3.22.52 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^2} \, dx\) [2152]

Optimal. Leaf size=426 \[ \frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{e^9 (d+e x)}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^2}{e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^3}{3 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^4}{e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^6}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^9} \]

[Out]

2*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*x/e^8-(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)-2*(-
b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^2/e^9+1/3*(70*c^4*d^4+b^4*e^4-4*
b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^3/e^9-
c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^4/e^9+2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+
7*b*d))*(e*x+d)^5/e^9-2/3*c^3*(-b*e+2*c*d)*(e*x+d)^6/e^9+1/7*c^4*(e*x+d)^7/e^9-4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d
^2)^3*ln(e*x+d)/e^9

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} \frac {(d+e x)^3 \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 )}{3 e^9}+\frac {2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {c (d+e x)^4 (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}-\frac {2 (d+e x)^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}+\frac {2 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^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{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 )^3}{e^9}-\frac {2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 712

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

________________________________________________________________________________________

Mathematica [A]
time = 0.22, size = 780, normalized size = 1.83 \begin {gather*} \frac {c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )+35 e^4 \left (12 a^3 b d e^3-3 a^4 e^4+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+35 c e^3 \left (12 a^3 e^3 \left (-d^2+d e x+e^2 x^2\right )+18 a^2 b e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+12 a b^2 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+b^3 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+21 c^2 e^2 \left (10 a^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^2 \left (-30 d^6+150 d^5 e x+90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4-9 d e^5 x^5+6 e^6 x^6\right )\right )+7 c^3 e \left (6 a e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+b \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )-420 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3 (d+e x) \log (d+e x)}{105 e^9 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^
6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 35*e^4*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x
^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^
3*x^3 + e^4*x^4)) + 35*c*e^3*(12*a^3*e^3*(-d^2 + d*e*x + e^2*x^2) + 18*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*
x^2 + e^3*x^3) + 12*a*b^2*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^3*(12*d^5 - 48*d^
4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 21*c^2*e^2*(10*a^2*e^2*(-3*d^4 + 9*d^3*e
*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 -
5*d*e^4*x^4 + 3*e^5*x^5) + b^2*(-30*d^6 + 150*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*
d*e^5*x^5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x
^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 +
21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)*Log[d
+ e*x])/(105*e^9*(d + e*x))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(978\) vs. \(2(418)=836\).
