3.2151 \(\int \frac {(a+b x+c x^2)^4}{d+e x} \, dx\)

Optimal. Leaf size=428 \[ \frac {(d+e x)^4 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{4 e^9}+\frac {c^2 (d+e x)^6 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac {4 c (d+e x)^5 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac {4 (d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^4}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9} \]

[Out]

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

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Rubi [A]  time = 0.71, antiderivative size = 428, 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} \[ \frac {(d+e x)^4 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{4 e^9}+\frac {c^2 (d+e x)^6 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac {4 c (d+e x)^5 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac {4 (d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^4}{e^9}-\frac {4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x),x]

[Out]

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

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Mathematica [A]  time = 0.39, size = 616, normalized size = 1.44 \[ \frac {x \left (84 c^2 e^2 \left (5 a^2 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 a b e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+56 c e^3 \left (30 a^3 e^3 (e x-2 d)+30 a^2 b e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+15 a b^2 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^3 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+70 b e^4 \left (48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+8 c^3 e \left (7 a e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+b \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )+c^4 \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac {\log (d+e x) \left (e (a e-b d)+c d^2\right )^4}{e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c^4*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*
d*e^6*x^6 + 105*e^7*x^7) + 70*b*e^4*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e
^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 56*c*e^3*(30*a^3*e^3*(-2*d + e*x) + 30*a^2*b*
e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 15*a*b^2*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^3*(60*d^4 -
30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 84*c^2*e^2*(5*a^2*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*
e^2*x^2 + 3*e^3*x^3) + 2*a*b*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^2*(-60*d
^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 8*c^3*e*(7*a*e*(-60*d^5 + 30
*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + b*(420*d^6 - 210*d^5*e*x + 140*d^4*e
^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6))))/(840*e^8) + ((c*d^2 + e*(-(b*d) + a*
e))^4*Log[d + e*x])/e^9

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fricas [A]  time = 0.67, size = 799, normalized size = 1.87 \[ \frac {105 \, c^{4} e^{8} x^{8} - 120 \, {\left (c^{4} d e^{7} - 4 \, b c^{3} e^{8}\right )} x^{7} + 140 \, {\left (c^{4} d^{2} e^{6} - 4 \, b c^{3} d e^{7} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} - 168 \, {\left (c^{4} d^{3} e^{5} - 4 \, b c^{3} d^{2} e^{6} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 210 \, {\left (c^{4} d^{4} e^{4} - 4 \, b c^{3} d^{3} e^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 4 \, {\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} - 280 \, {\left (c^{4} d^{5} e^{3} - 4 \, b c^{3} d^{4} e^{4} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 4 \, {\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} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e^{2} - 4 \, b c^{3} d^{5} e^{3} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} - 840 \, {\left (c^{4} d^{7} e - 4 \, b c^{3} d^{6} e^{2} - 4 \, a^{3} b e^{8} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x + 840 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} + a^{4} e^{8} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 4 \, {\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}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*c^4*e^8*x^8 - 120*(c^4*d*e^7 - 4*b*c^3*e^8)*x^7 + 140*(c^4*d^2*e^6 - 4*b*c^3*d*e^7 + 2*(3*b^2*c^2 +
2*a*c^3)*e^8)*x^6 - 168*(c^4*d^3*e^5 - 4*b*c^3*d^2*e^6 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 4*(b^3*c + 3*a*b*c^2
)*e^8)*x^5 + 210*(c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 4*(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 - 280*(c^4*d^5*e^3 - 4*b*c^3*d^4*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3
*e^5 - 4*(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 - 4*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 +
420*(c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 4*(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 - 4*(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 - 840*(c^4*d
^7*e - 4*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 4*(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 + 840*(c^
4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(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 - 4*(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
)*log(e*x + d))/e^9

