3.2156 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^6} \, dx\)

Optimal. Leaf size=414 \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac {c^3 x^2 (3 c d-2 b e)}{e^7}+\frac {c^4 x^3}{3 e^6} \]

[Out]

c^2*(21*c^2*d^2+6*b^2*e^2-4*c*e*(-a*e+6*b*d))*x/e^8-c^3*(-2*b*e+3*c*d)*x^2/e^7+1/3*c^4*x^3/e^6-1/5*(a*e^2-b*d*
e+c*d^2)^4/e^9/(e*x+d)^5+(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^4-2/3*(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))/e^9/(e*x+d)^3+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^9/(e*x+d)^2+(-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^9/(e*x+d)-4*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*ln(e*x+d)
/e^9

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Rubi [A]  time = 0.56, antiderivative size = 414, 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 = {698} \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac {c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac {c^3 x^2 (3 c d-2 b e)}{e^7}+\frac {c^4 x^3}{3 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^6,x]

[Out]

(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e)*x^2)/e^7 + (c^4*x^3)/(3*e^6)
- (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4)
- (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)))/(3*e^9*(d + e*x)^3) + (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)))/(e^9*(d + e*x)^2) - (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))/(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(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)^6} \, dx &=\int \left (\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right )}{e^8}-\frac {2 c^3 (3 c d-2 b e) x}{e^7}+\frac {c^4 x^2}{e^6}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^6}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^5}+\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)^4}+\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)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\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 )}{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 ) \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 {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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 419, normalized size = 1.01 \[ \frac {\frac {30 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{(d+e x)^2}-\frac {15 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{d+e x}+15 c^2 e x \left (4 c e (a e-6 b d)+6 b^2 e^2+21 c^2 d^2\right )-\frac {10 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-60 c (2 c d-b e) \log (d+e x) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )+\frac {15 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^4}-\frac {3 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^5}+15 c^3 e^2 x^2 (2 b e-3 c d)+5 c^4 e^3 x^3}{15 e^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^6,x]

[Out]

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

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fricas [B]  time = 0.88, size = 1262, normalized size = 3.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/15*(5*c^4*e^8*x^8 - 743*c^4*d^8 + 1377*b*c^3*d^7*e - 3*a^3*b*d*e^7 - 3*a^4*e^8 - 261*(3*b^2*c^2 + 2*a*c^3)*d
^6*e^2 + 137*(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 - 3*(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 - 10*(2*c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + 10*(14*c^4*d^2*e^6 - 21*b*c^3*d*e
^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 25*(47*c^4*d^3*e^5 - 60*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7)
*x^5 + 5*(335*c^4*d^4*e^4 - 240*b*c^3*d^3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 60*(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 - 10*(85*c^4*d^5*e^3 - 390*b*c^3*d^4*e^4 + 120*(3*b^2*c^2 + 2*a*
c^3)*d^3*e^5 - 90*(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 + 3*(a*b^3 + 3*a^2*b*c)
*e^8)*x^3 - 10*(365*c^4*d^6*e^2 - 810*b*c^3*d^5*e^3 + 180*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(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 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e
^8)*x^2 - 5*(575*c^4*d^7*e - 1125*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 225*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 125*(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 + 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 - 60*(14*c^4*d^8 - 21*b*c^3*d^7*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - (b^3*c + 3*a*b*c^2)*
d^5*e^3 + (14*c^4*d^3*e^5 - 21*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 +
5*(14*c^4*d^4*e^4 - 21*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - (b^3*c + 3*a*b*c^2)*d*e^7)*x^4 + 10*(
14*c^4*d^5*e^3 - 21*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - (b^3*c + 3*a*b*c^2)*d^2*e^6)*x^3 + 10*(1
4*c^4*d^6*e^2 - 21*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - (b^3*c + 3*a*b*c^2)*d^3*e^5)*x^2 + 5*(14*
c^4*d^7*e - 21*b*c^3*d^6*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - (b^3*c + 3*a*b*c^2)*d^4*e^4)*x)*log(e*x + d))
/(e^14*x^5 + 5*d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x + d^5*e^9)