time = 0.72, size = 979, normalized size = 2.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^8*(-24*a^2*b*c*d*e^5*x+36*a*b^2*c*d^2*e^4*x+4*a*b^2*c*e^6*x^3+4*a*c^3*d^2*e^4*x^3+30*b^2*c^2*d^4*e^2*x-24*
b*c^3*d^5*e*x+3*b*c^3*d^2*e^4*x^4-3*b^2*c^2*d*e^5*x^4+b^3*c*e^6*x^4+5/3*c^4*d^4*e^2*x^3+2*a*b^3*e^6*x^2-b^4*d*
e^5*x^2-3*c^4*d^5*e*x^2+4*e^6*c*a^3*x+3/5*c^4*d^2*e^4*x^5-c^4*d^3*e^3*x^4+2*a^2*c^2*e^6*x^3-8/5*b*c^3*d*e^5*x^
5+3*a*b*c^2*e^6*x^4-2*a*c^3*d*e^5*x^4+1/7*c^4*x^7*e^6+18*a*b*c^2*d^2*e^4*x^2-8*a*b^3*d*e^5*x+20*d^4*e^2*c^3*a*
x-8/3*b^3*c*d*e^5*x^3+1/3*b^4*e^6*x^3-12*a*b^2*c*d*e^5*x^2+7*d^6*c^4*x+6*b^2*c^2*d^2*e^4*x^3-16/3*b*c^3*d^3*e^
3*x^3+6*a^2*b*c*e^6*x^2-6*a^2*c^2*d*e^5*x^2-48*a*b*c^2*d^3*e^3*x-8*a*b*c^2*d*e^5*x^3+2/3*b*c^3*e^6*x^6-1/3*c^4
*d*e^5*x^6+4/5*a*c^3*e^6*x^5+6/5*b^2*c^2*e^6*x^5-8*a*c^3*d^3*e^3*x^2+6*b^3*c*d^2*e^4*x^2-12*b^2*c^2*d^3*e^3*x^
2+10*b*c^3*d^4*e^2*x^2+18*d^2*e^4*a^2*c^2*x+6*a^2*b^2*e^6*x+3*b^4*d^2*e^4*x-16*b^3*c*d^3*e^3*x)-(a^4*e^8-4*a^3
*b*d*e^7+4*a^3*c*d^2*e^6+6*a^2*b^2*d^2*e^6-12*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4-4*a*b^3*d^3*e^5+12*a*b^2*c*d^4
*e^4-12*a*b*c^2*d^5*e^3+4*a*c^3*d^6*e^2+b^4*d^4*e^4-4*b^3*c*d^5*e^3+6*b^2*c^2*d^6*e^2-4*b*c^3*d^7*e+c^4*d^8)/e
^9/(e*x+d)+(4*a^3*b*e^7-8*a^3*c*d*e^6-12*a^2*b^2*d*e^6+36*a^2*b*c*d^2*e^5-24*a^2*c^2*d^3*e^4+12*a*b^3*d^2*e^5-
48*a*b^2*c*d^3*e^4+60*a*b*c^2*d^4*e^3-24*a*c^3*d^5*e^2-4*b^4*d^3*e^4+20*b^3*c*d^4*e^3-36*b^2*c^2*d^5*e^2+28*b*
c^3*d^6*e-8*c^4*d^7)/e^9*ln(e*x+d)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 823, normalized size = 1.93 \begin {gather*} -4 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{5} - 5 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{4} - a^{3} b e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{3} - 3 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{2} + {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{105} \, {\left (15 \, c^{4} x^{7} e^{6} - 35 \, {\left (c^{4} d e^{5} - 2 \, b c^{3} e^{6}\right )} x^{6} + 21 \, {\left (3 \, c^{4} d^{2} e^{4} - 8 \, b c^{3} d e^{5} + 6 \, b^{2} c^{2} e^{6} + 4 \, a c^{3} e^{6}\right )} x^{5} - 105 \, {\left (c^{4} d^{3} e^{3} - 3 \, b c^{3} d^{2} e^{4} - b^{3} c e^{6} - 3 \, a b c^{2} e^{6} + {\left (3 \, b^{2} c^{2} e^{5} + 2 \, a c^{3} e^{5}\right )} d\right )} x^{4} + 35 \, {\left (5 \, c^{4} d^{4} e^{2} - 16 \, b c^{3} d^{3} e^{3} + b^{4} e^{6} + 12 \, a b^{2} c e^{6} + 6 \, a^{2} c^{2} e^{6} + 6 \, {\left (3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} d^{2} - 8 \, {\left (b^{3} c e^{5} + 3 \, a b c^{2} e^{5}\right )} d\right )} x^{3} - 105 \, {\left (3 \, c^{4} d^{5} e - 10 \, b c^{3} d^{4} e^{2} - 2 \, a b^{3} e^{6} - 6 \, a^{2} b c e^{6} + 4 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{3} - 6 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{2} + {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{6} - 24 \, b c^{3} d^{5} e + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{4} + 6 \, a^{2} b^{2} e^{6} + 4 \, a^{3} c e^{6} - 16 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{3} + 3 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{2} - 8 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d\right )} x\right )} e^{\left (-8\right )} - \frac {c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 2 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{6} - 4 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{5} - 4 \, a^{3} b d e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{4} + a^{4} e^{8} - 4 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{3} + 2 \, {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d^{2}}{x e^{10} + d e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^2,x, algorithm="maxima")

[Out]

