3.16.22 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^4} \, dx\) [1522]

Optimal. Leaf size=396 \[ -\frac {c \left (40 c^3 d^3-5 b^3 e^3-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)\right ) x}{e^7}+\frac {c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2}{2 e^6}-\frac {c^3 (8 c d-7 b e) x^3}{3 e^5}+\frac {c^4 x^4}{2 e^4}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^8 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^8 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^8} \]

[Out]

-c*(40*c^3*d^3-5*b^3*e^3-2*c^2*d*e*(-12*a*e+35*b*d)+3*b*c*e^2*(-5*a*e+12*b*d))*x/e^7+1/2*c^2*(6*a*c*e^2+9*b^2*
e^2-28*b*c*d*e+20*c^2*d^2)*x^2/e^6-1/3*c^3*(-7*b*e+8*c*d)*x^3/e^5+1/2*c^4*x^4/e^4+1/3*(-b*e+2*c*d)*(a*e^2-b*d*
e+c*d^2)^3/e^8/(e*x+d)^3-1/2*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^8/(e*x+d)^2+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^8/(e*x+d)+(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))*ln(e*x+d)/e^8

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Rubi [A]
time = 0.36, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785} \begin {gather*} \frac {\log (d+e x) \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 )}{e^8}-\frac {c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac {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 )}{e^8 (d+e x)}-\frac {\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 )}{2 e^8 (d+e x)^2}+\frac {c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac {c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac {c^4 x^4}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

-((c*(40*c^3*d^3 - 5*b^3*e^3 - 2*c^2*d*e*(35*b*d - 12*a*e) + 3*b*c*e^2*(12*b*d - 5*a*e))*x)/e^7) + (c^2*(20*c^
2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c*e^2)*x^2)/(2*e^6) - (c^3*(8*c*d - 7*b*e)*x^3)/(3*e^5) + (c^4*x^4)/(2*e^
4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^3) - ((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)))/(2*e^8*(d + e*x)^2) + (3*(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^8*(d + e*x)) + ((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))*Log[d + e*x])/e^8

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {c \left (-40 c^3 d^3+5 b^3 e^3+2 c^2 d e (35 b d-12 a e)-3 b c e^2 (12 b d-5 a e)\right )}{e^7}+\frac {c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x}{e^6}-\frac {c^3 (8 c d-7 b e) x^2}{e^5}+\frac {2 c^4 x^3}{e^4}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^4}+\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 )}{e^7 (d+e x)^3}+\frac {3 (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^7 (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^7 (d+e x)}\right ) \, dx\\ &=-\frac {c \left (40 c^3 d^3-5 b^3 e^3-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)\right ) x}{e^7}+\frac {c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2}{2 e^6}-\frac {c^3 (8 c d-7 b e) x^3}{3 e^5}+\frac {c^4 x^4}{2 e^4}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\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 )}{2 e^8 (d+e x)^2}+\frac {3 (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^8 (d+e x)}+\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 ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 404, normalized size = 1.02 \begin {gather*} \frac {-6 c e \left (40 c^3 d^3-5 b^3 e^3+3 b c e^2 (12 b d-5 a e)+2 c^2 d e (-35 b d+12 a e)\right ) x+3 c^2 e^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2-2 c^3 e^3 (8 c d-7 b e) x^3+3 c^4 e^4 x^4+\frac {2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {3 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {18 (2 c d-b e) \left (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 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{d+e x}+6 \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 ) \log (d+e x)}{6 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

(-6*c*e*(40*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(12*b*d - 5*a*e) + 2*c^2*d*e*(-35*b*d + 12*a*e))*x + 3*c^2*e^2*(20
*c^2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c*e^2)*x^2 - 2*c^3*e^3*(8*c*d - 7*b*e)*x^3 + 3*c^4*e^4*x^4 + (2*(2*c*d
- b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (3*(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)^2 + (18*(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) + 6*(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))*Log[d + e*x])/(6
*e^8)