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giac [B]  time = 0.17, size = 945, normalized size = 2.21 \[ {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} + 4 \, a c^{3} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} - 12 \, a b c^{2} d^{5} e^{3} + b^{4} d^{4} e^{4} + 12 \, a b^{2} c d^{4} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} - 12 \, 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} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{840} \, {\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, c^{4} d^{2} x^{6} e^{5} - 168 \, c^{4} d^{3} x^{5} e^{4} + 210 \, c^{4} d^{4} x^{4} e^{3} - 280 \, c^{4} d^{5} x^{3} e^{2} + 420 \, c^{4} d^{6} x^{2} e - 840 \, c^{4} d^{7} x + 480 \, b c^{3} x^{7} e^{7} - 560 \, b c^{3} d x^{6} e^{6} + 672 \, b c^{3} d^{2} x^{5} e^{5} - 840 \, b c^{3} d^{3} x^{4} e^{4} + 1120 \, b c^{3} d^{4} x^{3} e^{3} - 1680 \, b c^{3} d^{5} x^{2} e^{2} + 3360 \, b c^{3} d^{6} x e + 840 \, b^{2} c^{2} x^{6} e^{7} + 560 \, a c^{3} x^{6} e^{7} - 1008 \, b^{2} c^{2} d x^{5} e^{6} - 672 \, a c^{3} d x^{5} e^{6} + 1260 \, b^{2} c^{2} d^{2} x^{4} e^{5} + 840 \, a c^{3} d^{2} x^{4} e^{5} - 1680 \, b^{2} c^{2} d^{3} x^{3} e^{4} - 1120 \, a c^{3} d^{3} x^{3} e^{4} + 2520 \, b^{2} c^{2} d^{4} x^{2} e^{3} + 1680 \, a c^{3} d^{4} x^{2} e^{3} - 5040 \, b^{2} c^{2} d^{5} x e^{2} - 3360 \, a c^{3} d^{5} x e^{2} + 672 \, b^{3} c x^{5} e^{7} + 2016 \, a b c^{2} x^{5} e^{7} - 840 \, b^{3} c d x^{4} e^{6} - 2520 \, a b c^{2} d x^{4} e^{6} + 1120 \, b^{3} c d^{2} x^{3} e^{5} + 3360 \, a b c^{2} d^{2} x^{3} e^{5} - 1680 \, b^{3} c d^{3} x^{2} e^{4} - 5040 \, a b c^{2} d^{3} x^{2} e^{4} + 3360 \, b^{3} c d^{4} x e^{3} + 10080 \, a b c^{2} d^{4} x e^{3} + 210 \, b^{4} x^{4} e^{7} + 2520 \, a b^{2} c x^{4} e^{7} + 1260 \, a^{2} c^{2} x^{4} e^{7} - 280 \, b^{4} d x^{3} e^{6} - 3360 \, a b^{2} c d x^{3} e^{6} - 1680 \, a^{2} c^{2} d x^{3} e^{6} + 420 \, b^{4} d^{2} x^{2} e^{5} + 5040 \, a b^{2} c d^{2} x^{2} e^{5} + 2520 \, a^{2} c^{2} d^{2} x^{2} e^{5} - 840 \, b^{4} d^{3} x e^{4} - 10080 \, a b^{2} c d^{3} x e^{4} - 5040 \, a^{2} c^{2} d^{3} x e^{4} + 1120 \, a b^{3} x^{3} e^{7} + 3360 \, a^{2} b c x^{3} e^{7} - 1680 \, a b^{3} d x^{2} e^{6} - 5040 \, a^{2} b c d x^{2} e^{6} + 3360 \, a b^{3} d^{2} x e^{5} + 10080 \, a^{2} b c d^{2} x e^{5} + 2520 \, a^{2} b^{2} x^{2} e^{7} + 1680 \, a^{3} c x^{2} e^{7} - 5040 \, a^{2} b^{2} d x e^{6} - 3360 \, a^{3} c d x e^{6} + 3360 \, a^{3} b x e^{7}\right )} e^{\left (-8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^
4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*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 - 4*a^3*b*d*e^7 + a^4*e^8)*e^(-9)*log(abs(x*e + d)) + 1/840*(105*c^4*x^8*e^7 - 120*c^4*d*x^7*e^6
+ 140*c^4*d^2*x^6*e^5 - 168*c^4*d^3*x^5*e^4 + 210*c^4*d^4*x^4*e^3 - 280*c^4*d^5*x^3*e^2 + 420*c^4*d^6*x^2*e -
840*c^4*d^7*x + 480*b*c^3*x^7*e^7 - 560*b*c^3*d*x^6*e^6 + 672*b*c^3*d^2*x^5*e^5 - 840*b*c^3*d^3*x^4*e^4 + 112
0*b*c^3*d^4*x^3*e^3 - 1680*b*c^3*d^5*x^2*e^2 + 3360*b*c^3*d^6*x*e + 840*b^2*c^2*x^6*e^7 + 560*a*c^3*x^6*e^7 -
1008*b^2*c^2*d*x^5*e^6 - 672*a*c^3*d*x^5*e^6 + 1260*b^2*c^2*d^2*x^4*e^5 + 840*a*c^3*d^2*x^4*e^5 - 1680*b^2*c^2
*d^3*x^3*e^4 - 1120*a*c^3*d^3*x^3*e^4 + 2520*b^2*c^2*d^4*x^2*e^3 + 1680*a*c^3*d^4*x^2*e^3 - 5040*b^2*c^2*d^5*x
*e^2 - 3360*a*c^3*d^5*x*e^2 + 672*b^3*c*x^5*e^7 + 2016*a*b*c^2*x^5*e^7 - 840*b^3*c*d*x^4*e^6 - 2520*a*b*c^2*d*
x^4*e^6 + 1120*b^3*c*d^2*x^3*e^5 + 3360*a*b*c^2*d^2*x^3*e^5 - 1680*b^3*c*d^3*x^2*e^4 - 5040*a*b*c^2*d^3*x^2*e^
4 + 3360*b^3*c*d^4*x*e^3 + 10080*a*b*c^2*d^4*x*e^3 + 210*b^4*x^4*e^7 + 2520*a*b^2*c*x^4*e^7 + 1260*a^2*c^2*x^4
*e^7 - 280*b^4*d*x^3*e^6 - 3360*a*b^2*c*d*x^3*e^6 - 1680*a^2*c^2*d*x^3*e^6 + 420*b^4*d^2*x^2*e^5 + 5040*a*b^2*
c*d^2*x^2*e^5 + 2520*a^2*c^2*d^2*x^2*e^5 - 840*b^4*d^3*x*e^4 - 10080*a*b^2*c*d^3*x*e^4 - 5040*a^2*c^2*d^3*x*e^
4 + 1120*a*b^3*x^3*e^7 + 3360*a^2*b*c*x^3*e^7 - 1680*a*b^3*d*x^2*e^6 - 5040*a^2*b*c*d*x^2*e^6 + 3360*a*b^3*d^2
*x*e^5 + 10080*a^2*b*c*d^2*x*e^5 + 2520*a^2*b^2*x^2*e^7 + 1680*a^3*c*x^2*e^7 - 5040*a^2*b^2*d*x*e^6 - 3360*a^3
*c*d*x*e^6 + 3360*a^3*b*x*e^7)*e^(-8)

