3.19.31 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=440 \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9 (d+e x)^9}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac {\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 440, 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} \begin {gather*} -\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}{7 e^9 (d+e x)^7}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 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)^6}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\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 )}{9 e^9 (d+e x)^9}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac {\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-(c*d^2 - b*d*e + a*e^2)^4/(11*e^9*(d + e*x)^11) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^9*(d + e*x
)^10) - (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)))/(9*e^9*(d + e*x)^9) + ((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)))/(2*e^9*(d + e*x)^8) - (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))/(7*e^9*(d + e*x)^7) + (2*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)^6) - (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(5*e^9*(d + e*x)^5) + (c^3*(2*c*d - b*e))/
(e^9*(d + e*x)^4) - c^4/(3*e^9*(d + e*x)^3)

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)^{12}} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^{12}}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^{11}}+\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)^{10}}+\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)^9}+\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)^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 )}{e^8 (d+e x)^7}+\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)^6}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^5}+\frac {c^4}{e^8 (d+e x)^4}\right ) \, dx\\ &=-\frac {\left (c d^2-b d e+a e^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^{10}}-\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 )}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{2 e^9 (d+e x)^8}-\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 )}{7 e^9 (d+e x)^7}+\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^6}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{5 e^9 (d+e x)^5}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 731, normalized size = 1.66 \begin {gather*} -\frac {6 c^2 e^2 \left (3 a^2 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+3 b^2 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+c e^3 \left (56 a^3 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+63 a^2 b e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+36 a b^2 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+10 b^3 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+3 e^4 \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+3 c^3 e \left (4 a e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 b \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )+14 c^4 \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )}{6930 e^9 (d+e x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-1/6930*(14*c^4*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462
*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 3*e^4*(210*a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a^2*b^2*e^2*
(d^2 + 11*d*e*x + 55*e^2*x^2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + b^4*(d^4 + 11*d^3*
e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + c*e^3*(56*a^3*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) + 63*a^
2*b*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 36*a*b^2*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*
d*e^3*x^3 + 330*e^4*x^4) + 10*b^3*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e
^5*x^5)) + 6*c^2*e^2*(3*a^2*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b*e*(d
^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 3*b^2*(d^6 + 11*d^5*e*x +
55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + 3*c^3*e*(4*a*e*(d^6 + 11*
d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*b*(d^7 + 11*d^
6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)))/
(e^9*(d + e*x)^11)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^12, x]

________________________________________________________________________________________

fricas [B]  time = 0.38, size = 924, normalized size = 2.10 \begin {gather*} -\frac {2310 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 21 \, b c^{3} d^{7} e + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 2310 \, {\left (2 \, c^{4} d e^{7} + 3 \, b c^{3} e^{8}\right )} x^{7} + 462 \, {\left (14 \, c^{4} d^{2} e^{6} + 21 \, b c^{3} d e^{7} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 462 \, {\left (14 \, c^{4} d^{3} e^{5} + 21 \, b c^{3} d^{2} e^{6} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 330 \, {\left (14 \, c^{4} d^{4} e^{4} + 21 \, b c^{3} d^{3} e^{5} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 165 \, {\left (14 \, c^{4} d^{5} e^{3} + 21 \, b c^{3} d^{4} e^{4} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 55 \, {\left (14 \, c^{4} d^{6} e^{2} + 21 \, b c^{3} d^{5} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 11 \, {\left (14 \, c^{4} d^{7} e + 21 \, b c^{3} d^{6} e^{2} + 252 \, a^{3} b e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{6930 \, {\left (e^{20} x^{11} + 11 \, d e^{19} x^{10} + 55 \, d^{2} e^{18} x^{9} + 165 \, d^{3} e^{17} x^{8} + 330 \, d^{4} e^{16} x^{7} + 462 \, d^{5} e^{15} x^{6} + 462 \, d^{6} e^{14} x^{5} + 330 \, d^{7} e^{13} x^{4} + 165 \, d^{8} e^{12} x^{3} + 55 \, d^{9} e^{11} x^{2} + 11 \, d^{10} e^{10} x + d^{11} 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)^12,x, algorithm="fricas")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 28*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d^2*e^6 + 2
1*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5 + 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a
*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^
3)*d^2*e^6 + 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 165*(14*c^4*d^5*e^3 +
21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d*e^7 + 21*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 21*(a*b^3 + 3*a^2*b
*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2
*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 21*(a*b^
3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11 + 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 16
5*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3
+ 55*d^9*e^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

