3.2153 \(\int \frac{(a+b x+c x^2)^4}{(d+e x)^3} \, dx\)

Optimal. Leaf size=430 \[ \frac{(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^9}+\frac{c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac{4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}-\frac{4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{2 \log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac{4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac{c^4 (d+e x)^6}{6 e^9} \]

[Out]

(-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))*x)/e^8 - (c*d^2 - b*d*e
+ a*e^2)^4/(2*e^9*(d + e*x)^2) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)) + ((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)^2)/(2*e^9) - (4*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)^3)/(3*
e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^4)/(2*e^9) - (4*c^3*(2*c*d - b*e)*(d + e*
x)^5)/(5*e^9) + (c^4*(d + e*x)^6)/(6*e^9) + (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))*Log[d + e*x])/e^9

________________________________________________________________________________________

Rubi [A]  time = 0.666783, antiderivative size = 430, 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} \[ \frac{(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^9}+\frac{c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac{4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}-\frac{4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{2 \log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac{4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac{c^4 (d+e x)^6}{6 e^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^3,x]

[Out]

(-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))*x)/e^8 - (c*d^2 - b*d*e
+ a*e^2)^4/(2*e^9*(d + e*x)^2) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)) + ((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)^2)/(2*e^9) - (4*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)^3)/(3*
e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^4)/(2*e^9) - (4*c^3*(2*c*d - b*e)*(d + e*
x)^5)/(5*e^9) + (c^4*(d + e*x)^6)/(6*e^9) + (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))*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)^3} \, dx &=\int \left (\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}+\frac{\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^3}+\frac{4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (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^8 (d+e x)}+\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)}{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)^2}{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)^3}{e^8}-\frac{4 c^3 (2 c d-b e) (d+e x)^4}{e^8}+\frac{c^4 (d+e x)^5}{e^8}\right ) \, dx\\ &=-\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 ) x}{e^8}-\frac{\left (c d^2-b d e+a e^2\right )^4}{2 e^9 (d+e x)^2}+\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)}+\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}{2 e^9}-\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}{3 e^9}+\frac{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}{2 e^9}-\frac{4 c^3 (2 c d-b e) (d+e x)^5}{5 e^9}+\frac{c^4 (d+e x)^6}{6 e^9}+\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 ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.204605, size = 440, normalized size = 1.02 \[ \frac{15 e^2 x^2 \left (6 c^2 e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-12 b^2 c e^3 (b d-a e)-8 c^3 d^2 e (5 b d-3 a e)+b^4 e^4+15 c^4 d^4\right )+30 e x \left (-6 c^2 d e^2 \left (3 a^2 e^2-12 a b d e+10 b^2 d^2\right )+12 b c e^3 \left (a^2 e^2-3 a b d e+2 b^2 d^2\right )+b^3 e^4 (4 a e-3 b d)+20 c^3 d^3 e (3 b d-2 a e)-21 c^4 d^5\right )+15 c^2 e^4 x^4 \left (2 c e (a e-3 b d)+3 b^2 e^2+3 c^2 d^2\right )+20 c e^3 x^3 (b e-c d) \left (c e (6 a e-7 b d)+2 b^2 e^2+5 c^2 d^2\right )+60 \log (d+e x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+\frac{120 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{d+e x}-\frac{15 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^2}+6 c^3 e^5 x^5 (4 b e-3 c d)+5 c^4 e^6 x^6}{30 e^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^3,x]

[Out]

(30*e*(-21*c^4*d^5 + 20*c^3*d^3*e*(3*b*d - 2*a*e) + b^3*e^4*(-3*b*d + 4*a*e) + 12*b*c*e^3*(2*b^2*d^2 - 3*a*b*d
*e + a^2*e^2) - 6*c^2*d*e^2*(10*b^2*d^2 - 12*a*b*d*e + 3*a^2*e^2))*x + 15*e^2*(15*c^4*d^4 + b^4*e^4 - 8*c^3*d^
2*e*(5*b*d - 3*a*e) - 12*b^2*c*e^3*(b*d - a*e) + 6*c^2*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*x^2 + 20*c*e^3*(
-(c*d) + b*e)*(5*c^2*d^2 + 2*b^2*e^2 + c*e*(-7*b*d + 6*a*e))*x^3 + 15*c^2*e^4*(3*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(
-3*b*d + a*e))*x^4 + 6*c^3*e^5*(-3*c*d + 4*b*e)*x^5 + 5*c^4*e^6*x^6 - (15*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e
*x)^2 + (120*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x) + 60*(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*Log[d + e*x])/(30*e^9)