-4*(2*c^4*d^7 - 7*b*c^3*d^6*e + 3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^5 - 5*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^4 - a^3*
b*e^7 + (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^3 - 3*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^2 + (3*a^2*b^2*e^6 +
2*a^3*c*e^6)*d)*e^(-9)*log(x*e + d) + 1/105*(15*c^4*x^7*e^6 - 35*(c^4*d*e^5 - 2*b*c^3*e^6)*x^6 + 21*(3*c^4*d^2
*e^4 - 8*b*c^3*d*e^5 + 6*b^2*c^2*e^6 + 4*a*c^3*e^6)*x^5 - 105*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4 - b^3*c*e^6 - 3*a
*b*c^2*e^6 + (3*b^2*c^2*e^5 + 2*a*c^3*e^5)*d)*x^4 + 35*(5*c^4*d^4*e^2 - 16*b*c^3*d^3*e^3 + b^4*e^6 + 12*a*b^2*
c*e^6 + 6*a^2*c^2*e^6 + 6*(3*b^2*c^2*e^4 + 2*a*c^3*e^4)*d^2 - 8*(b^3*c*e^5 + 3*a*b*c^2*e^5)*d)*x^3 - 105*(3*c^
4*d^5*e - 10*b*c^3*d^4*e^2 - 2*a*b^3*e^6 - 6*a^2*b*c*e^6 + 4*(3*b^2*c^2*e^3 + 2*a*c^3*e^3)*d^3 - 6*(b^3*c*e^4
+ 3*a*b*c^2*e^4)*d^2 + (b^4*e^5 + 12*a*b^2*c*e^5 + 6*a^2*c^2*e^5)*d)*x^2 + 105*(7*c^4*d^6 - 24*b*c^3*d^5*e + 1
0*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^4 + 6*a^2*b^2*e^6 + 4*a^3*c*e^6 - 16*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^3 + 3*(b^
4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^2 - 8*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d)*x)*e^(-8) - (c^4*d^8 - 4*b*c^3*
d^7*e + 2*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^6 - 4*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^5 - 4*a^3*b*d*e^7 + (b^4*e^4 + 1
2*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^4 + a^4*e^8 - 4*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^3 + 2*(3*a^2*b^2*e^6 + 2*a^3*c*
e^6)*d^2)/(x*e^10 + d*e^9)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1060 vs. \(2 (425) = 850\).
time = 4.58, size = 1060, normalized size = 2.49 \begin {gather*} -\frac {105 \, c^{4} d^{8} - {\left (15 \, c^{4} x^{8} + 70 \, b c^{3} x^{7} + 42 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 105 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} + 35 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} - 105 \, a^{4} + 210 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} + 210 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} + {\left (20 \, c^{4} d x^{7} + 98 \, b c^{3} d x^{6} + 63 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 175 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} - 420 \, a^{3} b d + 70 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 630 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} - 210 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} - 7 \, {\left (4 \, c^{4} d^{2} x^{6} + 21 \, b c^{3} d^{2} x^{5} + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} + 50 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} + 30 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} - 120 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x - 30 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} + 7 \, {\left (6 \, c^{4} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{4} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 150 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} - 45 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x - 60 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - 35 \, {\left (2 \, c^{4} d^{4} x^{4} + 14 \, b c^{3} d^{4} x^{3} + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} - 48 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} + 70 \, {\left (2 \, c^{4} d^{5} x^{3} + 21 \, b c^{3} d^{5} x^{2} - 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x - 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} - 210 \, {\left (2 \, c^{4} d^{6} x^{2} - 12 \, b c^{3} d^{6} x - {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} - 105 \, {\left (7 \, c^{4} d^{7} x + 4 \, b c^{3} d^{7}\right )} e + 420 \, {\left (2 \, c^{4} d^{8} - a^{3} b x e^{8} - {\left (a^{3} b d - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} - {\left (3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} + {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} + {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} - {\left (7 \, b c^{3} d^{6} x - 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + {\left (2 \, c^{4} d^{7} x - 7 \, b c^{3} d^{7}\right )} e\right )} \log \left (x e + d\right )}{105 \, {\left (x e^{10} + d e^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^2,x, algorithm="fricas")