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maple [B]  time = 0.05, size = 1096, normalized size = 2.56 \[ \frac {c^{4} x^{8}}{8 e}+\frac {4 b \,c^{3} x^{7}}{7 e}-\frac {c^{4} d \,x^{7}}{7 e^{2}}+\frac {2 a \,c^{3} x^{6}}{3 e}+\frac {b^{2} c^{2} x^{6}}{e}-\frac {2 b \,c^{3} d \,x^{6}}{3 e^{2}}+\frac {c^{4} d^{2} x^{6}}{6 e^{3}}+\frac {12 a b \,c^{2} x^{5}}{5 e}-\frac {4 a \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {4 b^{3} c \,x^{5}}{5 e}-\frac {6 b^{2} c^{2} d \,x^{5}}{5 e^{2}}+\frac {4 b \,c^{3} d^{2} x^{5}}{5 e^{3}}-\frac {c^{4} d^{3} x^{5}}{5 e^{4}}+\frac {3 a^{2} c^{2} x^{4}}{2 e}+\frac {3 a \,b^{2} c \,x^{4}}{e}-\frac {3 a b \,c^{2} d \,x^{4}}{e^{2}}+\frac {a \,c^{3} d^{2} x^{4}}{e^{3}}+\frac {b^{4} x^{4}}{4 e}-\frac {b^{3} c d \,x^{4}}{e^{2}}+\frac {3 b^{2} c^{2} d^{2} x^{4}}{2 e^{3}}-\frac {b \,c^{3} d^{3} x^{4}}{e^{4}}+\frac {c^{4} d^{4} x^{4}}{4 e^{5}}+\frac {4 a^{2} b c \,x^{3}}{e}-\frac {2 a^{2} c^{2} d \,x^{3}}{e^{2}}+\frac {4 a \,b^{3} x^{3}}{3 e}-\frac {4 a \,b^{2} c d \,x^{3}}{e^{2}}+\frac {4 a b \,c^{2} d^{2} x^{3}}{e^{3}}-\frac {4 a \,c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {b^{4} d \,x^{3}}{3 e^{2}}+\frac {4 b^{3} c \,d^{2} x^{3}}{3 e^{3}}-\frac {2 b^{2} c^{2} d^{3} x^{3}}{e^{4}}+\frac {4 b \,c^{3} d^{4} x^{3}}{3 e^{5}}-\frac {c^{4} d^{5} x^{3}}{3 e^{6}}+\frac {2 a^{3} c \,x^{2}}{e}+\frac {3 a^{2} b^{2} x^{2}}{e}-\frac {6 a^{2} b c d \,x^{2}}{e^{2}}+\frac {3 a^{2} c^{2} d^{2} x^{2}}{e^{3}}-\frac {2 a \,b^{3} d \,x^{2}}{e^{2}}+\frac {6 a \,b^{2} c \,d^{2} x^{2}}{e^{3}}-\frac {6 a b \,c^{2} d^{3} x^{2}}{e^{4}}+\frac {2 a \,c^{3} d^{4} x^{2}}{e^{5}}+\frac {b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {2 b^{3} c \,d^{3} x^{2}}{e^{4}}+\frac {3 b^{2} c^{2} d^{4} x^{2}}{e^{5}}-\frac {2 b \,c^{3} d^{5} x^{2}}{e^{6}}+\frac {c^{4} d^{6} x^{2}}{2 e^{7}}+\frac {a^{4} \ln \left (e x +d \right )}{e}-\frac {4 a^{3} b d \ln \left (e x +d \right )}{e^{2}}+\frac {4 a^{3} b x}{e}+\frac {4 a^{3} c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {4 a^{3} c d x}{e^{2}}+\frac {6 a^{2} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 a^{2} b^{2} d x}{e^{2}}-\frac {12 a^{2} b c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {12 a^{2} b c \,d^{2} x}{e^{3}}+\frac {6 a^{2} c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {6 a^{2} c^{2} d^{3} x}{e^{4}}-\frac {4 a \,b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {4 