*a^4-4/7/e^2/(e*x+d)^7*a^3*b+4/7/e^5/(e*x+d)^7*b^4*d^3+8
/7/e^9/(e*x+d)^7*c^4*d^7-3/2/e^5/(e*x+d)^4*c^2*a^2-35/2/e^9/(e*x+d)^4*c^4*d^4-4/3*c/e^6/(e*x+d)^3*b^3+56/3*c^4
/e^9/(e*x+d)^3*d^3-2*c^3/e^7/(e*x+d)^2*a-3*c^2/e^7/(e*x+d)^2*b^2-14*c^4/e^9/(e*x+d)^2*d^2-1/8/e^5/(e*x+d)^8*b^
4*d^4-1/8/e^9/(e*x+d)^8*c^4*d^8-2/3/e^3/(e*x+d)^6*a^3*c-1/e^3/(e*x+d)^6*b^2*a^2-1/e^5/(e*x+d)^6*b^4*d^2-14/3/e
^9/(e*x+d)^6*c^4*d^6-4/5/e^4/(e*x+d)^5*a*b^3+4/5/e^5/(e*x+d)^5*d*b^4+56/5/e^9/(e*x+d)^5*c^4*d^5+16/e^7/(e*x+d)
^5*d^3*a*c^3-8/e^6/(e*x+d)^5*d^2*b^3*c+24/e^7/(e*x+d)^5*d^3*b^2*c^2-28/e^8/(e*x+d)^5*d^4*b*c^3-12/7/e^4/(e*x+d
)^7*d^2*a*b^3+24/7/e^7/(e*x+d)^7*a*c^3*d^5-20/7/e^6/(e*x+d)^7*d^4*b^3*c-24/e^6/(e*x+d)^5*d^2*a*b*c^2+6/e^4/(e*
x+d)^6*d*a^2*b*c-3/2/e^5/(e*x+d)^8*d^4*a*b^2*c+3/2/e^6/(e*x+d)^8*d^5*a*b*c^2+3/2/e^4/(e*x+d)^8*d^3*a^2*b*c+48/
7/e^5/(e*x+d)^7*d^3*a*b^2*c-60/7/e^6/(e*x+d)^7*d^4*a*b*c^2-36/7/e^4/(e*x+d)^7*c*b*a^2*d^2+15/e^6/(e*x+d)^4*a*b
*c^2*d+c^4*ln(e*x+d)/e^9

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Maxima [B]  time = 1.13881, size = 1207, 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)^9,x, algorithm="maxima")

[Out]

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(
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 - 12*(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 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*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 + 560*(154*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(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 + 70*(1750*c^4*d^4*e^4 - 420*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 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4
- 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(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 -
12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058*c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(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 - 12*(a*b^3 + 3*a^2*b*c)*d*e^7 -
10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a
*c^3)*d^5*e^3 - 20*(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 - 12*(a*b^3 + 3*a^2*
b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e^15*x^6 + 56*d^3*e^14*x^5
+ 70*d^4*e^13*x^4 + 56*d^5*e^12*x^3 + 28*d^6*e^11*x^2 + 8*d^7*e^10*x + d^8*e^9) + c^4*log(e*x + d)/e^9

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Fricas [B]  time = 1.78933, size = 2138, normalized size = 4.91 \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)^9,x, algorithm="fricas")

[Out]

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(
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 - 12*(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 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*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 + 560*(154*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(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 + 70*(1750*c^4*d^4*e^4 - 420*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 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4
- 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(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 -
12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058*c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(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 - 12*(a*b^3 + 3*a^2*b*c)*d*e^7 -
10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a
*c^3)*d^5*e^3 - 20*(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 - 12*(a*b^3 + 3*a^2*
b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 840*(c^4*e^8*x^8 + 8*c^4*d*e^7*x^7 + 28*c^4*d^2*e^6*x^6 + 5
6*c^4*d^3*e^5*x^5 + 70*c^4*d^4*e^4*x^4 + 56*c^4*d^5*e^3*x^3 + 28*c^4*d^6*e^2*x^2 + 8*c^4*d^7*e*x + c^4*d^8)*lo
g(e*x + d))/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e^15*x^6 + 56*d^3*e^14*x^5 + 70*d^4*e^13*x^4 + 56*d^5*e^12*x^3 +
28*d^6*e^11*x^2 + 8*d^7*e^10*x + d^8*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)**9,x)