________________________________________________________________________________________

giac [B]  time = 0.20, size = 945, normalized size = 2.15 \begin {gather*} -\frac {{\left (2310 \, c^{4} x^{8} e^{8} + 4620 \, c^{4} d x^{7} e^{7} + 6468 \, c^{4} d^{2} x^{6} e^{6} + 6468 \, c^{4} d^{3} x^{5} e^{5} + 4620 \, c^{4} d^{4} x^{4} e^{4} + 2310 \, c^{4} d^{5} x^{3} e^{3} + 770 \, c^{4} d^{6} x^{2} e^{2} + 154 \, c^{4} d^{7} x e + 14 \, c^{4} d^{8} + 6930 \, b c^{3} x^{7} e^{8} + 9702 \, b c^{3} d x^{6} e^{7} + 9702 \, b c^{3} d^{2} x^{5} e^{6} + 6930 \, b c^{3} d^{3} x^{4} e^{5} + 3465 \, b c^{3} d^{4} x^{3} e^{4} + 1155 \, b c^{3} d^{5} x^{2} e^{3} + 231 \, b c^{3} d^{6} x e^{2} + 21 \, b c^{3} d^{7} e + 8316 \, b^{2} c^{2} x^{6} e^{8} + 5544 \, a c^{3} x^{6} e^{8} + 8316 \, b^{2} c^{2} d x^{5} e^{7} + 5544 \, a c^{3} d x^{5} e^{7} + 5940 \, b^{2} c^{2} d^{2} x^{4} e^{6} + 3960 \, a c^{3} d^{2} x^{4} e^{6} + 2970 \, b^{2} c^{2} d^{3} x^{3} e^{5} + 1980 \, a c^{3} d^{3} x^{3} e^{5} + 990 \, b^{2} c^{2} d^{4} x^{2} e^{4} + 660 \, a c^{3} d^{4} x^{2} e^{4} + 198 \, b^{2} c^{2} d^{5} x e^{3} + 132 \, a c^{3} d^{5} x e^{3} + 18 \, b^{2} c^{2} d^{6} e^{2} + 12 \, a c^{3} d^{6} e^{2} + 4620 \, b^{3} c x^{5} e^{8} + 13860 \, a b c^{2} x^{5} e^{8} + 3300 \, b^{3} c d x^{4} e^{7} + 9900 \, a b c^{2} d x^{4} e^{7} + 1650 \, b^{3} c d^{2} x^{3} e^{6} + 4950 \, a b c^{2} d^{2} x^{3} e^{6} + 550 \, b^{3} c d^{3} x^{2} e^{5} + 1650 \, a b c^{2} d^{3} x^{2} e^{5} + 110 \, b^{3} c d^{4} x e^{4} + 330 \, a b c^{2} d^{4} x e^{4} + 10 \, b^{3} c d^{5} e^{3} + 30 \, a b c^{2} d^{5} e^{3} + 990 \, b^{4} x^{4} e^{8} + 11880 \, a b^{2} c x^{4} e^{8} + 5940 \, a^{2} c^{2} x^{4} e^{8} + 495 \, b^{4} d x^{3} e^{7} + 5940 \, a b^{2} c d x^{3} e^{7} + 2970 \, a^{2} c^{2} d x^{3} e^{7} + 165 \, b^{4} d^{2} x^{2} e^{6} + 1980 \, a b^{2} c d^{2} x^{2} e^{6} + 990 \, a^{2} c^{2} d^{2} x^{2} e^{6} + 33 \, b^{4} d^{3} x e^{5} + 396 \, a b^{2} c d^{3} x e^{5} + 198 \, a^{2} c^{2} d^{3} x e^{5} + 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} + 3465 \, a b^{3} x^{3} e^{8} + 10395 \, a^{2} b c x^{3} e^{8} + 1155 \, a b^{3} d x^{2} e^{7} + 3465 \, a^{2} b c d x^{2} e^{7} + 231 \, a