a \,b^{3} d^{2} x}{e^{3}}+\frac {12 a \,b^{2} c \,d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {12 a \,b^{2} c \,d^{3} x}{e^{4}}-\frac {12 a b \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {12 a b \,c^{2} d^{4} x}{e^{5}}+\frac {4 a \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {4 a \,c^{3} d^{5} x}{e^{6}}+\frac {b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {b^{4} d^{3} x}{e^{4}}-\frac {4 b^{3} c \,d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {4 b^{3} c \,d^{4} x}{e^{5}}+\frac {6 b^{2} c^{2} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {6 b^{2} c^{2} d^{5} x}{e^{6}}-\frac {4 b \,c^{3} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {4 b \,c^{3} d^{6} x}{e^{7}}+\frac {c^{4} d^{8} \ln \left (e x +d \right )}{e^{9}}-\frac {c^{4} d^{7} x}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/e^4*x^2*a*b*c^2*d^3-6/e^2*x^2*a^2*b*c*d-4/e^2*x^3*a*b^2*c*d+4/e^3*x^3*a*b*c^2*d^2-3/e^2*x^4*a*b*c^2*d-12/e^
4*x*a*b^2*c*d^3+12/e^5*x*a*b*c^2*d^4+12/e^3*x*a^2*b*c*d^2-12/e^6*ln(e*x+d)*a*b*c^2*d^5-12/e^4*ln(e*x+d)*a^2*b*
c*d^3+12/e^5*ln(e*x+d)*a*b^2*c*d^4+6/e^3*x^2*a*b^2*c*d^2+1/2/e^7*x^2*c^4*d^6+1/2/e^3*x^2*b^4*d^2-1/5/e^4*x^5*c
^4*d^3-1/e^4*x*b^4*d^3+1/e*x^6*b^2*c^2-1/e^8*x*c^4*d^7+2/3/e*x^6*a*c^3+4/7/e*x^7*b*c^3+3/e*x^2*a^2*b^2-1/3/e^6
*x^3*c^4*d^5+2/e*x^2*a^3*c+4/5/e*x^5*b^3*c+1/6/e^3*x^6*c^4*d^2+4/3/e*x^3*a*b^3-1/7/e^2*x^7*c^4*d+4/e*x*a^3*b-1
/3/e^2*x^3*b^4*d+3/2/e*x^4*a^2*c^2+1/4/e^5*x^4*c^4*d^4+1/e^5*ln(e*x+d)*b^4*d^4+1/e^9*ln(e*x+d)*c^4*d^8+1/4/e*x
^4*b^4+1/8/e*c^4*x^8+1/e*ln(e*x+d)*a^4+4/e*x^3*a^2*b*c+4/5/e^3*x^5*b*c^3*d^2-1/e^2*x^4*b^3*c*d+3/2/e^3*x^4*b^2
*c^2*d^2-2/e^2*x^3*a^2*c^2*d-4/3/e^4*x^3*a*c^3*d^3-2/3/e^2*x^6*b*c^3*d-2/e^4*x^2*b^3*c*d^3-4/5/e^2*x^5*a*c^3*d
+4/3/e^5*x^3*b*c^3*d^4+4/3/e^3*x^3*b^3*c*d^2-2/e^4*x^3*b^2*c^2*d^3-6/5/e^2*x^5*b^2*c^2*d-1/e^4*x^4*b*c^3*d^3+3
/e*x^4*a*b^2*c+12/5/e*x^5*a*b*c^2+4/e^7*x*b*c^3*d^6-4/e^2*ln(e*x+d)*a^3*b*d+4/e^3*ln(e*x+d)*a^3*c*d^2-6/e^4*x*
a^2*c^2*d^3+4/e^3*x*a*b^3*d^2-4/e^8*ln(e*x+d)*b*c^3*d^7+6/e^3*ln(e*x+d)*a^2*b^2*d^2+3/e^5*x^2*b^2*c^2*d^4-2/e^
6*x^2*b*c^3*d^5+1/e^3*x^4*a*c^3*d^2-6/e^2*x*a^2*b^2*d+6/e^5*ln(e*x+d)*a^2*c^2*d^4-4/e^4*ln(e*x+d)*a*b^3*d^3+4/
e^7*ln(e*x+d)*a*c^3*d^6-4/e^6*ln(e*x+d)*b^3*c*d^5+6/e^7*ln(e*x+d)*b^2*c^2*d^6-6/e^6*x*b^2*c^2*d^5-4/e^2*x*a^3*
c*d+3/e^3*x^2*a^2*c^2*d^2-2/e^2*x^2*a*b^3*d+2/e^5*x^2*a*c^3*d^4-4/e^6*x*a*c^3*d^5+4/e^5*x*b^3*c*d^4