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Maple [A]
time = 0.84, size = 725, normalized size = 1.83 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

c/e^7*(1/2*c^3*x^4*e^3+7/3*b*c^2*e^3*x^3-8/3*c^3*d*e^2*x^3+3*a*c^2*e^3*x^2+9/2*b^2*c*e^3*x^2-14*b*c^2*d*e^2*x^
2+10*c^3*d^2*e*x^2+15*a*b*c*e^3*x-24*d*e^2*c^2*a*x+5*b^3*e^3*x-36*b^2*d*e^2*c*x+70*b*c^2*d^2*e*x-40*c^3*d^3*x)
-(9*a^2*b*c*e^5-18*a^2*c^2*d*e^4+3*a*b^3*e^5-36*a*b^2*c*d*e^4+90*a*b*c^2*d^2*e^3-60*a*c^3*d^3*e^2-3*b^4*d*e^4+
30*b^3*c*d^2*e^3-90*b^2*c^2*d^3*e^2+105*b*c^3*d^4*e-42*c^4*d^5)/e^8/(e*x+d)+1/e^8*(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)*ln(e
*x+d)-1/2/e^8*(2*a^3*c*e^6+3*a^2*b^2*e^6-18*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4-6*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4-
60*a*b*c^2*d^3*e^3+30*a*c^3*d^4*e^2+3*b^4*d^2*e^4-20*b^3*c*d^3*e^3+45*b^2*c^2*d^4*e^2-42*b*c^3*d^5*e+14*c^4*d^
6)/(e*x+d)^2-1/3*(a^3*b*e^7-2*a^3*c*d*e^6-3*a^2*b^2*d*e^6+9*a^2*b*c*d^2*e^5-6*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5-
12*a*b^2*c*d^3*e^4+15*a*b*c^2*d^4*e^3-6*a*c^3*d^5*e^2-b^4*d^3*e^4+5*b^3*c*d^4*e^3-9*b^2*c^2*d^5*e^2+7*b*c^3*d^
6*e-2*c^4*d^7)/e^8/(e*x+d)^3

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Maxima [A]
time = 0.31, size = 682, normalized size = 1.72 \begin {gather*} {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} e^{\left (-8\right )} \log \left (x e + d\right ) + \frac {1}{6} \, {\left (3 \, c^{4} x^{4} e^{3} - 2 \, {\left (8 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (20 \, c^{4} d^{2} e - 28 \, b c^{3} d e^{2} + 9 \, b^{2} c^{2} e^{3} + 6 \, a c^{3} e^{3}\right )} x^{2} - 6 \, {\left (40 \, c^{4} d^{3} - 70 \, b c^{3} d^{2} e - 5 \, b^{3} c e^{3} - 15 \, a b c^{2} e^{3} + 12 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} x\right )} e^{\left (-7\right )} + \frac {214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e + 141 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{5} - 130 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{4} - 2 \, a^{3} b e^{7} + 11 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{3} - 6 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{2} + 18 \, {\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} - a b^{3} e^{7} - 3 \, a^{2} b c e^{7} + 10 \, {\left (3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} d^{3} - 10 \, {\left (b^{3} c e^{5} + 3 \, a b c^{2} e^{5}\right )} d^{2} + {\left (b^{4} e^{6} + 12 \, a b^{2} c e^{6} + 6 \, a^{2} c^{2} e^{6}\right )} d\right )} x^{2} - {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d + 3 \, {\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 105 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{4} - 3 \, a^{2} b^{2} e^{7} - 2 \, a^{3} c e^{7} - 100 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{3} + 9 \, {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d^{2} - 6 \, {\left (a b^{3} e^{6} + 3 \, a^{2} b c e^{6}\right )} d\right )} x}{6 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