b^{3} d^{2} x e^{6} + 693 \, a^{2} b c d^{2} x e^{6} + 21 \, a b^{3} d^{3} e^{5} + 63 \, a^{2} b c d^{3} e^{5} + 4620 \, a^{2} b^{2} x^{2} e^{8} + 3080 \, a^{3} c x^{2} e^{8} + 924 \, a^{2} b^{2} d x e^{7} + 616 \, a^{3} c d x e^{7} + 84 \, a^{2} b^{2} d^{2} e^{6} + 56 \, a^{3} c d^{2} e^{6} + 2772 \, a^{3} b x e^{8} + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8}\right )} e^{\left (-9\right )}}{6930 \, {\left (x e + d\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/6930*(2310*c^4*x^8*e^8 + 4620*c^4*d*x^7*e^7 + 6468*c^4*d^2*x^6*e^6 + 6468*c^4*d^3*x^5*e^5 + 4620*c^4*d^4*x^
4*e^4 + 2310*c^4*d^5*x^3*e^3 + 770*c^4*d^6*x^2*e^2 + 154*c^4*d^7*x*e + 14*c^4*d^8 + 6930*b*c^3*x^7*e^8 + 9702*
b*c^3*d*x^6*e^7 + 9702*b*c^3*d^2*x^5*e^6 + 6930*b*c^3*d^3*x^4*e^5 + 3465*b*c^3*d^4*x^3*e^4 + 1155*b*c^3*d^5*x^
2*e^3 + 231*b*c^3*d^6*x*e^2 + 21*b*c^3*d^7*e + 8316*b^2*c^2*x^6*e^8 + 5544*a*c^3*x^6*e^8 + 8316*b^2*c^2*d*x^5*
e^7 + 5544*a*c^3*d*x^5*e^7 + 5940*b^2*c^2*d^2*x^4*e^6 + 3960*a*c^3*d^2*x^4*e^6 + 2970*b^2*c^2*d^3*x^3*e^5 + 19
80*a*c^3*d^3*x^3*e^5 + 990*b^2*c^2*d^4*x^2*e^4 + 660*a*c^3*d^4*x^2*e^4 + 198*b^2*c^2*d^5*x*e^3 + 132*a*c^3*d^5
*x*e^3 + 18*b^2*c^2*d^6*e^2 + 12*a*c^3*d^6*e^2 + 4620*b^3*c*x^5*e^8 + 13860*a*b*c^2*x^5*e^8 + 3300*b^3*c*d*x^4
*e^7 + 9900*a*b*c^2*d*x^4*e^7 + 1650*b^3*c*d^2*x^3*e^6 + 4950*a*b*c^2*d^2*x^3*e^6 + 550*b^3*c*d^3*x^2*e^5 + 16
50*a*b*c^2*d^3*x^2*e^5 + 110*b^3*c*d^4*x*e^4 + 330*a*b*c^2*d^4*x*e^4 + 10*b^3*c*d^5*e^3 + 30*a*b*c^2*d^5*e^3 +
990*b^4*x^4*e^8 + 11880*a*b^2*c*x^4*e^8 + 5940*a^2*c^2*x^4*e^8 + 495*b^4*d*x^3*e^7 + 5940*a*b^2*c*d*x^3*e^7 +
2970*a^2*c^2*d*x^3*e^7 + 165*b^4*d^2*x^2*e^6 + 1980*a*b^2*c*d^2*x^2*e^6 + 990*a^2*c^2*d^2*x^2*e^6 + 33*b^4*d^
3*x*e^5 + 396*a*b^2*c*d^3*x*e^5 + 198*a^2*c^2*d^3*x*e^5 + 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 + 3465*a*b^3*x^3*e^8 + 10395*a^2*b*c*x^3*e^8 + 1155*a*b^3*d*x^2*e^7 + 3465*a^2*b*c*d*x^2*e^7 + 231*a*b^3*d
^2*x*e^6 + 693*a^2*b*c*d^2*x*e^6 + 21*a*b^3*d^3*e^5 + 63*a^2*b*c*d^3*e^5 + 4620*a^2*b^2*x^2*e^8 + 3080*a^3*c*x
^2*e^8 + 924*a^2*b^2*d*x*e^7 + 616*a^3*c*d*x*e^7 + 84*a^2*b^2*d^2*e^6 + 56*a^3*c*d^2*e^6 + 2772*a^3*b*x*e^8 +
252*a^3*b*d*e^7 + 630*a^4*e^8)*e^(-9)/(x*e + d)^11