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giac [B]  time = 0.19, size = 841, normalized size = 2.03 \[ -4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 9 \, b^{2} c^{2} d e^{2} + 6 \, a c^{3} d e^{2} - b^{3} c e^{3} - 3 \, a b c^{2} e^{3}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (c^{4} x^{3} e^{12} - 9 \, c^{4} d x^{2} e^{11} + 63 \, c^{4} d^{2} x e^{10} + 6 \, b c^{3} x^{2} e^{12} - 72 \, b c^{3} d x e^{11} + 18 \, b^{2} c^{2} x e^{12} + 12 \, a c^{3} x e^{12}\right )} e^{\left (-18\right )} - \frac {{\left (743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 783 \, b^{2} c^{2} d^{6} e^{2} + 522 \, a c^{3} d^{6} e^{2} - 137 \, b^{3} c d^{5} e^{3} - 411 \, 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} + 3 \, a b^{3} d^{3} e^{5} + 9 \, a^{2} b c d^{3} e^{5} + 3 \, a^{2} b^{2} d^{2} e^{6} + 2 \, a^{3} c d^{2} e^{6} + 3 \, a^{3} b d e^{7} + 15 \, {\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 )} x^{4} + 3 \, a^{4} e^{8} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + 150 \, b^{2} c^{2} d^{3} e^{5} + 100 \, a c^{3} d^{3} e^{5} - 30 \, b^{3} c d^{2} e^{6} - 90 \, 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} + a b^{3} e^{8} + 3 \, a^{2} b c e^{8}\right )} x^{3} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 585 \, b^{2} c^{2} d^{4} e^{4} + 390 \, a c^{3} d^{4} e^{4} - 110 \, b^{3} c d^{3} e^{5} - 330 \, 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} + 3 \, a b^{3} d e^{7} + 9 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 693 \, b^{2} c^{2} d^{5} e^{3} + 462 \, a c^{3} d^{5} e^{3} - 125 \, b^{3} c d^{4} e^{4} - 375 \, a b c^{2} d^{4} e^{4} + 3 \, b^{4} d^{3} e^{5} + 36 \, a b^{2} c d^{3} e^{5} + 18 \, 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} + 3 \, a^{3} b e^{8}\right )} x\right )} e^{\left (-9\right )}}{15 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="giac")

[Out]

-4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*e^(-9)*log(abs(
x*e + d)) + 1/3*(c^4*x^3*e^12 - 9*c^4*d*x^2*e^11 + 63*c^4*d^2*x*e^10 + 6*b*c^3*x^2*e^12 - 72*b*c^3*d*x*e^11 +
18*b^2*c^2*x*e^12 + 12*a*c^3*x*e^12)*e^(-18) - 1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 783*b^2*c^2*d^6*e^2 + 52
2*a*c^3*d^6*e^2 - 137*b^3*c*d^5*e^3 - 411*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 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 3*a^2*b^2*d^2*e^6 + 2*a^3*c*d^2*e^6 + 3*a^3*b*d*e^7 + 15*(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)*x^4 + 3*a^4*e^8 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 150*b^2*c^2
*d^3*e^5 + 100*a*c^3*d^3*e^5 - 30*b^3*c*d^2*e^6 - 90*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 + a*b^3*e^8 + 3*a^2*b*c*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 585*b^2*c^2*d^4*e^4 + 390
*a*c^3*d^4*e^4 - 110*b^3*c*d^3*e^5 - 330*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 + 3*a*b^3*d*e^7 + 9*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^2 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*
e^2 + 693*b^2*c^2*d^5*e^3 + 462*a*c^3*d^5*e^3 - 125*b^3*c*d^4*e^4 - 375*a*b*c^2*d^4*e^4 + 3*b^4*d^3*e^5 + 36*a
*b^2*c*d^3*e^5 + 18*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 +
3*a^3*b*e^8)*x)*e^(-9)/(x*e + d)^5

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maple [B]  time = 0.07, size = 1341, normalized size = 3.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^6,x)

[Out]