(70*c^4*d^4 - 140*b*c^3*d^3*e + b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4 + 30*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^
2 - 20*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d)*e^(-8)*log(x*e + d) + 1/6*(3*c^4*x^4*e^3 - 2*(8*c^4*d*e^2 - 7*b*c^3*e^3)
*x^3 + 3*(20*c^4*d^2*e - 28*b*c^3*d*e^2 + 9*b^2*c^2*e^3 + 6*a*c^3*e^3)*x^2 - 6*(40*c^4*d^3 - 70*b*c^3*d^2*e -
5*b^3*c*e^3 - 15*a*b*c^2*e^3 + 12*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d)*x)*e^(-7) + 1/6*(214*c^4*d^7 - 518*b*c^3*d^
6*e + 141*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^5 - 130*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^4 - 2*a^3*b*e^7 + 11*(b^4*e^4
+ 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^3 - 6*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^2 + 18*(14*c^4*d^5*e^2 - 35*b*c^3*d^4*
e^3 - a*b^3*e^7 - 3*a^2*b*c*e^7 + 10*(3*b^2*c^2*e^4 + 2*a*c^3*e^4)*d^3 - 10*(b^3*c*e^5 + 3*a*b*c^2*e^5)*d^2 +
(b^4*e^6 + 12*a*b^2*c*e^6 + 6*a^2*c^2*e^6)*d)*x^2 - (3*a^2*b^2*e^6 + 2*a^3*c*e^6)*d + 3*(154*c^4*d^6*e - 378*b
*c^3*d^5*e^2 + 105*(3*b^2*c^2*e^3 + 2*a*c^3*e^3)*d^4 - 3*a^2*b^2*e^7 - 2*a^3*c*e^7 - 100*(b^3*c*e^4 + 3*a*b*c^
2*e^4)*d^3 + 9*(b^4*e^5 + 12*a*b^2*c*e^5 + 6*a^2*c^2*e^5)*d^2 - 6*(a*b^3*e^6 + 3*a^2*b*c*e^6)*d)*x)/(x^3*e^11
+ 3*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (387) = 774\).
time = 2.20, size = 994, normalized size = 2.51 \begin {gather*} \frac {214 \, c^{4} d^{7} + {\left (3 \, c^{4} x^{7} + 14 \, b c^{3} x^{6} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{5} + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{4} - 2 \, a^{3} b - 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} - 3 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x\right )} e^{7} - {\left (7 \, c^{4} d x^{6} + 42 \, b c^{3} d x^{5} + 45 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{4} - 90 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{3} - 18 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{2} + 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} e^{6} + 3 \, {\left (7 \, c^{4} d^{2} x^{5} + 70 \, b c^{3} d^{2} x^{4} - 63 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{3} - 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{2} + 9 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2}\right )} e^{5} - {\left (105 \, c^{4} d^{3} x^{4} - 1022 \, b c^{3} d^{3} x^{3} + 27 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{2} + 270 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x - 11 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} - {\left (556 \, c^{4} d^{4} x^{3} - 546 \, b c^{3} d^{4} x^{2} - 243 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x + 130 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4}\right )} e^{3} - 3 \, {\left (136 \, c^{4} d^{5} x^{2} + 238 \, b c^{3} d^{5} x - 47 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5}\right )} e^{2} + 74 \, {\left (3 \, c^{4} d^{6} x - 7 \, b c^{3} d^{6}\right )} e + 6 \, {\left (70 \, c^{4} d^{7} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{3} e^{7} - {\left (20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{3} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{2}\right )} e^{6} + 3 \, {\left (10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{3} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x\right )} e^{5} - {\left (140 \, b c^{3} d^{3} x^{3} - 90 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{2} + 60 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} + 10 \, {\left (7 \, c^{4} d^{4} x^{3} - 42 \, b c^{3} d^{4} x^{2} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4}\right )} e^{3} + 30 \, {\left (7 \, c^{4} d^{5} x^{2} - 14 \, b c^{3} d^{5} x + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5}\right )} e^{2} + 70 \, {\left (3 \, c^{4} d^{6} x - 2 \, b c^{3} d^{6}\right )} e\right )} \log \left (x e + d\right )}{6 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(214*c^4*d^7 + (3*c^4*x^7 + 14*b*c^3*x^6 + 9*(3*b^2*c^2 + 2*a*c^3)*x^5 + 30*(b^3*c + 3*a*b*c^2)*x^4 - 2*a^
3*b - 18*(a*b^3 + 3*a^2*b*c)*x^2 - 3*(3*a^2*b^2 + 2*a^3*c)*x)*e^7 - (7*c^4*d*x^6 + 42*b*c^3*d*x^5 + 45*(3*b^2*
c^2 + 2*a*c^3)*d*x^4 - 90*(b^3*c + 3*a*b*c^2)*d*x^3 - 18*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^2 + 18*(a*b^3 + 3*
a^2*b*c)*d*x + (3*a^2*b^2 + 2*a^3*c)*d)*e^6 + 3*(7*c^4*d^2*x^5 + 70*b*c^3*d^2*x^4 - 63*(3*b^2*c^2 + 2*a*c^3)*d
^2*x^3 - 30*(b^3*c + 3*a*b*c^2)*d^2*x^2 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x - 2*(a*b^3 + 3*a^2*b*c)*d^2)*
e^5 - (105*c^4*d^3*x^4 - 1022*b*c^3*d^3*x^3 + 27*(3*b^2*c^2 + 2*a*c^3)*d^3*x^2 + 270*(b^3*c + 3*a*b*c^2)*d^3*x
- 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3)*e^4 - (556*c^4*d^4*x^3 - 546*b*c^3*d^4*x^2 - 243*(3*b^2*c^2 + 2*a*c^
3)*d^4*x + 130*(b^3*c + 3*a*b*c^2)*d^4)*e^3 - 3*(136*c^4*d^5*x^2 + 238*b*c^3*d^5*x - 47*(3*b^2*c^2 + 2*a*c^3)*
d^5)*e^2 + 74*(3*c^4*d^6*x - 7*b*c^3*d^6)*e + 6*(70*c^4*d^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^3*e^7 - (20*(b^
3*c + 3*a*b*c^2)*d*x^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^2)*e^6 + 3*(10*(3*b^2*c^2 + 2*a*c^3)*d^2*x^3 - 2
0*(b^3*c + 3*a*b*c^2)*d^2*x^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x)*e^5 - (140*b*c^3*d^3*x^3 - 90*(3*b^2*c^2
+ 2*a*c^3)*d^3*x^2 + 60*(b^3*c + 3*a*b*c^2)*d^3*x - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3)*e^4 + 10*(7*c^4*d^4*x
^3 - 42*b*c^3*d^4*x^2 + 9*(3*b^2*c^2 + 2*a*c^3)*d^4*x - 2*(b^3*c + 3*a*b*c^2)*d^4)*e^3 + 30*(7*c^4*d^5*x^2 - 1
4*b*c^3*d^5*x + (3*b^2*c^2 + 2*a*c^3)*d^5)*e^2 + 70*(3*c^4*d^6*x - 2*b*c^3*d^6)*e)*log(x*e + d))/(x^3*e^11 + 3
*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Sympy [F(-1)] Timed out
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((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.24, size = 678, normalized size = 1.71 \begin {gather*} {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (3 \, c^{4} x^{4} e^{12} - 16 \, c^{4} d x^{3} e^{11} + 60 \, c^{4} d^{2} x^{2} e^{10} - 240 \, c^{4} d^{3} x e^{9} + 14 \, b c^{3} x^{3} e^{12} - 84 \, b c^{3} d x^{2} e^{11} + 420 \, b c^{3} d^{2} x e^{10} + 27 \, b^{2} c^{2} x^{2} e^{12} + 18 \, a c^{3} x^{2} e^{12} - 216 \, b^{2} c^{2} d x e^{11} - 144 \, a c^{3} d x e^{11} + 30 \, b^{3} c x e^{12} + 90 \, a b c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e + 423 \, b^{2} c^{2} d^{5} e^{2} + 282 \, a c^{3} d^{5} e^{2} - 130 \, b^{3} c d^{4} e^{3} - 390 \, a b c^{2} d^{4} e^{3} + 11 \, b^{4} d^{3} e^{4} + 132 \, a b^{2} c d^{3} e^{4} + 66 \, a^{2} c^{2} d^{3} e^{4} - 6 \, a b^{3} d^{2} e^{5} - 18 \, a^{2} b c d^{2} e^{5} - 3 \, a^{2} b^{2} d e^{6} - 2 \, a^{3} c d e^{6} - 2 \, a^{3} b e^{7} + 18 \, {\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 30 \, b^{2} c^{2} d^{3} e^{4} + 20 \, a c^{3} d^{3} e^{4} - 10 \, b^{3} c d^{2} e^{5} - 30 \, a b c^{2} d^{2} e^{5} + b^{4} d e^{6} + 12 \, a b^{2} c d e^{6} + 6 \, a^{2} c^{2} d e^{6} - a b^{3} e^{7} - 3 \, a^{2} b c e^{7}\right )} x^{2} + 3 \, {\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 315 \, b^{2} c^{2} d^{4} e^{3} + 210 \, a c^{3} d^{4} e^{3} - 100 \, b^{3} c d^{3} e^{4} - 300 \, a b c^{2} d^{3} e^{4} + 9 \, b^{4} d^{2} e^{5} + 108 \, a b^{2} c d^{2} e^{5} + 54 \, a^{2} c^{2} d^{2} e^{5} - 6 \, a b^{3} d e^{6} - 18 \, a^{2} b c d e^{6} - 3 \, a^{2} b^{2} e^{7} - 2 \, a^{3} c e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