________________________________________________________________________________________

maple [B]  time = 0.05, size = 914, normalized size = 2.08 \begin {gather*} -\frac {c^{4}}{3 \left (e x +d \right )^{3} e^{9}}-\frac {\left (b e -2 c d \right ) c^{3}}{\left (e x +d \right )^{4} e^{9}}-\frac {2 \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) c^{2}}{5 \left (e x +d \right )^{5} e^{9}}-\frac {2 \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} c d \,e^{2}+21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) c}{3 \left (e x +d \right )^{6} e^{9}}-\frac {4 a^{3} b \,e^{7}-8 a^{3} c d \,e^{6}-12 d \,a^{2} b^{2} e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 a^{2} c^{2} d^{3} e^{4}+12 d^{2} a \,b^{3} e^{5}-48 d^{3} a \,b^{2} c \,e^{4}+60 d^{4} a b \,c^{2} e^{3}-24 a \,c^{3} d^{5} e^{2}-4 d^{3} b^{4} e^{4}+20 d^{4} b^{3} c \,e^{3}-36 d^{5} b^{2} c^{2} e^{2}+28 b \,c^{3} d^{6} e -8 c^{4} d^{7}}{10 \left (e x +d \right )^{10} e^{9}}-\frac {6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}}{7 \left (e x +d \right )^{7} e^{9}}-\frac {4 e^{6} a^{3} c +6 a^{2} b^{2} e^{6}-36 d \,a^{2} b c \,e^{5}+36 a^{2} c^{2} d^{2} e^{4}-12 d a \,b^{3} e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 d^{3} a b \,c^{2} e^{3}+60 a \,c^{3} d^{4} e^{2}+6 b^{4} d^{2} e^{4}-40 d^{3} b^{3} c \,e^{3}+90 d^{4} b^{2} c^{2} e^{2}-84 b \,c^{3} d^{5} e +28 c^{4} d^{6}}{9 \left (e x +d \right )^{9} e^{9}}-\frac {12 a^{2} b c \,e^{5}-24 d \,a^{2} c^{2} e^{4}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 d^{2} a b \,c^{2} e^{3}-80 d^{3} a \,c^{3} e^{2}-4 b^{4} d \,e^{4}+40 d^{2} b^{3} c \,e^{3}-120 d^{3} b^{2} c^{2} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{8 \left (e x +d \right )^{8} e^{9}}-\frac {a^{4} e^{8}-4 d \,a^{3} b \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 d^{2} a^{2} b^{2} e^{6}-12 d^{3} a^{2} b c \,e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 d^{3} a \,b^{3} e^{5}+12 d^{4} a \,b^{2} c \,e^{4}-12 d^{5} a b \,c^{2} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 d^{5} b^{3} c \,e^{3}+6 d^{6} b^{2} c^{2} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{11 \left (e x +d \right )^{11} e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^12,x)

[Out]

-1/10*(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/(e*x+d)^10-1/7*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*
b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d^3*e+70*c^4*d^4)/e^9/(e*x+d)^7-2/3*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*
e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/e^9/(e*x+d)^6-2/5*c^2*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)
/e^9/(e*x+d)^5-1/3*c^4/e^9/(e*x+d)^3-c^3*(b*e-2*c*d)/e^9/(e*x+d)^4-1/9*(4*a^3*c*e^6+6*a^2*b^2*e^6-36*a^2*b*c*d
*e^5+36*a^2*c^2*d^2*e^4-12*a*b^3*d*e^5+72*a*b^2*c*d^2*e^4-120*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^2+6*b^4*d^2*e^4-4
0*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2-84*b*c^3*d^5*e+28*c^4*d^6)/e^9/(e*x+d)^9-1/8*(12*a^2*b*c*e^5-24*a^2*c^2*d*e
^4+4*a*b^3*e^5-48*a*b^2*c*d*e^4+120*a*b*c^2*d^2*e^3-80*a*c^3*d^3*e^2-4*b^4*d*e^4+40*b^3*c*d^2*e^3-120*b^2*c^2*
d^3*e^2+140*b*c^3*d^4*e-56*c^4*d^5)/e^9/(e*x+d)^8-1/11*(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)^11