[Out]

-1/105*(105*c^4*d^8 - (15*c^4*x^8 + 70*b*c^3*x^7 + 42*(3*b^2*c^2 + 2*a*c^3)*x^6 + 105*(b^3*c + 3*a*b*c^2)*x^5
+ 35*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^4 - 105*a^4 + 210*(a*b^3 + 3*a^2*b*c)*x^3 + 210*(3*a^2*b^2 + 2*a^3*c)*x^
2)*e^8 + (20*c^4*d*x^7 + 98*b*c^3*d*x^6 + 63*(3*b^2*c^2 + 2*a*c^3)*d*x^5 + 175*(b^3*c + 3*a*b*c^2)*d*x^4 - 420
*a^3*b*d + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^3 + 630*(a*b^3 + 3*a^2*b*c)*d*x^2 - 210*(3*a^2*b^2 + 2*a^3*c)
*d*x)*e^7 - 7*(4*c^4*d^2*x^6 + 21*b*c^3*d^2*x^5 + 15*(3*b^2*c^2 + 2*a*c^3)*d^2*x^4 + 50*(b^3*c + 3*a*b*c^2)*d^
2*x^3 + 30*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x^2 - 120*(a*b^3 + 3*a^2*b*c)*d^2*x - 30*(3*a^2*b^2 + 2*a^3*c)*d
^2)*e^6 + 7*(6*c^4*d^3*x^5 + 35*b*c^3*d^3*x^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 + 150*(b^3*c + 3*a*b*c^2)*d^3
*x^2 - 45*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*x - 60*(a*b^3 + 3*a^2*b*c)*d^3)*e^5 - 35*(2*c^4*d^4*x^4 + 14*b*c^
3*d^4*x^3 + 18*(3*b^2*c^2 + 2*a*c^3)*d^4*x^2 - 48*(b^3*c + 3*a*b*c^2)*d^4*x - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*d^4)*e^4 + 70*(2*c^4*d^5*x^3 + 21*b*c^3*d^5*x^2 - 15*(3*b^2*c^2 + 2*a*c^3)*d^5*x - 6*(b^3*c + 3*a*b*c^2)*d^5)
*e^3 - 210*(2*c^4*d^6*x^2 - 12*b*c^3*d^6*x - (3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 - 105*(7*c^4*d^7*x + 4*b*c^3*d^7)*
e + 420*(2*c^4*d^8 - a^3*b*x*e^8 - (a^3*b*d - (3*a^2*b^2 + 2*a^3*c)*d*x)*e^7 - (3*(a*b^3 + 3*a^2*b*c)*d^2*x -
(3*a^2*b^2 + 2*a^3*c)*d^2)*e^6 + ((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*x - 3*(a*b^3 + 3*a^2*b*c)*d^3)*e^5 - (5*(
b^3*c + 3*a*b*c^2)*d^4*x - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4)*e^4 + (3*(3*b^2*c^2 + 2*a*c^3)*d^5*x - 5*(b^3*c
+ 3*a*b*c^2)*d^5)*e^3 - (7*b*c^3*d^6*x - 3*(3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 + (2*c^4*d^7*x - 7*b*c^3*d^7)*e)*lo
g(x*e + d))/(x*e^10 + d*e^9)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (423) = 846\).
time = 2.42, size = 847, normalized size = 1.99 \begin {gather*} \frac {c^{4} x^{7}}{7 e^{2}} + x^{6} \cdot \left (\frac {2 b c^{3}}{3 e^{2}} - \frac {c^{4} d}{3 e^{3}}\right ) + x^{5} \cdot \left (\frac {4 a c^{3}}{5 e^{2}} + \frac {6 b^{2} c^{2}}{5 e^{2}} - \frac {8 b c^{3} d}{5 e^{3}} + \frac {3 c^{4} d^{2}}{5 e^{4}}\right ) + x^{4} \cdot \left (\frac {3 a b c^{2}}{e^{2}} - \frac {2 a c^{3} d}{e^{3}} + \frac {b^{3} c}{e^{2}} - \frac {3 b^{2} c^{2} d}{e^{3}} + \frac {3 b c^{3} d^{2}}{e^{4}} - \frac {c^{4} d^{3}}{e^{5}}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c^{2}}{e^{2}} + \frac {4 a b^{2} c}{e^{2}} - \frac {8 a b c^{2} d}{e^{3}} + \frac {4 a c^{3} d^{2}}{e^{4}} + \frac {b^{4}}{3 e^{2}} - \frac {8 b^{3} c d}{3 e^{3}} + \frac {6 b^{2} c^{2} d^{2}}{e^{4}} - \frac {16 b c^{3} d^{3}}{3 e^{5}} + \frac {5 c^{4} d^{4}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {6 a^{2} b c}{e^{2}} - \frac {6 a^{2} c^{2} d}{e^{3}} + \frac {2 a b^{3}}{e^{2}} - \frac {12 a b^{2} c d}{e^{3}} + \frac {18 a b c^{2} d^{2}}{e^{4}} - \frac {8 a c^{3} d^{3}}{e^{5}} - \frac {b^{4} d}{e^{3}} + \frac {6 b^{3} c d^{2}}{e^{4}} - \frac {12 b^{2} c^{2} d^{3}}{e^{5}} + \frac {10 b c^{3} d^{4}}{e^{6}} - \frac {3 c^{4} d^{5}}{e^{7}}\right ) + x \left (\frac {4 a^{3} c}{e^{2}} + \frac {6 a^{2} b^{2}}{e^{2}} - \frac {24 a^{2} b c d}{e^{3}} + \frac {18 a^{2} c^{2} d^{2}}{e^{4}} - \frac {8 a b^{3} d}{e^{3}} + \frac {36 a b^{2} c d^{2}}{e^{4}} - \frac {48 a b c^{2} d^{3}}{e^{5}} + \frac {20 a c^{3} d^{4}}{e^{6}} + \frac {3 b^{4} d^{2}}{e^{4}} - \frac {16 b^{3} c d^{3}}{e^{5}} + \frac {30 b^{2} c^{2} d^{4}}{e^{6}} - \frac {24 b c^{3} d^{5}}{e^{7}} + \frac {7 c^{4} d^{6}}{e^{8}}\right ) + \frac {- a^{4} e^{8} + 4 a^{3} b d e^{7} - 4 a^{3} c d^{2} e^{6} - 6 a^{2} b^{2} d^{2} e^{6} + 12 a^{2} b c d^{3} e^{5} - 6 a^{2} c^{2} d^{4} e^{4} + 4 a b^{3} d^{3} e^{5} - 12 a b^{2} c d^{4} e^{4} + 12 a b c^{2} d^{5} e^{3} - 4 a c^{3} d^{6} e^{2} - b^{4} d^{4} e^{4} + 4 b^{3} c d^{5} e^{3} - 6 b^{2} c^{2} d^{6} e^{2} + 4 b c^{3} d^{7} e - c^{4} d^{8}}{d e^{9} + e^{10} x} + \frac {4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**4*x**7/(7*e**2) + x**6*(2*b*c**3/(3*e**2) - c**4*d/(3*e**3)) + x**5*(4*a*c**3/(5*e**2) + 6*b**2*c**2/(5*e**
2) - 8*b*c**3*d/(5*e**3) + 3*c**4*d**2/(5*e**4)) + x**4*(3*a*b*c**2/e**2 - 2*a*c**3*d/e**3 + b**3*c/e**2 - 3*b
**2*c**2*d/e**3 + 3*b*c**3*d**2/e**4 - c**4*d**3/e**5) + x**3*(2*a**2*c**2/e**2 + 4*a*b**2*c/e**2 - 8*a*b*c**2
*d/e**3 + 4*a*c**3*d**2/e**4 + b**4/(3*e**2) - 8*b**3*c*d/(3*e**3) + 6*b**2*c**2*d**2/e**4 - 16*b*c**3*d**3/(3
*e**5) + 5*c**4*d**4/(3*e**6)) + x**2*(6*a**2*b*c/e**2 - 6*a**2*c**2*d/e**3 + 2*a*b**3/e**2 - 12*a*b**2*c*d/e*
*3 + 18*a*b*c**2*d**2/e**4 - 8*a*c**3*d**3/e**5 - b**4*d/e**3 + 6*b**3*c*d**2/e**4 - 12*b**2*c**2*d**3/e**5 +
10*b*c**3*d**4/e**6 - 3*c**4*d**5/e**7) + x*(4*a**3*c/e**2 + 6*a**2*b**2/e**2 - 24*a**2*b*c*d/e**3 + 18*a**2*c
**2*d**2/e**4 - 8*a*b**3*d/e**3 + 36*a*b**2*c*d**2/e**4 - 48*a*b*c**2*d**3/e**5 + 20*a*c**3*d**4/e**6 + 3*b**4
*d**2/e**4 - 16*b**3*c*d**3/e**5 + 30*b**2*c**2*d**4/e**6 - 24*b*c**3*d**5/e**7 + 7*c**4*d**6/e**8) + (-a**4*e
**8 + 4*a**3*b*d*e**7 - 4*a**3*c*d**2*e**6 - 6*a**2*b**2*d**2*e**6 + 12*a**2*b*c*d**3*e**5 - 6*a**2*c**2*d**4*
e**4 + 4*a*b**3*d**3*e**5 - 12*a*b**2*c*d**4*e**4 + 12*a*b*c**2*d**5*e**3 - 4*a*c**3*d**6*e**2 - b**4*d**4*e**
4 + 4*b**3*c*d**5*e**3 - 6*b**2*c**2*d**6*e**2 + 4*b*c**3*d**7*e - c**4*d**8)/(d*e**9 + e**10*x) + 4*(b*e - 2*
c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**9

