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

Optimal. Leaf size=417 \[ \frac {x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac {2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac {c^3 x^4 (c d-b e)}{e^5}+\frac {c^4 x^5}{5 e^4} \]

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Rubi [A]  time = 0.65, antiderivative size = 417, 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 {x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}+\frac {2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\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 )}{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 ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac {c^3 x^4 (c d-b e)}{e^5}+\frac {c^4 x^5}{5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*
a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*
x^2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)/e^5 + (c^
4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(
e^9*(d + e*x)^2) - (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)))/(e^9*(d + e*x)
) - (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))*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)^4} \, dx &=\int \left (\frac {35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )}{e^8}+\frac {4 c \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)-3 b c e^2 (2 b d-a e)\right ) x}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^2}{e^6}-\frac {4 c^3 (c d-b e) x^3}{e^5}+\frac {c^4 x^4}{e^4}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^4}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{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 )^3}{e^9 (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^9 (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+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 = 425, normalized size = 1.02 \begin {gather*} \frac {-60 (2 c d-b e) \log (d+e x) \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 )+15 e x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (a e-2 b d)+b^4 e^4+35 c^4 d^4\right )+30 c e^2 x^2 \left (2 c^2 d e (5 b d-2 a e)+3 b c e^2 (a e-2 b d)+b^3 e^3-5 c^3 d^3\right )-\frac {30 \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}+10 c^2 e^3 x^3 \left (2 c e (a e-4 b d)+3 b^2 e^2+5 c^2 d^2\right )+\frac {30 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}-\frac {5 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^3}+15 c^3 e^4 x^4 (b e-c d)+3 c^4 e^5 x^5}{15 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^
2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d +
a*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3
*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/
(d + e*x)^2 - (30*(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) - 60
*(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))*Log[d + e*x])/(15*e^9)

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 1282, normalized size = 3.07

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)^4,x, algorithm="fricas")

[Out]

1/15*(3*c^4*e^8*x^8 - 365*c^4*d^8 + 1070*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 5*a^4*e^8 - 370*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 + 470*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 65*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 110*(a*b^3 + 3*a^2*b*c
)*d^3*e^5 - 10*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 3*(2*c^4*d*e^7 - 5*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 - 35*b*c^3*
d*e^7 + 10*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 3*(14*c^4*d^3*e^5 - 35*b*c^3*d^2*e^6 + 10*(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 + 15*(14*c^4*d^4*e^4 - 35*b*c^3*d^3*e^5 + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 5*(235*c^4*d^5*e^3 - 556*b*c^3*d
^4*e^4 + 146*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 126*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d*e^7)*x^3 + 15*(67*c^4*d^6*e^2 - 136*b*c^3*d^5*e^3 + 26*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 6*(b^3*c + 3*a*b*c^
2)*d^3*e^5 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 12*(a*b^3 + 3*a^2*b*c)*d*e^7 - 2*(3*a^2*b^2 + 2*a^3*c)
*e^8)*x^2 - 15*(17*c^4*d^7*e - 74*b*c^3*d^6*e^2 + 2*a^3*b*e^8 + 34*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 54*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 18*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2
+ 2*a^3*c)*d*e^7)*x - 60*(14*c^4*d^8 - 35*b*c^3*d^7*e + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - (a*b^3 + 3*a^2*b*c)*d^3*e^5 + (14*c^4*d^5*e^3 - 35*b*c
^3*d^4*e^4 + 10*(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 + (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 + 3*(14*c^4*d^6*e^2 - 35*b*c^3*d^5*e^3 + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - (a*b^3 + 3*a^2*b*c)*d*e^7)*x^2
+ 3*(14*c^4*d^7*e - 35*b*c^3*d^6*e^2 + 10*(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 + (b^
4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - (a*b^3 + 3*a^2*b*c)*d^2*e^6)*x)*log(e*x + d))/(e^12*x^3 + 3*d*e^11*x^2 +
3*d^2*e^10*x + d^3*e^9)