________________________________________________________________________________________

maxima [B]  time = 1.75, size = 924, normalized size = 2.10 \begin {gather*} -\frac {2310 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 21 \, b c^{3} d^{7} e + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 2310 \, {\left (2 \, c^{4} d e^{7} + 3 \, b c^{3} e^{8}\right )} x^{7} + 462 \, {\left (14 \, c^{4} d^{2} e^{6} + 21 \, b c^{3} d e^{7} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 462 \, {\left (14 \, c^{4} d^{3} e^{5} + 21 \, b c^{3} d^{2} e^{6} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 330 \, {\left (14 \, c^{4} d^{4} e^{4} + 21 \, b c^{3} d^{3} e^{5} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 165 \, {\left (14 \, c^{4} d^{5} e^{3} + 21 \, b c^{3} d^{4} e^{4} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 55 \, {\left (14 \, c^{4} d^{6} e^{2} + 21 \, b c^{3} d^{5} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 11 \, {\left (14 \, c^{4} d^{7} e + 21 \, b c^{3} d^{6} e^{2} + 252 \, a^{3} b e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 10 \, {\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} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{6930 \, {\left (e^{20} x^{11} + 11 \, d e^{19} x^{10} + 55 \, d^{2} e^{18} x^{9} + 165 \, d^{3} e^{17} x^{8} + 330 \, d^{4} e^{16} x^{7} + 462 \, d^{5} e^{15} x^{6} + 462 \, d^{6} e^{14} x^{5} + 330 \, d^{7} e^{13} x^{4} + 165 \, d^{8} e^{12} x^{3} + 55 \, d^{9} e^{11} x^{2} + 11 \, d^{10} e^{10} x + d^{11} 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)^12,x, algorithm="maxima")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 28*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d^2*e^6 + 2
1*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5 + 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a
*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^
3)*d^2*e^6 + 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 165*(14*c^4*d^5*e^3 +
21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d*e^7 + 21*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 21*(a*b^3 + 3*a^2*b
*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2
*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 21*(a*b^
3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11 + 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 16
5*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3
+ 55*d^9*e^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

________________________________________________________________________________________

mupad [B]  time = 1.09, size = 997, normalized size = 2.27 \begin {gather*} -\frac {\frac {630\,a^4\,e^8+252\,a^3\,b\,d\,e^7+56\,a^3\,c\,d^2\,e^6+84\,a^2\,b^2\,d^2\,e^6+63\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+21\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4+30\,a\,b\,c^2\,d^5\,e^3+12\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4+10\,b^3\,c\,d^5\,e^3+18\,b^2\,c^2\,d^6\,e^2+21\,b\,c^3\,d^7\,e+14\,c^4\,d^8}{6930\,e^9}+\frac {x^3\,\left (63\,a^2\,b\,c\,e^5+18\,a^2\,c^2\,d\,e^4+21\,a\,b^3\,e^5+36\,a\,b^2\,c\,d\,e^4+30\,a\,b\,c^2\,d^2\,e^3+12\,a\,c^3\,d^3\,e^2+3\,b^4\,d\,e^4+10\,b^3\,c\,d^2\,e^3+18\,b^2\,c^2\,d^3\,e^2+21\,b\,c^3\,d^4\,e+14\,c^4\,d^5\right )}{42\,e^6}+\frac {x^4\,\left (18\,a^2\,c^2\,e^4+36\,a\,b^2\,c\,e^4+30\,a\,b\,c^2\,d\,e^3+12\,a\,c^3\,d^2\,e^2+3\,b^4\,e^4+10\,b^3\,c\,d\,e^3+18\,b^2\,c^2\,d^2\,e^2+21\,b\,c^3\,d^3\,e+14\,c^4\,d^4\right )}{21\,e^5}+\frac {x\,\left (252\,a^3\,b\,e^7+56\,a^3\,c\,d\,e^6+84\,a^2\,b^2\,d\,e^6+63\,a^2\,b\,c\,d^2\,e^5+18\,a^2\,c^2\,d^3\,e^4+21\,a\,b^3\,d^2\,e^5+36\,a\,b^2\,c\,d^3\,e^4+30\,a\,b\,c^2\,d^4\,e^3+12\,a\,c^3\,d^5\,e^2+3\,b^4\,d^3\,e^4+10\,b^3\,c\,d^4\,e^3+18\,b^2\,c^2\,d^5\,e^2+21\,b\,c^3\,d^6\,e+14\,c^4\,d^7\right )}{630\,e^8}+\frac {c^4\,x^8}{3\,e}+\frac {x^2\,\left (56\,a^3\,c\,e^6+84\,a^2\,b^2\,e^6+63\,a^2\,b\,c\,d\,e^5+18\,a^2\,c^2\,d^2\,e^4+21\,a\,b^3\,d\,e^5+36\,a\,b^2\,c\,d^2\,e^4+30\,a\,b\,c^2\,d^3\,e^3+12\,a\,c^3\,d^4\,e^2+3\,b^4\,d^2\,e^4+10\,b^3\,c\,d^3\,e^3+18\,b^2\,c^2\,d^4\,e^2+21\,b\,c^3\,d^5\,e+14\,c^4\,d^6\right )}{126\,e^7}+\frac {c^3\,x^7\,\left (3\,b\,e+2\,c\,d\right )}{3\,e^2}+\frac {c^2\,x^6\,\left (18\,b^2\,e^2+21\,b\,c\,d\,e+14\,c^2\,d^2+12\,a\,c\,e^2\right )}{15\,e^3}+\frac {c\,x^5\,\left (10\,b^3\,e^3+18\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+30\,a\,b\,c\,e^3+14\,c^3\,d^3+12\,a\,c^2\,d\,e^2\right )}{15\,e^4}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^12,x)