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (425) = 850\).
time = 1.75, size = 1013, normalized size = 2.38 \begin {gather*} \frac {1}{105} \, {\left (15 \, c^{4} - \frac {70 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {42 \, {\left (14 \, c^{4} d^{2} e^{2} - 14 \, b c^{3} d e^{3} + 3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {105 \, {\left (14 \, c^{4} d^{3} e^{3} - 21 \, b c^{3} d^{2} e^{4} + 9 \, b^{2} c^{2} d e^{5} + 6 \, a c^{3} d e^{5} - b^{3} c e^{6} - 3 \, a b c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {35 \, {\left (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}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {210 \, {\left (14 \, c^{4} d^{5} e^{5} - 35 \, b c^{3} d^{4} e^{6} + 30 \, b^{2} c^{2} d^{3} e^{7} + 20 \, a c^{3} d^{3} e^{7} - 10 \, b^{3} c d^{2} e^{8} - 30 \, a b c^{2} d^{2} e^{8} + b^{4} d e^{9} + 12 \, a b^{2} c d e^{9} + 6 \, a^{2} c^{2} d e^{9} - a b^{3} e^{10} - 3 \, a^{2} b c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac {210 \, {\left (14 \, c^{4} d^{6} e^{6} - 42 \, b c^{3} d^{5} e^{7} + 45 \, b^{2} c^{2} d^{4} e^{8} + 30 \, a c^{3} d^{4} e^{8} - 20 \, b^{3} c d^{3} e^{9} - 60 \, a b c^{2} d^{3} e^{9} + 3 \, b^{4} d^{2} e^{10} + 36 \, a b^{2} c d^{2} e^{10} + 18 \, a^{2} c^{2} d^{2} e^{10} - 6 \, a b^{3} d e^{11} - 18 \, a^{2} b c d e^{11} + 3 \, a^{2} b^{2} e^{12} + 2 \, a^{3} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )} {\left (x e + d\right )}^{7} e^{\left (-9\right )} + 4 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 9 \, b^{2} c^{2} d^{5} e^{2} + 6 \, a c^{3} d^{5} e^{2} - 5 \, b^{3} c d^{4} e^{3} - 15 \, a b c^{2} d^{4} e^{3} + b^{4} d^{3} e^{4} + 12 \, a b^{2} c d^{3} e^{4} + 6 \, a^{2} c^{2} d^{3} e^{4} - 3 \, a b^{3} d^{2} e^{5} - 9 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} - a^{3} b e^{7}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{4} d^{8} e^{7}}{x e + d} - \frac {4 \, b c^{3} d^{7} e^{8}}{x e + d} + \frac {6 \, b^{2} c^{2} d^{6} e^{9}}{x e + d} + \frac {4 \, a c^{3} d^{6} e^{9}}{x e + d} - \frac {4 \, b^{3} c d^{5} e^{10}}{x e + d} - \frac {12 \, a b c^{2} d^{5} e^{10}}{x e + d} + \frac {b^{4} d^{4} e^{11}}{x e + d} + \frac {12 \, a b^{2} c d^{4} e^{11}}{x e + d} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{x e + d} - \frac {4 \, a b^{3} d^{3} e^{12}}{x e + d} - \frac {12 \, a^{2} b c d^{3} e^{12}}{x e + d} + \frac {6 \, a^{2} b^{2} d^{2} e^{13}}{x e + d} + \frac {4 \, a^{3} c d^{2} e^{13}}{x e + d} - \frac {4 \, a^{3} b d e^{14}}{x e + d} + \frac {a^{4} e^{15}}{x e + d}\right )} e^{\left (-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^2,x, algorithm="giac")