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giac [B]  time = 0.17, size = 865, normalized size = 2.07 \begin {gather*} -4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 30 \, b^{2} c^{2} d^{3} e^{2} + 20 \, a c^{3} d^{3} e^{2} - 10 \, b^{3} c d^{2} e^{3} - 30 \, a b c^{2} d^{2} e^{3} + b^{4} d e^{4} + 12 \, a b^{2} c d e^{4} + 6 \, a^{2} c^{2} d e^{4} - a b^{3} e^{5} - 3 \, a^{2} b c e^{5}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{15} \, {\left (3 \, c^{4} x^{5} e^{16} - 15 \, c^{4} d x^{4} e^{15} + 50 \, c^{4} d^{2} x^{3} e^{14} - 150 \, c^{4} d^{3} x^{2} e^{13} + 525 \, c^{4} d^{4} x e^{12} + 15 \, b c^{3} x^{4} e^{16} - 80 \, b c^{3} d x^{3} e^{15} + 300 \, b c^{3} d^{2} x^{2} e^{14} - 1200 \, b c^{3} d^{3} x e^{13} + 30 \, b^{2} c^{2} x^{3} e^{16} + 20 \, a c^{3} x^{3} e^{16} - 180 \, b^{2} c^{2} d x^{2} e^{15} - 120 \, a c^{3} d x^{2} e^{15} + 900 \, b^{2} c^{2} d^{2} x e^{14} + 600 \, a c^{3} d^{2} x e^{14} + 30 \, b^{3} c x^{2} e^{16} + 90 \, a b c^{2} x^{2} e^{16} - 240 \, b^{3} c d x e^{15} - 720 \, a b c^{2} d x e^{15} + 15 \, b^{4} x e^{16} + 180 \, a b^{2} c x e^{16} + 90 \, a^{2} c^{2} x e^{16}\right )} e^{\left (-20\right )} - \frac {{\left (73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 222 \, b^{2} c^{2} d^{6} e^{2} + 148 \, a c^{3} d^{6} e^{2} - 94 \, b^{3} c d^{5} e^{3} - 282 \, a b c^{2} d^{5} e^{3} + 13 \, b^{4} d^{4} e^{4} + 156 \, a b^{2} c d^{4} e^{4} + 78 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a b^{3} d^{3} e^{5} - 66 \, a^{2} b c d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} + 2 \, a^{3} b d e^{7} + a^{4} e^{8} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 45 \, b^{2} c^{2} d^{4} e^{4} + 30 \, a c^{3} d^{4} e^{4} - 20 \, b^{3} c d^{3} e^{5} - 60 \, 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} - 6 \, a b^{3} d e^{7} - 18 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + 81 \, b^{2} c^{2} d^{5} e^{3} + 54 \, a c^{3} d^{5} e^{3} - 35 \, b^{3} c d^{4} e^{4} - 105 \, a b c^{2} d^{4} e^{4} + 5 \, b^{4} d^{3} e^{5} + 60 \, a b^{2} c d^{3} e^{5} + 30 \, a^{2} c^{2} d^{3} e^{5} - 9 \, a b^{3} d^{2} e^{6} - 27 \, 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}\right )} x\right )} e^{\left (-9\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*e^(-9)*log(abs(x*e + d)) + 1/1
5*(3*c^4*x^5*e^16 - 15*c^4*d*x^4*e^15 + 50*c^4*d^2*x^3*e^14 - 150*c^4*d^3*x^2*e^13 + 525*c^4*d^4*x*e^12 + 15*b
*c^3*x^4*e^16 - 80*b*c^3*d*x^3*e^15 + 300*b*c^3*d^2*x^2*e^14 - 1200*b*c^3*d^3*x*e^13 + 30*b^2*c^2*x^3*e^16 + 2
0*a*c^3*x^3*e^16 - 180*b^2*c^2*d*x^2*e^15 - 120*a*c^3*d*x^2*e^15 + 900*b^2*c^2*d^2*x*e^14 + 600*a*c^3*d^2*x*e^
14 + 30*b^3*c*x^2*e^16 + 90*a*b*c^2*x^2*e^16 - 240*b^3*c*d*x*e^15 - 720*a*b*c^2*d*x*e^15 + 15*b^4*x*e^16 + 180
*a*b^2*c*x*e^16 + 90*a^2*c^2*x*e^16)*e^(-20) - 1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 222*b^2*c^2*d^6*e^2 + 148*a
*c^3*d^6*e^2 - 94*b^3*c*d^5*e^3 - 282*a*b*c^2*d^5*e^3 + 13*b^4*d^4*e^4 + 156*a*b^2*c*d^4*e^4 + 78*a^2*c^2*d^4*
e^4 - 22*a*b^3*d^3*e^5 - 66*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 2*a^3*b*d*e^7 + a^4*e^8 +
6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 45*b^2*c^2*d^4*e^4 + 30*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 - 60*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 - 6*a*b^3*d*e^7 - 18*a^2*b*c*d*e^7 + 3*a^2*b^
2*e^8 + 2*a^3*c*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + 81*b^2*c^2*d^5*e^3 + 54*a*c^3*d^5*e^3 - 35*b^3
*c*d^4*e^4 - 105*a*b*c^2*d^4*e^4 + 5*b^4*d^3*e^5 + 60*a*b^2*c*d^3*e^5 + 30*a^2*c^2*d^3*e^5 - 9*a*b^3*d^2*e^6 -
27*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)^3