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maxima [A]  time = 1.13, size = 797, normalized size = 1.86 \[ \frac {105 \, c^{4} e^{7} x^{8} - 120 \, {\left (c^{4} d e^{6} - 4 \, b c^{3} e^{7}\right )} x^{7} + 140 \, {\left (c^{4} d^{2} e^{5} - 4 \, b c^{3} d e^{6} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{6} - 168 \, {\left (c^{4} d^{3} e^{4} - 4 \, b c^{3} d^{2} e^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{5} + 210 \, {\left (c^{4} d^{4} e^{3} - 4 \, b c^{3} d^{3} e^{4} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{4} - 280 \, {\left (c^{4} d^{5} e^{2} - 4 \, b c^{3} d^{4} e^{3} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e - 4 \, b c^{3} d^{5} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x^{2} - 840 \, {\left (c^{4} d^{7} - 4 \, b c^{3} d^{6} e - 4 \, a^{3} b e^{7} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} x}{840 \, e^{8}} + \frac {{\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} + a^{4} e^{8} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 4 \, {\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}\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),x, algorithm="maxima")

[Out]

1/840*(105*c^4*e^7*x^8 - 120*(c^4*d*e^6 - 4*b*c^3*e^7)*x^7 + 140*(c^4*d^2*e^5 - 4*b*c^3*d*e^6 + 2*(3*b^2*c^2 +
2*a*c^3)*e^7)*x^6 - 168*(c^4*d^3*e^4 - 4*b*c^3*d^2*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 4*(b^3*c + 3*a*b*c^2
)*e^7)*x^5 + 210*(c^4*d^4*e^3 - 4*b*c^3*d^3*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 4*(b^3*c + 3*a*b*c^2)*d*e^
6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^4 - 280*(c^4*d^5*e^2 - 4*b*c^3*d^4*e^3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3
*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^2*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 4*(a*b^3 + 3*a^2*b*c)*e^7)*x^3 +
420*(c^4*d^6*e - 4*b*c^3*d^5*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^3*e^4 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*e^7)*x^2 - 840*(c^4*d^7
- 4*b*c^3*d^6*e - 4*a^3*b*e^7 + 2*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a
*b^2*c + 6*a^2*c^2)*d^3*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^6)*x)/e^8 + (c^4*d^8
- 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*(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 - 4*(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)*log
(e*x + d)/e^9

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mupad [B]  time = 0.71, size = 870, normalized size = 2.03 \[ x^4\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{4\,e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{e}\right )}{5\,e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{5\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{3\,e}-\frac {4\,a\,b\,\left (b^2+3\,a\,c\right )}{3\,e}\right )+x^7\,\left (\frac {4\,b\,c^3}{7\,e}-\frac {c^4\,d}{7\,e^2}\right )-x\,\left (\frac {d\,\left (\frac {4\,c\,a^3+6\,a^2\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{e}-\frac {4\,a\,b\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{e}-\frac {4\,a^3\,b}{e}\right )+x^2\,\left (\frac {4\,c\,a^3+6\,a^2\,b^2}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{e}-\frac {4\,a\,b\,\left (b^2+3\,a\,c\right )}{e}\right )}{2\,e}\right )+x^6\,\left (\frac {6\,b^2\,c^2+4\,a\,c^3}{6\,e}-\frac {d\,\left (\frac {4\,b\,c^3}{e}-\frac {c^4\,d}{e^2}\right )}{6\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (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\right )}{e^9}+\frac {c^4\,x^8}{8\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x),x)