12/e^5/(e*x+d)^4*a*b^2*c*d^3+60/e^6/(e*x+d)*a*b*c^2*d-24/e^5/(e*x+d)^3*a*b^2*c*d^2+40/e^6/(e*x+d)^3*a*b*c^2*d^
3-9/e^4/(e*x+d)^4*a^2*b*c*d^2+4*c^3/e^6*a*x+6*c^2/e^6*b^2*x+21*c^4/e^8*d^2*x+12/e^5/(e*x+d)^2*a^2*c^2*d+40/e^7
/(e*x+d)^2*a*c^3*d^3-20/e^6/(e*x+d)^2*b^3*c*d^2+60/e^7/(e*x+d)^2*b^2*c^2*d^3-24*c^3/e^7*b*d*x+12*c^2/e^6*ln(e*
x+d)*a*b-24*c^3/e^7*ln(e*x+d)*a*d-36*c^2/e^7*ln(e*x+d)*b^2*d+84*c^3/e^8*ln(e*x+d)*b*d^2-3/e^4/(e*x+d)^4*a*b^3*
d^2+6/e^7/(e*x+d)^4*a*c^3*d^5+28/e^8/(e*x+d)^3*b*c^3*d^5+2/e^3/(e*x+d)^4*a^3*c*d+3/e^3/(e*x+d)^4*a^2*b^2*d+6/e
^5/(e*x+d)^4*a^2*c^2*d^3-1/e^5/(e*x+d)*b^4-1/5/e/(e*x+d)^5*a^4-12/5/e^5/(e*x+d)^5*d^4*a*b^2*c+12/5/e^6/(e*x+d)
^5*d^5*a*b*c^2+24/e^5/(e*x+d)^2*a*b^2*c*d-60/e^6/(e*x+d)^2*a*b*c^2*d^2+12/e^4/(e*x+d)^3*a^2*b*c*d-15/e^6/(e*x+
d)^4*a*b*c^2*d^4+12/5/e^4/(e*x+d)^5*d^3*a^2*b*c-2/e^5/(e*x+d)^3*b^4*d^2+4*c/e^6*ln(e*x+d)*b^3-56*c^4/e^9*ln(e*
x+d)*d^3+2*c^3/e^6*x^2*b-3*c^4/e^7*x^2*d-28/3/e^9/(e*x+d)^3*c^4*d^6-1/e^2/(e*x+d)^4*a^3*b+1/e^5/(e*x+d)^4*b^4*
d^3+2/e^9/(e*x+d)^4*c^4*d^7-6/e^5/(e*x+d)*c^2*a^2-70/e^9/(e*x+d)*c^4*d^4-1/5/e^5/(e*x+d)^5*d^4*b^4-1/5/e^9/(e*
x+d)^5*c^4*d^8-2/e^4/(e*x+d)^2*a*b^3+2/e^5/(e*x+d)^2*b^4*d+28/e^9/(e*x+d)^2*c^4*d^5-4/3/e^3/(e*x+d)^3*a^3*c-2/
e^3/(e*x+d)^3*a^2*b^2+1/3*c^4*x^3/e^6-70/e^8/(e*x+d)^2*b*c^3*d^4-12/e^5/(e*x+d)^3*a^2*c^2*d^2-30/e^7/(e*x+d)^3
*b^2*c^2*d^4-5/e^6/(e*x+d)^4*b^3*c*d^4+9/e^7/(e*x+d)^4*b^2*c^2*d^5-7/e^8/(e*x+d)^4*b*c^3*d^6-12/e^5/(e*x+d)*a*
b^2*c-60/e^7/(e*x+d)*a*c^3*d^2+20/e^6/(e*x+d)*b^3*c*d-90/e^7/(e*x+d)*b^2*c^2*d^2+140/e^8/(e*x+d)*b*c^3*d^3+4/5
/e^2/(e*x+d)^5*d*a^3*b-4/5/e^3/(e*x+d)^5*a^3*c*d^2-6/5/e^3/(e*x+d)^5*d^2*a^2*b^2-6/5/e^5/(e*x+d)^5*a^2*c^2*d^4
+4/5/e^4/(e*x+d)^5*d^3*a*b^3-4/5/e^7/(e*x+d)^5*a*c^3*d^6+4/5/e^6/(e*x+d)^5*d^5*b^3*c-6/5/e^7/(e*x+d)^5*b^2*c^2
*d^6+4/5/e^8/(e*x+d)^5*b*c^3*d^7-6/e^4/(e*x+d)^2*a^2*b*c+4/e^4/(e*x+d)^3*a*b^3*d-20/e^7/(e*x+d)^3*a*c^3*d^4+40
/3/e^6/(e*x+d)^3*b^3*c*d^3