(70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 + b^
4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*e^(-8)*log(abs(x*e + d)) + 1/6*(3*c^4*x^4*e^12 - 16*c^4*d*x^3*e^11 + 6
0*c^4*d^2*x^2*e^10 - 240*c^4*d^3*x*e^9 + 14*b*c^3*x^3*e^12 - 84*b*c^3*d*x^2*e^11 + 420*b*c^3*d^2*x*e^10 + 27*b
^2*c^2*x^2*e^12 + 18*a*c^3*x^2*e^12 - 216*b^2*c^2*d*x*e^11 - 144*a*c^3*d*x*e^11 + 30*b^3*c*x*e^12 + 90*a*b*c^2
*x*e^12)*e^(-16) + 1/6*(214*c^4*d^7 - 518*b*c^3*d^6*e + 423*b^2*c^2*d^5*e^2 + 282*a*c^3*d^5*e^2 - 130*b^3*c*d^
4*e^3 - 390*a*b*c^2*d^4*e^3 + 11*b^4*d^3*e^4 + 132*a*b^2*c*d^3*e^4 + 66*a^2*c^2*d^3*e^4 - 6*a*b^3*d^2*e^5 - 18
*a^2*b*c*d^2*e^5 - 3*a^2*b^2*d*e^6 - 2*a^3*c*d*e^6 - 2*a^3*b*e^7 + 18*(14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 30*
b^2*c^2*d^3*e^4 + 20*a*c^3*d^3*e^4 - 10*b^3*c*d^2*e^5 - 30*a*b*c^2*d^2*e^5 + b^4*d*e^6 + 12*a*b^2*c*d*e^6 + 6*
a^2*c^2*d*e^6 - a*b^3*e^7 - 3*a^2*b*c*e^7)*x^2 + 3*(154*c^4*d^6*e - 378*b*c^3*d^5*e^2 + 315*b^2*c^2*d^4*e^3 +
210*a*c^3*d^4*e^3 - 100*b^3*c*d^3*e^4 - 300*a*b*c^2*d^3*e^4 + 9*b^4*d^2*e^5 + 108*a*b^2*c*d^2*e^5 + 54*a^2*c^2
*d^2*e^5 - 6*a*b^3*d*e^6 - 18*a^2*b*c*d*e^6 - 3*a^2*b^2*e^7 - 2*a^3*c*e^7)*x)*e^(-8)/(x*e + d)^3