[Out]

x^4*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(4*e) + (d*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/
e))/e - (4*b*c*(3*a*c + b^2))/e))/(4*e)) - x^5*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/
e))/(5*e) - (4*b*c*(3*a*c + b^2))/(5*e)) - x^3*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((4*a*c^3 + 6*b^
2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*b*c*(3*a*c + b^2))/e))/e))/(3*e) - (4*a*b*(3*a*c + b^2))
/(3*e)) + x^7*((4*b*c^3)/(7*e) - (c^4*d)/(7*e^2)) - x*((d*((4*a^3*c + 6*a^2*b^2)/e + (d*((d*((b^4 + 6*a^2*c^2
+ 12*a*b^2*c)/e + (d*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*b*c*(3*a*c + b^
2))/e))/e))/e - (4*a*b*(3*a*c + b^2))/e))/e))/e - (4*a^3*b)/e) + x^2*((4*a^3*c + 6*a^2*b^2)/(2*e) + (d*((d*((b
^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((4*a*c^3 + 6*b^2*c^2)/e - (d*((4*b*c^3)/e - (c^4*d)/e^2))/e))/e - (4*
b*c*(3*a*c + b^2))/e))/e))/e - (4*a*b*(3*a*c + b^2))/e))/(2*e)) + x^6*((4*a*c^3 + 6*b^2*c^2)/(6*e) - (d*((4*b*
c^3)/e - (c^4*d)/e^2))/(6*e)) + (log(d + e*x)*(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^9 + (c^4*x^8)/(8*e)

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sympy [A]  time = 1.54, size = 808, normalized size = 1.89 \[ \frac {c^{4} x^{8}}{8 e} + x^{7} \left (\frac {4 b c^{3}}{7 e} - \frac {c^{4} d}{7 e^{2}}\right ) + x^{6} \left (\frac {2 a c^{3}}{3 e} + \frac {b^{2} c^{2}}{e} - \frac {2 b c^{3} d}{3 e^{2}} + \frac {c^{4} d^{2}}{6 e^{3}}\right ) + x^{5} \left (\frac {12 a b c^{2}}{5 e} - \frac {4 a c^{3} d}{5 e^{2}} + \frac {4 b^{3} c}{5 e} - \frac {6 b^{2} c^{2} d}{5 e^{2}} + \frac {4 b c^{3} d^{2}}{5 e^{3}} - \frac {c^{4} d^{3}}{5 e^{4}}\right ) + x^{4} \left (\frac {3 a^{2} c^{2}}{2 e} + \frac {3 a b^{2} c}{e} - \frac {3 a b c^{2} d}{e^{2}} + \frac {a c^{3} d^{2}}{e^{3}} + \frac {b^{4}}{4 e} - \frac {b^{3} c d}{e^{2}} + \frac {3 b^{2} c^{2} d^{2}}{2 e^{3}} - \frac {b c^{3} d^{3}}{e^{4}} + \frac {c^{4} d^{4}}{4 e^{5}}\right ) + x^{3} \left (\frac {4 a^{2} b c}{e} - \frac {2 a^{2} c^{2} d}{e^{2}} + \frac {4 a b^{3}}{3 e} - \frac {4 a b^{2} c d}{e^{2}} + \frac {4 a b c^{2} d^{2}}{e^{3}} - \frac {4 a c^{3} d^{3}}{3 e^{4}} - \frac {b^{4} d}{3 e^{2}} + \frac {4 b^{3} c d^{2}}{3 e^{3}} - \frac {2 b^{2} c^{2} d^{3}}{e^{4}} + \frac {4 b c^{3} d^{4}}{3 e^{5}} - \frac {c^{4} d^{5}}{3 e^{6}}\right ) + x^{2} \left (\frac {2 a^{3} c}{e} + \frac {3 a^{2} b^{2}}{e} - \frac {6 a^{2} b c d}{e^{2}} + \frac {3 a^{2} c^{2} d^{2}}{e^{3}} - \frac {2 a b^{3} d}{e^{2}} + \frac {6 a b^{2} c d^{2}}{e^{3}} - \frac {6 a b c^{2} d^{3}}{e^{4}} + \frac {2 a c^{3} d^{4}}{e^{5}} + \frac {b^{4} d^{2}}{2 e^{3}} - \frac {2 b^{3} c d^{3}}{e^{4}} + \frac {3 b^{2} c^{2} d^{4}}{e^{5}} - \frac {2 b c^{3} d^{5}}{e^{6}} + \frac {c^{4} d^{6}}{2 e^{7}}\right ) + x \left (\frac {4 a^{3} b}{e} - \frac {4 a^{3} c d}{e^{2}} - \frac {6 a^{2} b^{2} d}{e^{2}} + \frac {12 a^{2} b c d^{2}}{e^{3}} - \frac {6 a^{2} c^{2} d^{3}}{e^{4}} + \frac {4 a b^{3} d^{2}}{e^{3}} - \frac {12 a b^{2} c d^{3}}{e^{4}} + \frac {12 a b c^{2} d^{4}}{e^{5}} - \frac {4 a c^{3} d^{5}}{e^{6}} - \frac {b^{4} d^{3}}{e^{4}} + \frac {4 b^{3} c d^{4}}{e^{5}} - \frac {6 b^{2} c^{2} d^{5}}{e^{6}} + \frac {4 b c^{3} d^{6}}{e^{7}} - \frac {c^{4} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{2} - b d e + c d^{2}\right )^{4} \log {\left (d + e x \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),x)