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Maple [B]  time = 0.058, size = 1216, 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]

int((c*x^2+b*x+a)^4/(e*x+d)^3,x)

[Out]

4/e^3*x^3*a*b*c^2-4/e^4*x^3*a*c^3*d-3/e^3/(e*x+d)^2*d^2*a^2*b^2+90/e^7*ln(e*x+d)*b^2*c^2*d^4-84/e^8*ln(e*x+d)*
b*c^3*d^5+8/e^3/(e*x+d)*a^3*c*d+12/e^3/(e*x+d)*a^2*b^2*d+24/e^5/(e*x+d)*a^2*c^2*d^3-12/e^4/(e*x+d)*a*b^3*d^2+2
4/e^7/(e*x+d)*a*c^3*d^5-20/e^6/(e*x+d)*b^3*c*d^4+36/e^7/(e*x+d)*b^2*c^2*d^5-28/e^8/(e*x+d)*b*c^3*d^6-3/e^5/(e*
x+d)^2*a^2*c^2*d^4+2/e^4/(e*x+d)^2*a*b^3*d^3-2/e^7/(e*x+d)^2*a*c^3*d^6+2/e^6/(e*x+d)^2*b^3*c*d^5-3/e^7/(e*x+d)
^2*b^2*c^2*d^6+2/e^8/(e*x+d)^2*b*c^3*d^7+36/e^5*ln(e*x+d)*a^2*c^2*d^2-6/e^4*x^3*b^2*c^2*d-3/e^4*x^4*b*c^3*d+12
/e^3*a^2*b*c*x-18/e^4*a^2*c^2*d*x-40/e^6*a*c^3*d^3*x+24/e^5*b^3*c*d^2*x-60/e^6*b^2*c^2*d^3*x+60/e^7*d^4*b*c^3*
x+12/e^5*x^2*a*c^3*d^2-6/e^4*x^2*b^3*c*d+18/e^5*x^2*b^2*c^2*d^2-20/e^6*x^2*b*c^3*d^3+8/e^5*x^3*b*c^3*d^2+6/e^3
*x^2*a*b^2*c+2/e^2/(e*x+d)^2*d*a^3*b-2/e^3/(e*x+d)^2*a^3*c*d^2-12/e^4*ln(e*x+d)*a*b^3*d+60/e^7*ln(e*x+d)*a*c^3
*d^4-40/e^6*ln(e*x+d)*b^3*c*d^3+1/6/e^3*c^4*x^6-1/2/e/(e*x+d)^2*a^4+1/2*b^4*x^2/e^3-3/5/e^4*x^5*c^4*d+3/2/e^3*
x^4*b^2*c^2+3/2/e^5*x^4*c^4*d^2+4/3/e^3*x^3*b^3*c-10/3/e^6*x^3*c^4*d^3+3/e^3*x^2*a^2*c^2-3/e^4*b^4*d*x+15/2/e^
7*x^2*c^4*d^4+4/e^3*a*b^3*x-21/e^8*c^4*d^5*x-1/2/e^5/(e*x+d)^2*b^4*d^4-1/2/e^9/(e*x+d)^2*c^4*d^8+4/e^3*ln(e*x+
d)*a^3*c+6/e^3*ln(e*x+d)*a^2*b^2+6/e^5*ln(e*x+d)*b^4*d^2+28/e^9*ln(e*x+d)*c^4*d^6-4/e^2/(e*x+d)*a^3*b+4/e^5/(e
*x+d)*b^4*d^3+8/e^9/(e*x+d)*c^4*d^7+1/e^3*x^4*a*c^3+4/5/e^3*x^5*b*c^3-36/e^4/(e*x+d)*a^2*b*c*d^2+48/e^5/(e*x+d
)*a*b^2*c*d^3-60/e^6/(e*x+d)*a*b*c^2*d^4+72/e^5*a*b*c^2*d^2*x-36/e^4*a*b^2*c*d*x-18/e^4*x^2*a*b*c^2*d+6/e^4/(e
*x+d)^2*a^2*b*c*d^3-6/e^5/(e*x+d)^2*a*b^2*c*d^4+6/e^6/(e*x+d)^2*a*b*c^2*d^5-36/e^4*ln(e*x+d)*a^2*b*c*d+72/e^5*
ln(e*x+d)*a*b^2*c*d^2-120/e^6*ln(e*x+d)*a*b*c^2*d^3