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maxima [B]  time = 1.40, size = 850, normalized size = 2.05 \[ -\frac {743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 3 \, a^{3} b d e^{7} + 3 \, a^{4} e^{8} + 261 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 137 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 15 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, b c^{3} d^{3} e^{5} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + 50 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 195 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 110 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 3 \, a^{3} b e^{8} + 231 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 125 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{15 \, {\left (e^{14} x^{5} + 5 \, d e^{13} x^{4} + 10 \, d^{2} e^{12} x^{3} + 10 \, d^{3} e^{11} x^{2} + 5 \, d^{4} e^{10} x + d^{5} e^{9}\right )}} + \frac {c^{4} e^{2} x^{3} - 3 \, {\left (3 \, c^{4} d e - 2 \, b c^{3} e^{2}\right )} x^{2} + 3 \, {\left (21 \, c^{4} d^{2} - 24 \, b c^{3} d e + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x}{3 \, e^{8}} - \frac {4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 3*a^3*b*d*e^7 + 3*a^4*e^8 + 261*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 137*(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 + 3*(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 + 15*(70*c^4*d^4*e^4 - 140*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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 50*(
3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 30*(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 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 195*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(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 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2
+ 2*a^3*c)*e^8)*x^2 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3
- 125*(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 + 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^14*x^5 + 5*d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x +
d^5*e^9) + 1/3*(c^4*e^2*x^3 - 3*(3*c^4*d*e - 2*b*c^3*e^2)*x^2 + 3*(21*c^4*d^2 - 24*b*c^3*d*e + 2*(3*b^2*c^2 +
2*a*c^3)*e^2)*x)/e^8 - 4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e
^3)*log(e*x + d)/e^9

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mupad [B]  time = 0.21, size = 959, normalized size = 2.32 \[ x^2\,\left (\frac {2\,b\,c^3}{e^6}-\frac {3\,c^4\,d}{e^7}\right )-x\,\left (\frac {6\,d\,\left (\frac {4\,b\,c^3}{e^6}-\frac {6\,c^4\,d}{e^7}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^6}+\frac {15\,c^4\,d^2}{e^8}\right )-\frac {x^3\,\left (6\,a^2\,b\,c\,e^7+12\,a^2\,c^2\,d\,e^6+2\,a\,b^3\,e^7+24\,a\,b^2\,c\,d\,e^6-180\,a\,b\,c^2\,d^2\,e^5+200\,a\,c^3\,d^3\,e^4+2\,b^4\,d\,e^6-60\,b^3\,c\,d^2\,e^5+300\,b^2\,c^2\,d^3\,e^4-490\,b\,c^3\,d^4\,e^3+252\,c^4\,d^5\,e^2\right )+x\,\left (a^3\,b\,e^7+\frac {2\,a^3\,c\,d\,e^6}{3}+a^2\,b^2\,d\,e^6+3\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-125\,a\,b\,c^2\,d^4\,e^3+154\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-\frac {125\,b^3\,c\,d^4\,e^3}{3}+231\,b^2\,c^2\,d^5\,e^2-399\,b\,c^3\,d^6\,e+\frac {638\,c^4\,d^7}{3}\right )+x^4\,\left (6\,a^2\,c^2\,e^7+12\,a\,b^2\,c\,e^7-60\,a\,b\,c^2\,d\,e^6+60\,a\,c^3\,d^2\,e^5+b^4\,e^7-20\,b^3\,c\,d\,e^6+90\,b^2\,c^2\,d^2\,e^5-140\,b\,c^3\,d^3\,e^4+70\,c^4\,d^4\,e^3\right )+\frac {3\,a^4\,e^8+3\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+9\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+3\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4-411\,a\,b\,c^2\,d^5\,e^3+522\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4-137\,b^3\,c\,d^5\,e^3+783\,b^2\,c^2\,d^6\,e^2-1377\,b\,c^3\,d^7\,e+743\,c^4\,d^8}{15\,e}+x^2\,\left (\frac {4\,a^3\,c\,e^7}{3}+2\,a^2\,b^2\,e^7+6\,a^2\,b\,c\,d\,e^6+12\,a^2\,c^2\,d^2\,e^5+2\,a\,b^3\,d\,e^6+24\,a\,b^2\,c\,d^2\,e^5-220\,a\,b\,c^2\,d^3\,e^4+260\,a\,c^3\,d^4\,e^3+2\,b^4\,d^2\,e^5-\frac {220\,b^3\,c\,d^3\,e^4}{3}+390\,b^2\,c^2\,d^4\,e^3-658\,b\,c^3\,d^5\,e^2+\frac {1036\,c^4\,d^6\,e}{3}\right )}{d^5\,e^8+5\,d^4\,e^9\,x+10\,d^3\,e^{10}\,x^2+10\,d^2\,e^{11}\,x^3+5\,d\,e^{12}\,x^4+e^{13}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,b^3\,c\,e^3+36\,b^2\,c^2\,d\,e^2-84\,b\,c^3\,d^2\,e-12\,a\,b\,c^2\,e^3+56\,c^4\,d^3+24\,a\,c^3\,d\,e^2\right )}{e^9}+\frac {c^4\,x^3}{3\,e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^6,x)