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maple [B]  time = 0.06, size = 1265, normalized size = 3.03

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)^4,x)

[Out]

-36/e^5/(e*x+d)*a^2*c^2*d^2+12/e^4/(e*x+d)*a*b^3*d-16/3/e^5*x^3*b*c^3*d+6/e^4*x^2*a*b*c^2+1/5*c^4*x^5/e^4-1/3/
e/(e*x+d)^3*a^4+1/e^4*x^4*b*c^3-1/e^5*x^4*c^4*d+4/3/e^4*x^3*a*c^3+2/e^4*x^3*b^2*c^2+10/3/e^6*x^3*c^4*d^2+2/e^4
*x^2*b^3*c-10/e^7*x^2*c^4*d^3+6/e^4*c^2*a^2*x+35/e^8*c^4*d^4*x-4/e^3/(e*x+d)*a^3*c-6/e^3/(e*x+d)*a^2*b^2-6/e^5
/(e*x+d)*b^4*d^2-28/e^9/(e*x+d)*c^4*d^6-2/e^2/(e*x+d)^2*a^3*b+4/e^6/(e*x+d)^3*a*b*c^2*d^5-48/e^5*ln(e*x+d)*a*b
^2*c*d+120/e^6*ln(e*x+d)*a*b*c^2*d^2-48/e^5*a*b*c^2*d*x+36/e^4/(e*x+d)*a^2*b*c*d-72/e^5/(e*x+d)*a*b^2*c*d^2+12
0/e^6/(e*x+d)*a*b*c^2*d^3-18/e^4/(e*x+d)^2*a^2*b*c*d^2+24/e^5/(e*x+d)^2*a*b^2*c*d^3-30/e^6/(e*x+d)^2*a*b*c^2*d
^4+4/e^4/(e*x+d)^3*d^3*a^2*b*c-4/e^5/(e*x+d)^3*a*b^2*c*d^4+b^4/e^4*x+4/e^4*ln(e*x+d)*a*b^3-4/e^5*ln(e*x+d)*b^4
*d-56/e^9*ln(e*x+d)*c^4*d^5+4/e^9/(e*x+d)^2*c^4*d^7-1/3/e^5/(e*x+d)^3*b^4*d^4-1/3/e^9/(e*x+d)^3*c^4*d^8+2/e^5/
(e*x+d)^2*b^4*d^3-90/e^7/(e*x+d)*b^2*c^2*d^4+84/e^8/(e*x+d)*b*c^3*d^5+4/e^3/(e*x+d)^2*a^3*c*d+6/e^3/(e*x+d)^2*
a^2*b^2*d+12/e^5/(e*x+d)^2*a^2*c^2*d^3-6/e^4/(e*x+d)^2*a*b^3*d^2+12/e^7/(e*x+d)^2*a*c^3*d^5-10/e^6/(e*x+d)^2*b
^3*c*d^4+18/e^7/(e*x+d)^2*b^2*c^2*d^5-14/e^8/(e*x+d)^2*b*c^3*d^6+4/3/e^2/(e*x+d)^3*d*a^3*b-4/3/e^3/(e*x+d)^3*a
^3*c*d^2-2/e^3/(e*x+d)^3*d^2*a^2*b^2-2/e^5/(e*x+d)^3*a^2*c^2*d^4+4/3/e^4/(e*x+d)^3*d^3*a*b^3-4/3/e^7/(e*x+d)^3
*a*c^3*d^6+4/3/e^6/(e*x+d)^3*b^3*c*d^5-2/e^7/(e*x+d)^3*b^2*c^2*d^6+4/3/e^8/(e*x+d)^3*b*c^3*d^7+12/e^4*ln(e*x+d
)*a^2*b*c-24/e^5*ln(e*x+d)*a^2*c^2*d-80/e^7*ln(e*x+d)*a*c^3*d^3+40/e^6*ln(e*x+d)*b^3*c*d^2-120/e^7*ln(e*x+d)*b
^2*c^2*d^3+140/e^8*ln(e*x+d)*b*c^3*d^4-8/e^5*x^2*a*c^3*d-12/e^5*x^2*b^2*c^2*d+20/e^6*x^2*b*c^3*d^2+12/e^4*a*b^
2*c*x+40/e^6*a*c^3*d^2*x-16/e^5*b^3*c*d*x+60/e^6*b^2*c^2*d^2*x-80/e^7*b*c^3*d^3*x-60/e^7/(e*x+d)*a*c^3*d^4+40/
e^6/(e*x+d)*b^3*c*d^3