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Maxima [A]  time = 1.12175, size = 1106, normalized size = 2.57 \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)^3,x, algorithm="maxima")

[Out]

1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e - 4*a^3*b*d*e^7 - a^4*e^8 + 22*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 36*(b^3*c + 3*
a*b*c^2)*d^5*e^3 + 7*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 20*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 6*(3*a^2*b^2 +
2*a^3*c)*d^2*e^6 + 8*(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)/(e^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/30*(5*c^4*e^5*x^6 - 6*(3*c^4*d*e^4 - 4*b*c^3*e^5)*x^5 +
15*(3*c^4*d^2*e^3 - 6*b*c^3*d*e^4 + (3*b^2*c^2 + 2*a*c^3)*e^5)*x^4 - 20*(5*c^4*d^3*e^2 - 12*b*c^3*d^2*e^3 + 3
*(3*b^2*c^2 + 2*a*c^3)*d*e^4 - 2*(b^3*c + 3*a*b*c^2)*e^5)*x^3 + 15*(15*c^4*d^4*e - 40*b*c^3*d^3*e^2 + 12*(3*b^
2*c^2 + 2*a*c^3)*d^2*e^3 - 12*(b^3*c + 3*a*b*c^2)*d*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^5)*x^2 - 30*(21*c^4
*d^5 - 60*b*c^3*d^4*e + 20*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 24*(b^3*c + 3*a*b*c^2)*d^2*e^3 + 3*(b^4 + 12*a*b^2*
c + 6*a^2*c^2)*d*e^4 - 4*(a*b^3 + 3*a^2*b*c)*e^5)*x)/e^8 + 2*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*b^2*c^2 + 2*
a*c^3)*d^4*e^2 - 20*(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 - 6*(a*b^3 + 3*a^2*
b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*log(e*x + d)/e^9

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Fricas [B]  time = 1.96845, size = 2573, normalized size = 5.98 \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)^3,x, algorithm="fricas")

[Out]

1/30*(5*c^4*e^8*x^8 + 225*c^4*d^8 - 780*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 15*a^4*e^8 + 330*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 - 540*(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 - 300*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 90*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 8*(c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 - 42*b*c^3*d
*e^7 + 15*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 2*(14*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 + 15*(3*b^2*c^2 + 2*a*c^3)*d*e
^7 - 20*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 5*(14*c^4*d^4*e^4 - 42*b*c^3*d^3*e^5 + 15*(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 - 20*(14*c^4*d^5*e^3 - 42*b*c^3*d^
4*e^4 + 15*(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 - 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 15*(69*c^4*d^6*e^2 - 200*b*c^3*d^5*e^3 + 68*(3*b^2*c^2 + 2*a*c^3)*d^4*
e^4 - 84*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 16*(a*b^3 + 3*a^2*b*c)*d*e^
7)*x^2 - 30*(13*c^4*d^7*e - 32*b*c^3*d^6*e^2 + 4*a^3*b*e^8 + 8*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(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 + 8*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 4*(3*a^2*b^2 + 2*a^3
*c)*d*e^7)*x + 60*(14*c^4*d^8 - 42*b*c^3*d^7*e + 15*(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 - 6*(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 + (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 + 2*(14
*c^4*d^7*e - 42*b*c^3*d^6*e^2 + 15*(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 + 1
2*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 6*(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^11*x^2 + 2*d*e^10*x + d^2*e^9)