[Out]

1/105*(15*c^4 - 70*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 42*(14*c^4*d^2*e^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*
e^4 + 2*a*c^3*e^4)*e^(-2)/(x*e + d)^2 - 105*(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e
^5 - b^3*c*e^6 - 3*a*b*c^2*e^6)*e^(-3)/(x*e + d)^3 + 35*(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)*e^(-4)/(
x*e + d)^4 - 210*(14*c^4*d^5*e^5 - 35*b*c^3*d^4*e^6 + 30*b^2*c^2*d^3*e^7 + 20*a*c^3*d^3*e^7 - 10*b^3*c*d^2*e^8
- 30*a*b*c^2*d^2*e^8 + b^4*d*e^9 + 12*a*b^2*c*d*e^9 + 6*a^2*c^2*d*e^9 - a*b^3*e^10 - 3*a^2*b*c*e^10)*e^(-5)/(
x*e + d)^5 + 210*(14*c^4*d^6*e^6 - 42*b*c^3*d^5*e^7 + 45*b^2*c^2*d^4*e^8 + 30*a*c^3*d^4*e^8 - 20*b^3*c*d^3*e^9
- 60*a*b*c^2*d^3*e^9 + 3*b^4*d^2*e^10 + 36*a*b^2*c*d^2*e^10 + 18*a^2*c^2*d^2*e^10 - 6*a*b^3*d*e^11 - 18*a^2*b
*c*d*e^11 + 3*a^2*b^2*e^12 + 2*a^3*c*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-9) + 4*(2*c^4*d^7 - 7*b*c^3*d^6
*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3
*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e^6 + 2*a^3*c*d*e^6 - a^3*b*e^7)*
e^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^4*d^8*e^7/(x*e + d) - 4*b*c^3*d^7*e^8/(x*e + d) + 6*b^2*c^2*d
^6*e^9/(x*e + d) + 4*a*c^3*d^6*e^9/(x*e + d) - 4*b^3*c*d^5*e^10/(x*e + d) - 12*a*b*c^2*d^5*e^10/(x*e + d) + b^
4*d^4*e^11/(x*e + d) + 12*a*b^2*c*d^4*e^11/(x*e + d) + 6*a^2*c^2*d^4*e^11/(x*e + d) - 4*a*b^3*d^3*e^12/(x*e +
d) - 12*a^2*b*c*d^3*e^12/(x*e + d) + 6*a^2*b^2*d^2*e^13/(x*e + d) + 4*a^3*c*d^2*e^13/(x*e + d) - 4*a^3*b*d*e^1
4/(x*e + d) + a^4*e^15/(x*e + d))*e^(-16)