[Out]

-((630*a^4*e^8 + 14*c^4*d^8 + 3*b^4*d^4*e^4 + 21*a*b^3*d^3*e^5 + 12*a*c^3*d^6*e^2 + 56*a^3*c*d^2*e^6 + 10*b^3*
c*d^5*e^3 + 84*a^2*b^2*d^2*e^6 + 18*a^2*c^2*d^4*e^4 + 18*b^2*c^2*d^6*e^2 + 252*a^3*b*d*e^7 + 21*b*c^3*d^7*e +
30*a*b*c^2*d^5*e^3 + 36*a*b^2*c*d^4*e^4 + 63*a^2*b*c*d^3*e^5)/(6930*e^9) + (x^3*(14*c^4*d^5 + 21*a*b^3*e^5 + 3
*b^4*d*e^4 + 12*a*c^3*d^3*e^2 + 18*a^2*c^2*d*e^4 + 10*b^3*c*d^2*e^3 + 18*b^2*c^2*d^3*e^2 + 63*a^2*b*c*e^5 + 21
*b*c^3*d^4*e + 36*a*b^2*c*d*e^4 + 30*a*b*c^2*d^2*e^3))/(42*e^6) + (x^4*(3*b^4*e^4 + 14*c^4*d^4 + 18*a^2*c^2*e^
4 + 12*a*c^3*d^2*e^2 + 18*b^2*c^2*d^2*e^2 + 36*a*b^2*c*e^4 + 21*b*c^3*d^3*e + 10*b^3*c*d*e^3 + 30*a*b*c^2*d*e^
3))/(21*e^5) + (x*(14*c^4*d^7 + 252*a^3*b*e^7 + 3*b^4*d^3*e^4 + 21*a*b^3*d^2*e^5 + 84*a^2*b^2*d*e^6 + 12*a*c^3
*d^5*e^2 + 10*b^3*c*d^4*e^3 + 18*a^2*c^2*d^3*e^4 + 18*b^2*c^2*d^5*e^2 + 56*a^3*c*d*e^6 + 21*b*c^3*d^6*e + 30*a
*b*c^2*d^4*e^3 + 36*a*b^2*c*d^3*e^4 + 63*a^2*b*c*d^2*e^5))/(630*e^8) + (c^4*x^8)/(3*e) + (x^2*(14*c^4*d^6 + 56
*a^3*c*e^6 + 84*a^2*b^2*e^6 + 3*b^4*d^2*e^4 + 12*a*c^3*d^4*e^2 + 10*b^3*c*d^3*e^3 + 18*a^2*c^2*d^2*e^4 + 18*b^
2*c^2*d^4*e^2 + 21*a*b^3*d*e^5 + 21*b*c^3*d^5*e + 63*a^2*b*c*d*e^5 + 30*a*b*c^2*d^3*e^3 + 36*a*b^2*c*d^2*e^4))
/(126*e^7) + (c^3*x^7*(3*b*e + 2*c*d))/(3*e^2) + (c^2*x^6*(18*b^2*e^2 + 14*c^2*d^2 + 12*a*c*e^2 + 21*b*c*d*e))
/(15*e^3) + (c*x^5*(10*b^3*e^3 + 14*c^3*d^3 + 30*a*b*c*e^3 + 12*a*c^2*d*e^2 + 21*b*c^2*d^2*e + 18*b^2*c*d*e^2)
)/(15*e^4))/(d^11 + e^11*x^11 + 11*d*e^10*x^10 + 55*d^9*e^2*x^2 + 165*d^8*e^3*x^3 + 330*d^7*e^4*x^4 + 462*d^6*
e^5*x^5 + 462*d^5*e^6*x^6 + 330*d^4*e^7*x^7 + 165*d^3*e^8*x^8 + 55*d^2*e^9*x^9 + 11*d^10*e*x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**12,x)

[Out]

Timed out

________________________________________________________________________________________