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Sympy [B]  time = 22.8442, size = 892, normalized size = 2.07 \begin{align*} \frac{c^{4} x^{6}}{6 e^{3}} - \frac{a^{4} e^{8} + 4 a^{3} b d e^{7} - 12 a^{3} c d^{2} e^{6} - 18 a^{2} b^{2} d^{2} e^{6} + 60 a^{2} b c d^{3} e^{5} - 42 a^{2} c^{2} d^{4} e^{4} + 20 a b^{3} d^{3} e^{5} - 84 a b^{2} c d^{4} e^{4} + 108 a b c^{2} d^{5} e^{3} - 44 a c^{3} d^{6} e^{2} - 7 b^{4} d^{4} e^{4} + 36 b^{3} c d^{5} e^{3} - 66 b^{2} c^{2} d^{6} e^{2} + 52 b c^{3} d^{7} e - 15 c^{4} d^{8} + x \left (8 a^{3} b e^{8} - 16 a^{3} c d e^{7} - 24 a^{2} b^{2} d e^{7} + 72 a^{2} b c d^{2} e^{6} - 48 a^{2} c^{2} d^{3} e^{5} + 24 a b^{3} d^{2} e^{6} - 96 a b^{2} c d^{3} e^{5} + 120 a b c^{2} d^{4} e^{4} - 48 a c^{3} d^{5} e^{3} - 8 b^{4} d^{3} e^{5} + 40 b^{3} c d^{4} e^{4} - 72 b^{2} c^{2} d^{5} e^{3} + 56 b c^{3} d^{6} e^{2} - 16 c^{4} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac{x^{5} \left (4 b c^{3} e - 3 c^{4} d\right )}{5 e^{4}} + \frac{x^{4} \left (2 a c^{3} e^{2} + 3 b^{2} c^{2} e^{2} - 6 b c^{3} d e + 3 c^{4} d^{2}\right )}{2 e^{5}} + \frac{x^{3} \left (12 a b c^{2} e^{3} - 12 a c^{3} d e^{2} + 4 b^{3} c e^{3} - 18 b^{2} c^{2} d e^{2} + 24 b c^{3} d^{2} e - 10 c^{4} d^{3}\right )}{3 e^{6}} + \frac{x^{2} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 36 a b c^{2} d e^{3} + 24 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 12 b^{3} c d e^{3} + 36 b^{2} c^{2} d^{2} e^{2} - 40 b c^{3} d^{3} e + 15 c^{4} d^{4}\right )}{2 e^{7}} + \frac{x \left (12 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 4 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 72 a b c^{2} d^{2} e^{3} - 40 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 24 b^{3} c d^{2} e^{3} - 60 b^{2} c^{2} d^{3} e^{2} + 60 b c^{3} d^{4} e - 21 c^{4} d^{5}\right )}{e^{8}} + \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{2} \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 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)**3,x)

[Out]

c**4*x**6/(6*e**3) - (a**4*e**8 + 4*a**3*b*d*e**7 - 12*a**3*c*d**2*e**6 - 18*a**2*b**2*d**2*e**6 + 60*a**2*b*c
*d**3*e**5 - 42*a**2*c**2*d**4*e**4 + 20*a*b**3*d**3*e**5 - 84*a*b**2*c*d**4*e**4 + 108*a*b*c**2*d**5*e**3 - 4
4*a*c**3*d**6*e**2 - 7*b**4*d**4*e**4 + 36*b**3*c*d**5*e**3 - 66*b**2*c**2*d**6*e**2 + 52*b*c**3*d**7*e - 15*c
**4*d**8 + x*(8*a**3*b*e**8 - 16*a**3*c*d*e**7 - 24*a**2*b**2*d*e**7 + 72*a**2*b*c*d**2*e**6 - 48*a**2*c**2*d*
*3*e**5 + 24*a*b**3*d**2*e**6 - 96*a*b**2*c*d**3*e**5 + 120*a*b*c**2*d**4*e**4 - 48*a*c**3*d**5*e**3 - 8*b**4*
d**3*e**5 + 40*b**3*c*d**4*e**4 - 72*b**2*c**2*d**5*e**3 + 56*b*c**3*d**6*e**2 - 16*c**4*d**7*e))/(2*d**2*e**9
+ 4*d*e**10*x + 2*e**11*x**2) + x**5*(4*b*c**3*e - 3*c**4*d)/(5*e**4) + x**4*(2*a*c**3*e**2 + 3*b**2*c**2*e**
2 - 6*b*c**3*d*e + 3*c**4*d**2)/(2*e**5) + x**3*(12*a*b*c**2*e**3 - 12*a*c**3*d*e**2 + 4*b**3*c*e**3 - 18*b**2
*c**2*d*e**2 + 24*b*c**3*d**2*e - 10*c**4*d**3)/(3*e**6) + x**2*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 36*a*b*
c**2*d*e**3 + 24*a*c**3*d**2*e**2 + b**4*e**4 - 12*b**3*c*d*e**3 + 36*b**2*c**2*d**2*e**2 - 40*b*c**3*d**3*e +
15*c**4*d**4)/(2*e**7) + x*(12*a**2*b*c*e**5 - 18*a**2*c**2*d*e**4 + 4*a*b**3*e**5 - 36*a*b**2*c*d*e**4 + 72*
a*b*c**2*d**2*e**3 - 40*a*c**3*d**3*e**2 - 3*b**4*d*e**4 + 24*b**3*c*d**2*e**3 - 60*b**2*c**2*d**3*e**2 + 60*b
*c**3*d**4*e - 21*c**4*d**5)/e**8 + 2*(a*e**2 - b*d*e + c*d**2)**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14
*c**2*d**2)*log(d + e*x)/e**9