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maxima [B]  time = 1.36, size = 827, normalized size = 1.98 \begin {gather*} -\frac {73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 2 \, a^{3} b d e^{7} + a^{4} e^{8} + 74 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 94 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 13 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 22 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 20 \, {\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} - 6 \, {\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} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + a^{3} b e^{8} + 27 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 35 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 5 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 9 \, {\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}{3 \, {\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} + \frac {3 \, c^{4} e^{4} x^{5} - 15 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{2} e^{2} - 8 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{3} - 30 \, {\left (5 \, c^{4} d^{3} e - 10 \, b c^{3} d^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{2} + 15 \, {\left (35 \, c^{4} d^{4} - 80 \, b c^{3} d^{3} e + 20 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x}{15 \, e^{8}} - \frac {4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} \log \left (e x + d\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)^4,x, algorithm="maxima")

[Out]

-1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 2*a^3*b*d*e^7 + a^4*e^8 + 74*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 94*(b^3*c +
3*a*b*c^2)*d^5*e^3 + 13*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 22*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2
+ 2*a^3*c)*d^2*e^6 + 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 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + a^3*b*e^8 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 35*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 9*(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^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9) + 1/15*(3*c^4*e^4*x^5 - 15*(c^4*d*e^3 -
b*c^3*e^4)*x^4 + 10*(5*c^4*d^2*e^2 - 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 30*(5*c^4*d^3*e - 10*b*c
^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 - (b^3*c + 3*a*b*c^2)*e^4)*x^2 + 15*(35*c^4*d^4 - 80*b*c^3*d^3*e +
20*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 16*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x)/e^8 -
4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*log(e*x + d)/e^9