[Out]

Timed out

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Giac [A]  time = 1.10996, size = 1138, normalized size = 2.62 \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)^9,x, algorithm="giac")

[Out]

c^4*e^(-9)*log(abs(x*e + d)) + 1/840*(3360*(2*c^4*d*e^6 - b*c^3*e^7)*x^7 + 840*(42*c^4*d^2*e^5 - 14*b*c^3*d*e^
6 - 3*b^2*c^2*e^7 - 2*a*c^3*e^7)*x^6 + 560*(154*c^4*d^3*e^4 - 42*b*c^3*d^2*e^5 - 9*b^2*c^2*d*e^6 - 6*a*c^3*d*e
^6 - 2*b^3*c*e^7 - 6*a*b*c^2*e^7)*x^5 + 70*(1750*c^4*d^4*e^3 - 420*b*c^3*d^3*e^4 - 90*b^2*c^2*d^2*e^5 - 60*a*c
^3*d^2*e^5 - 20*b^3*c*d*e^6 - 60*a*b*c^2*d*e^6 - 3*b^4*e^7 - 36*a*b^2*c*e^7 - 18*a^2*c^2*e^7)*x^4 + 56*(1918*c
^4*d^5*e^2 - 420*b*c^3*d^4*e^3 - 90*b^2*c^2*d^3*e^4 - 60*a*c^3*d^3*e^4 - 20*b^3*c*d^2*e^5 - 60*a*b*c^2*d^2*e^5
- 3*b^4*d*e^6 - 36*a*b^2*c*d*e^6 - 18*a^2*c^2*d*e^6 - 12*a*b^3*e^7 - 36*a^2*b*c*e^7)*x^3 + 28*(2058*c^4*d^6*e
- 420*b*c^3*d^5*e^2 - 90*b^2*c^2*d^4*e^3 - 60*a*c^3*d^4*e^3 - 20*b^3*c*d^3*e^4 - 60*a*b*c^2*d^3*e^4 - 3*b^4*d
^2*e^5 - 36*a*b^2*c*d^2*e^5 - 18*a^2*c^2*d^2*e^5 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 30*a^2*b^2*e^7 - 20*a^3
*c*e^7)*x^2 + 8*(2178*c^4*d^7 - 420*b*c^3*d^6*e - 90*b^2*c^2*d^5*e^2 - 60*a*c^3*d^5*e^2 - 20*b^3*c*d^4*e^3 - 6
0*a*b*c^2*d^4*e^3 - 3*b^4*d^3*e^4 - 36*a*b^2*c*d^3*e^4 - 18*a^2*c^2*d^3*e^4 - 12*a*b^3*d^2*e^5 - 36*a^2*b*c*d^
2*e^5 - 30*a^2*b^2*d*e^6 - 20*a^3*c*d*e^6 - 60*a^3*b*e^7)*x + (2283*c^4*d^8 - 420*b*c^3*d^7*e - 90*b^2*c^2*d^6
*e^2 - 60*a*c^3*d^6*e^2 - 20*b^3*c*d^5*e^3 - 60*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 - 12*a*b^3*d^3*e^5 - 36*a^2*b*c*d^3*e^5 - 30*a^2*b^2*d^2*e^6 - 20*a^3*c*d^2*e^6 - 60*a^3*b*d*e^7 -
105*a^4*e^8)*e^(-1))*e^(-8)/(x*e + d)^8