[Out]

x^2*((2*b*c^3)/e^6 - (3*c^4*d)/e^7) - x*((6*d*((4*b*c^3)/e^6 - (6*c^4*d)/e^7))/e - (4*a*c^3 + 6*b^2*c^2)/e^6 +
(15*c^4*d^2)/e^8) - (x^3*(2*a*b^3*e^7 + 2*b^4*d*e^6 + 252*c^4*d^5*e^2 + 200*a*c^3*d^3*e^4 + 12*a^2*c^2*d*e^6
- 490*b*c^3*d^4*e^3 - 60*b^3*c*d^2*e^5 + 300*b^2*c^2*d^3*e^4 + 6*a^2*b*c*e^7 + 24*a*b^2*c*d*e^6 - 180*a*b*c^2*
d^2*e^5) + x*((638*c^4*d^7)/3 + a^3*b*e^7 + b^4*d^3*e^4 + a*b^3*d^2*e^5 + a^2*b^2*d*e^6 + 154*a*c^3*d^5*e^2 -
(125*b^3*c*d^4*e^3)/3 + 6*a^2*c^2*d^3*e^4 + 231*b^2*c^2*d^5*e^2 + (2*a^3*c*d*e^6)/3 - 399*b*c^3*d^6*e - 125*a*
b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 + 3*a^2*b*c*d^2*e^5) + x^4*(b^4*e^7 + 6*a^2*c^2*e^7 + 70*c^4*d^4*e^3 + 60*a
*c^3*d^2*e^5 - 140*b*c^3*d^3*e^4 + 90*b^2*c^2*d^2*e^5 + 12*a*b^2*c*e^7 - 20*b^3*c*d*e^6 - 60*a*b*c^2*d*e^6) +
(3*a^4*e^8 + 743*c^4*d^8 + 3*b^4*d^4*e^4 + 3*a*b^3*d^3*e^5 + 522*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 - 137*b^3*c*d
^5*e^3 + 3*a^2*b^2*d^2*e^6 + 18*a^2*c^2*d^4*e^4 + 783*b^2*c^2*d^6*e^2 + 3*a^3*b*d*e^7 - 1377*b*c^3*d^7*e - 411
*a*b*c^2*d^5*e^3 + 36*a*b^2*c*d^4*e^4 + 9*a^2*b*c*d^3*e^5)/(15*e) + x^2*((4*a^3*c*e^7)/3 + (1036*c^4*d^6*e)/3
+ 2*a^2*b^2*e^7 + 2*b^4*d^2*e^5 + 260*a*c^3*d^4*e^3 - 658*b*c^3*d^5*e^2 - (220*b^3*c*d^3*e^4)/3 + 12*a^2*c^2*d
^2*e^5 + 390*b^2*c^2*d^4*e^3 + 2*a*b^3*d*e^6 + 6*a^2*b*c*d*e^6 - 220*a*b*c^2*d^3*e^4 + 24*a*b^2*c*d^2*e^5))/(d
^5*e^8 + e^13*x^5 + 5*d^4*e^9*x + 5*d*e^12*x^4 + 10*d^3*e^10*x^2 + 10*d^2*e^11*x^3) - (log(d + e*x)*(56*c^4*d^
3 - 4*b^3*c*e^3 + 36*b^2*c^2*d*e^2 - 12*a*b*c^2*e^3 + 24*a*c^3*d*e^2 - 84*b*c^3*d^2*e))/e^9 + (c^4*x^3)/(3*e^6
)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**6,x)

[Out]

Timed out

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