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Giac [B]  time = 1.12606, size = 1202, 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)^3,x, algorithm="giac")

[Out]

2*(14*c^4*d^6 - 42*b*c^3*d^5*e + 45*b^2*c^2*d^4*e^2 + 30*a*c^3*d^4*e^2 - 20*b^3*c*d^3*e^3 - 60*a*b*c^2*d^3*e^3
+ 3*b^4*d^2*e^4 + 36*a*b^2*c*d^2*e^4 + 18*a^2*c^2*d^2*e^4 - 6*a*b^3*d*e^5 - 18*a^2*b*c*d*e^5 + 3*a^2*b^2*e^6
+ 2*a^3*c*e^6)*e^(-9)*log(abs(x*e + d)) + 1/30*(5*c^4*x^6*e^15 - 18*c^4*d*x^5*e^14 + 45*c^4*d^2*x^4*e^13 - 100
*c^4*d^3*x^3*e^12 + 225*c^4*d^4*x^2*e^11 - 630*c^4*d^5*x*e^10 + 24*b*c^3*x^5*e^15 - 90*b*c^3*d*x^4*e^14 + 240*
b*c^3*d^2*x^3*e^13 - 600*b*c^3*d^3*x^2*e^12 + 1800*b*c^3*d^4*x*e^11 + 45*b^2*c^2*x^4*e^15 + 30*a*c^3*x^4*e^15
- 180*b^2*c^2*d*x^3*e^14 - 120*a*c^3*d*x^3*e^14 + 540*b^2*c^2*d^2*x^2*e^13 + 360*a*c^3*d^2*x^2*e^13 - 1800*b^2
*c^2*d^3*x*e^12 - 1200*a*c^3*d^3*x*e^12 + 40*b^3*c*x^3*e^15 + 120*a*b*c^2*x^3*e^15 - 180*b^3*c*d*x^2*e^14 - 54
0*a*b*c^2*d*x^2*e^14 + 720*b^3*c*d^2*x*e^13 + 2160*a*b*c^2*d^2*x*e^13 + 15*b^4*x^2*e^15 + 180*a*b^2*c*x^2*e^15
+ 90*a^2*c^2*x^2*e^15 - 90*b^4*d*x*e^14 - 1080*a*b^2*c*d*x*e^14 - 540*a^2*c^2*d*x*e^14 + 120*a*b^3*x*e^15 + 3
60*a^2*b*c*x*e^15)*e^(-18) + 1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e + 66*b^2*c^2*d^6*e^2 + 44*a*c^3*d^6*e^2 - 36*b^3
*c*d^5*e^3 - 108*a*b*c^2*d^5*e^3 + 7*b^4*d^4*e^4 + 84*a*b^2*c*d^4*e^4 + 42*a^2*c^2*d^4*e^4 - 20*a*b^3*d^3*e^5
- 60*a^2*b*c*d^3*e^5 + 18*a^2*b^2*d^2*e^6 + 12*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 - a^4*e^8 + 8*(2*c^4*d^7*e - 7*b*
c^3*d^6*e^2 + 9*b^2*c^2*d^5*e^3 + 6*a*c^3*d^5*e^3 - 5*b^3*c*d^4*e^4 - 15*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 + 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)^2