________________________________________________________________________________________

Mupad [B]
time = 0.75, size = 1679, normalized size = 3.94 \begin {gather*} x\,\left (\frac {4\,c\,a^3+6\,a^2\,b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{e}\right )}{e}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{e^2}-\frac {4\,a\,b\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{e}\right )}{e^2}\right )+x^6\,\left (\frac {2\,b\,c^3}{3\,e^2}-\frac {c^4\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{e}\right )}{e}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{2\,e^2}-\frac {2\,a\,b\,\left (b^2+3\,a\,c\right )}{e^2}\right )+x^4\,\left (\frac {d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{2\,e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{4\,e^2}+\frac {b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )-x^5\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{5\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{5\,e^2}+\frac {c^4\,d^2}{5\,e^4}\right )+x^3\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{3\,e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{3\,e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^2}+\frac {c^4\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {4\,b\,c^3}{e^2}-\frac {2\,c^4\,d}{e^3}\right )}{e^2}+\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^2}\right )}{3\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^7+8\,a^3\,c\,d\,e^6+12\,a^2\,b^2\,d\,e^6-36\,a^2\,b\,c\,d^2\,e^5+24\,a^2\,c^2\,d^3\,e^4-12\,a\,b^3\,d^2\,e^5+48\,a\,b^2\,c\,d^3\,e^4-60\,a\,b\,c^2\,d^4\,e^3+24\,a\,c^3\,d^5\,e^2+4\,b^4\,d^3\,e^4-20\,b^3\,c\,d^4\,e^3+36\,b^2\,c^2\,d^5\,e^2-28\,b\,c^3\,d^6\,e+8\,c^4\,d^7\right )}{e^9}+\frac {c^4\,x^7}{7\,e^2}-\frac {a^4\,e^8-4\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6-12\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4-4\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4-12\,a\,b\,c^2\,d^5\,e^3+4\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}{e\,\left (x\,e^9+d\,e^8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((4*a^3*c + 6*a^2*b^2)/e^2 + (2*d*((2*d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2
*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d
)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c
*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2
+ (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e^2 - (4*a*b*(3*
a*c + b^2))/e^2))/e - (d^2*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e
- (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*
a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))
/e^2))/e))/e^2) + x^6*((2*b*c^3)/(3*e^2) - (c^4*d)/(3*e^3)) - x^2*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d
^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2
*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 -
(2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e
- (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c +
b^2))/e^2))/(2*e^2) - (2*a*b*(3*a*c + b^2))/e^2) + x^4*((d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3
+ 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/(2*e) - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/(4*e^2) + (b*c*(3*a*c + b^2)
)/e^2) - x^5*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/(5*e) - (4*a*c^3 + 6*b^2*c^2)/(5*e^2) + (c^4*d^2)/(5*e^4))
+ x^3*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(3*e^2) + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*
b^2*c^2)/e^2 + (c^4*d^2)/e^4))/(3*e^2) - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^
2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/(3*e)
) - (log(d + e*x)*(8*c^4*d^7 - 4*a^3*b*e^7 + 4*b^4*d^3*e^4 - 12*a*b^3*d^2*e^5 + 12*a^2*b^2*d*e^6 + 24*a*c^3*d^
5*e^2 - 20*b^3*c*d^4*e^3 + 24*a^2*c^2*d^3*e^4 + 36*b^2*c^2*d^5*e^2 + 8*a^3*c*d*e^6 - 28*b*c^3*d^6*e - 60*a*b*c
^2*d^4*e^3 + 48*a*b^2*c*d^3*e^4 - 36*a^2*b*c*d^2*e^5))/e^9 + (c^4*x^7)/(7*e^2) - (a^4*e^8 + c^4*d^8 + b^4*d^4*
e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^
4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b
*c*d^3*e^5)/(e*(d*e^8 + e^9*x))

________________________________________________________________________________________