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mupad [B]  time = 0.84, size = 1143, normalized size = 2.74 \begin {gather*} x^4\,\left (\frac {b\,c^3}{e^4}-\frac {c^4\,d}{e^5}\right )-x^2\,\left (\frac {2\,c^4\,d^3}{e^7}+\frac {3\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {2\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {2\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )-x^3\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{3\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{3\,e^4}+\frac {2\,c^4\,d^2}{e^6}\right )+x\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^4}+\frac {6\,d^2\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,c^4\,d^3}{e^7}+\frac {6\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )}{e}-\frac {c^4\,d^4}{e^8}-\frac {4\,d^3\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^3}\right )-\frac {x\,\left (2\,a^3\,b\,e^7+4\,a^3\,c\,d\,e^6+6\,a^2\,b^2\,d\,e^6-54\,a^2\,b\,c\,d^2\,e^5+60\,a^2\,c^2\,d^3\,e^4-18\,a\,b^3\,d^2\,e^5+120\,a\,b^2\,c\,d^3\,e^4-210\,a\,b\,c^2\,d^4\,e^3+108\,a\,c^3\,d^5\,e^2+10\,b^4\,d^3\,e^4-70\,b^3\,c\,d^4\,e^3+162\,b^2\,c^2\,d^5\,e^2-154\,b\,c^3\,d^6\,e+52\,c^4\,d^7\right )+\frac {a^4\,e^8+2\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6-66\,a^2\,b\,c\,d^3\,e^5+78\,a^2\,c^2\,d^4\,e^4-22\,a\,b^3\,d^3\,e^5+156\,a\,b^2\,c\,d^4\,e^4-282\,a\,b\,c^2\,d^5\,e^3+148\,a\,c^3\,d^6\,e^2+13\,b^4\,d^4\,e^4-94\,b^3\,c\,d^5\,e^3+222\,b^2\,c^2\,d^6\,e^2-214\,b\,c^3\,d^7\,e+73\,c^4\,d^8}{3\,e}+x^2\,\left (4\,a^3\,c\,e^7+6\,a^2\,b^2\,e^7-36\,a^2\,b\,c\,d\,e^6+36\,a^2\,c^2\,d^2\,e^5-12\,a\,b^3\,d\,e^6+72\,a\,b^2\,c\,d^2\,e^5-120\,a\,b\,c^2\,d^3\,e^4+60\,a\,c^3\,d^4\,e^3+6\,b^4\,d^2\,e^5-40\,b^3\,c\,d^3\,e^4+90\,b^2\,c^2\,d^4\,e^3-84\,b\,c^3\,d^5\,e^2+28\,c^4\,d^6\,e\right )}{d^3\,e^8+3\,d^2\,e^9\,x+3\,d\,e^{10}\,x^2+e^{11}\,x^3}+\frac {c^4\,x^5}{5\,e^4}-\frac {\ln \left (d+e\,x\right )\,\left (-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\right )}{e^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((b*c^3)/e^4 - (c^4*d)/e^5) - x^2*((2*c^4*d^3)/e^7 + (3*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (2*d*((
4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (2*b*c*(3*a*c + b^2
))/e^4) - x^3*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/(3*e) - (4*a*c^3 + 6*b^2*c^2)/(3*e^4) + (2*c^4*d^2)/e^6)
+ x*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^4 + (6*d^2*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c
^2)/e^4 + (6*c^4*d^2)/e^6))/e^2 + (4*d*((4*c^4*d^3)/e^7 + (6*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (4*d*(
(4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (4*b*c*(3*a*c + b^
2))/e^4))/e - (c^4*d^4)/e^8 - (4*d^3*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^3) - (x*(52*c^4*d^7 + 2*a^3*b*e^7 + 10
*b^4*d^3*e^4 - 18*a*b^3*d^2*e^5 + 6*a^2*b^2*d*e^6 + 108*a*c^3*d^5*e^2 - 70*b^3*c*d^4*e^3 + 60*a^2*c^2*d^3*e^4
+ 162*b^2*c^2*d^5*e^2 + 4*a^3*c*d*e^6 - 154*b*c^3*d^6*e - 210*a*b*c^2*d^4*e^3 + 120*a*b^2*c*d^3*e^4 - 54*a^2*b
*c*d^2*e^5) + (a^4*e^8 + 73*c^4*d^8 + 13*b^4*d^4*e^4 - 22*a*b^3*d^3*e^5 + 148*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6
- 94*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 78*a^2*c^2*d^4*e^4 + 222*b^2*c^2*d^6*e^2 + 2*a^3*b*d*e^7 - 214*b*c^3*
d^7*e - 282*a*b*c^2*d^5*e^3 + 156*a*b^2*c*d^4*e^4 - 66*a^2*b*c*d^3*e^5)/(3*e) + x^2*(4*a^3*c*e^7 + 28*c^4*d^6*
e + 6*a^2*b^2*e^7 + 6*b^4*d^2*e^5 + 60*a*c^3*d^4*e^3 - 84*b*c^3*d^5*e^2 - 40*b^3*c*d^3*e^4 + 36*a^2*c^2*d^2*e^
5 + 90*b^2*c^2*d^4*e^3 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 120*a*b*c^2*d^3*e^4 + 72*a*b^2*c*d^2*e^5))/(d^3*e
^8 + e^11*x^3 + 3*d^2*e^9*x + 3*d*e^10*x^2) + (c^4*x^5)/(5*e^4) - (log(d + e*x)*(56*c^4*d^5 - 4*a*b^3*e^5 + 4*
b^4*d*e^4 + 80*a*c^3*d^3*e^2 + 24*a^2*c^2*d*e^4 - 40*b^3*c*d^2*e^3 + 120*b^2*c^2*d^3*e^2 - 12*a^2*b*c*e^5 - 14
0*b*c^3*d^4*e + 48*a*b^2*c*d*e^4 - 120*a*b*c^2*d^2*e^3))/e^9