[Out]

c**4*x**8/(8*e) + x**7*(4*b*c**3/(7*e) - c**4*d/(7*e**2)) + x**6*(2*a*c**3/(3*e) + b**2*c**2/e - 2*b*c**3*d/(3
*e**2) + c**4*d**2/(6*e**3)) + x**5*(12*a*b*c**2/(5*e) - 4*a*c**3*d/(5*e**2) + 4*b**3*c/(5*e) - 6*b**2*c**2*d/
(5*e**2) + 4*b*c**3*d**2/(5*e**3) - c**4*d**3/(5*e**4)) + x**4*(3*a**2*c**2/(2*e) + 3*a*b**2*c/e - 3*a*b*c**2*
d/e**2 + a*c**3*d**2/e**3 + b**4/(4*e) - b**3*c*d/e**2 + 3*b**2*c**2*d**2/(2*e**3) - b*c**3*d**3/e**4 + c**4*d
**4/(4*e**5)) + x**3*(4*a**2*b*c/e - 2*a**2*c**2*d/e**2 + 4*a*b**3/(3*e) - 4*a*b**2*c*d/e**2 + 4*a*b*c**2*d**2
/e**3 - 4*a*c**3*d**3/(3*e**4) - b**4*d/(3*e**2) + 4*b**3*c*d**2/(3*e**3) - 2*b**2*c**2*d**3/e**4 + 4*b*c**3*d
**4/(3*e**5) - c**4*d**5/(3*e**6)) + x**2*(2*a**3*c/e + 3*a**2*b**2/e - 6*a**2*b*c*d/e**2 + 3*a**2*c**2*d**2/e
**3 - 2*a*b**3*d/e**2 + 6*a*b**2*c*d**2/e**3 - 6*a*b*c**2*d**3/e**4 + 2*a*c**3*d**4/e**5 + b**4*d**2/(2*e**3)
- 2*b**3*c*d**3/e**4 + 3*b**2*c**2*d**4/e**5 - 2*b*c**3*d**5/e**6 + c**4*d**6/(2*e**7)) + x*(4*a**3*b/e - 4*a*
*3*c*d/e**2 - 6*a**2*b**2*d/e**2 + 12*a**2*b*c*d**2/e**3 - 6*a**2*c**2*d**3/e**4 + 4*a*b**3*d**2/e**3 - 12*a*b
**2*c*d**3/e**4 + 12*a*b*c**2*d**4/e**5 - 4*a*c**3*d**5/e**6 - b**4*d**3/e**4 + 4*b**3*c*d**4/e**5 - 6*b**2*c*
*2*d**5/e**6 + 4*b*c**3*d**6/e**7 - c**4*d**7/e**8) + (a*e**2 - b*d*e + c*d**2)**4*log(d + e*x)/e**9

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