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sympy [B]  time = 97.98, size = 944, normalized size = 2.26 \begin {gather*} \frac {c^{4} x^{5}}{5 e^{4}} + x^{4} \left (\frac {b c^{3}}{e^{4}} - \frac {c^{4} d}{e^{5}}\right ) + x^{3} \left (\frac {4 a c^{3}}{3 e^{4}} + \frac {2 b^{2} c^{2}}{e^{4}} - \frac {16 b c^{3} d}{3 e^{5}} + \frac {10 c^{4} d^{2}}{3 e^{6}}\right ) + x^{2} \left (\frac {6 a b c^{2}}{e^{4}} - \frac {8 a c^{3} d}{e^{5}} + \frac {2 b^{3} c}{e^{4}} - \frac {12 b^{2} c^{2} d}{e^{5}} + \frac {20 b c^{3} d^{2}}{e^{6}} - \frac {10 c^{4} d^{3}}{e^{7}}\right ) + x \left (\frac {6 a^{2} c^{2}}{e^{4}} + \frac {12 a b^{2} c}{e^{4}} - \frac {48 a b c^{2} d}{e^{5}} + \frac {40 a c^{3} d^{2}}{e^{6}} + \frac {b^{4}}{e^{4}} - \frac {16 b^{3} c d}{e^{5}} + \frac {60 b^{2} c^{2} d^{2}}{e^{6}} - \frac {80 b c^{3} d^{3}}{e^{7}} + \frac {35 c^{4} d^{4}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 2 a^{3} b d e^{7} - 4 a^{3} c d^{2} e^{6} - 6 a^{2} b^{2} d^{2} e^{6} + 66 a^{2} b c d^{3} e^{5} - 78 a^{2} c^{2} d^{4} e^{4} + 22 a b^{3} d^{3} e^{5} - 156 a b^{2} c d^{4} e^{4} + 282 a b c^{2} d^{5} e^{3} - 148 a c^{3} d^{6} e^{2} - 13 b^{4} d^{4} e^{4} + 94 b^{3} c d^{5} e^{3} - 222 b^{2} c^{2} d^{6} e^{2} + 214 b c^{3} d^{7} e - 73 c^{4} d^{8} + x^{2} \left (- 12 a^{3} c e^{8} - 18 a^{2} b^{2} e^{8} + 108 a^{2} b c d e^{7} - 108 a^{2} c^{2} d^{2} e^{6} + 36 a b^{3} d e^{7} - 216 a b^{2} c d^{2} e^{6} + 360 a b c^{2} d^{3} e^{5} - 180 a c^{3} d^{4} e^{4} - 18 b^{4} d^{2} e^{6} + 120 b^{3} c d^{3} e^{5} - 270 b^{2} c^{2} d^{4} e^{4} + 252 b c^{3} d^{5} e^{3} - 84 c^{4} d^{6} e^{2}\right ) + x \left (- 6 a^{3} b e^{8} - 12 a^{3} c d e^{7} - 18 a^{2} b^{2} d e^{7} + 162 a^{2} b c d^{2} e^{6} - 180 a^{2} c^{2} d^{3} e^{5} + 54 a b^{3} d^{2} e^{6} - 360 a b^{2} c d^{3} e^{5} + 630 a b c^{2} d^{4} e^{4} - 324 a c^{3} d^{5} e^{3} - 30 b^{4} d^{3} e^{5} + 210 b^{3} c d^{4} e^{4} - 486 b^{2} c^{2} d^{5} e^{3} + 462 b c^{3} d^{6} e^{2} - 156 c^{4} d^{7} e\right )}{3 d^{3} e^{9} + 9 d^{2} e^{10} x + 9 d e^{11} x^{2} + 3 e^{12} x^{3}} + \frac {4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \left (3 a c e^{2} + b^{2} e^{2} - 7 b c d e + 7 c^{2} d^{2}\right ) \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)**4,x)

[Out]

c**4*x**5/(5*e**4) + x**4*(b*c**3/e**4 - c**4*d/e**5) + x**3*(4*a*c**3/(3*e**4) + 2*b**2*c**2/e**4 - 16*b*c**3
*d/(3*e**5) + 10*c**4*d**2/(3*e**6)) + x**2*(6*a*b*c**2/e**4 - 8*a*c**3*d/e**5 + 2*b**3*c/e**4 - 12*b**2*c**2*
d/e**5 + 20*b*c**3*d**2/e**6 - 10*c**4*d**3/e**7) + x*(6*a**2*c**2/e**4 + 12*a*b**2*c/e**4 - 48*a*b*c**2*d/e**
5 + 40*a*c**3*d**2/e**6 + b**4/e**4 - 16*b**3*c*d/e**5 + 60*b**2*c**2*d**2/e**6 - 80*b*c**3*d**3/e**7 + 35*c**
4*d**4/e**8) + (-a**4*e**8 - 2*a**3*b*d*e**7 - 4*a**3*c*d**2*e**6 - 6*a**2*b**2*d**2*e**6 + 66*a**2*b*c*d**3*e
**5 - 78*a**2*c**2*d**4*e**4 + 22*a*b**3*d**3*e**5 - 156*a*b**2*c*d**4*e**4 + 282*a*b*c**2*d**5*e**3 - 148*a*c
**3*d**6*e**2 - 13*b**4*d**4*e**4 + 94*b**3*c*d**5*e**3 - 222*b**2*c**2*d**6*e**2 + 214*b*c**3*d**7*e - 73*c**
4*d**8 + x**2*(-12*a**3*c*e**8 - 18*a**2*b**2*e**8 + 108*a**2*b*c*d*e**7 - 108*a**2*c**2*d**2*e**6 + 36*a*b**3
*d*e**7 - 216*a*b**2*c*d**2*e**6 + 360*a*b*c**2*d**3*e**5 - 180*a*c**3*d**4*e**4 - 18*b**4*d**2*e**6 + 120*b**
3*c*d**3*e**5 - 270*b**2*c**2*d**4*e**4 + 252*b*c**3*d**5*e**3 - 84*c**4*d**6*e**2) + x*(-6*a**3*b*e**8 - 12*a
**3*c*d*e**7 - 18*a**2*b**2*d*e**7 + 162*a**2*b*c*d**2*e**6 - 180*a**2*c**2*d**3*e**5 + 54*a*b**3*d**2*e**6 -
360*a*b**2*c*d**3*e**5 + 630*a*b*c**2*d**4*e**4 - 324*a*c**3*d**5*e**3 - 30*b**4*d**3*e**5 + 210*b**3*c*d**4*e
**4 - 486*b**2*c**2*d**5*e**3 + 462*b*c**3*d**6*e**2 - 156*c**4*d**7*e))/(3*d**3*e**9 + 9*d**2*e**10*x + 9*d*e
**11*x**2 + 3*e**12*x**3) + 4*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*
c**2*d**2)*log(d + e*x)/e**9

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