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Mupad [B]
time = 1.87, size = 807, normalized size = 2.04 \begin {gather*} x^3\,\left (\frac {7\,b\,c^3}{3\,e^4}-\frac {8\,c^4\,d}{3\,e^5}\right )-\frac {x\,\left (a^3\,c\,e^6+\frac {3\,a^2\,b^2\,e^6}{2}+9\,a^2\,b\,c\,d\,e^5-27\,a^2\,c^2\,d^2\,e^4+3\,a\,b^3\,d\,e^5-54\,a\,b^2\,c\,d^2\,e^4+150\,a\,b\,c^2\,d^3\,e^3-105\,a\,c^3\,d^4\,e^2-\frac {9\,b^4\,d^2\,e^4}{2}+50\,b^3\,c\,d^3\,e^3-\frac {315\,b^2\,c^2\,d^4\,e^2}{2}+189\,b\,c^3\,d^5\,e-77\,c^4\,d^6\right )-x^2\,\left (-9\,a^2\,b\,c\,e^6+18\,a^2\,c^2\,d\,e^5-3\,a\,b^3\,e^6+36\,a\,b^2\,c\,d\,e^5-90\,a\,b\,c^2\,d^2\,e^4+60\,a\,c^3\,d^3\,e^3+3\,b^4\,d\,e^5-30\,b^3\,c\,d^2\,e^4+90\,b^2\,c^2\,d^3\,e^3-105\,b\,c^3\,d^4\,e^2+42\,c^4\,d^5\,e\right )+\frac {2\,a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6+18\,a^2\,b\,c\,d^2\,e^5-66\,a^2\,c^2\,d^3\,e^4+6\,a\,b^3\,d^2\,e^5-132\,a\,b^2\,c\,d^3\,e^4+390\,a\,b\,c^2\,d^4\,e^3-282\,a\,c^3\,d^5\,e^2-11\,b^4\,d^3\,e^4+130\,b^3\,c\,d^4\,e^3-423\,b^2\,c^2\,d^5\,e^2+518\,b\,c^3\,d^6\,e-214\,c^4\,d^7}{6\,e}}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}-x^2\,\left (\frac {2\,d\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{2\,e^4}+\frac {6\,c^4\,d^2}{e^6}\right )-x\,\left (\frac {8\,c^4\,d^3}{e^7}+\frac {6\,d^2\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^4}+\frac {12\,c^4\,d^2}{e^6}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )+\frac {c^4\,x^4}{2\,e^4}+\frac {\ln \left (d+e\,x\right )\,\left (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\right )}{e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x)

[Out]

x^3*((7*b*c^3)/(3*e^4) - (8*c^4*d)/(3*e^5)) - (x*(a^3*c*e^6 - 77*c^4*d^6 + (3*a^2*b^2*e^6)/2 - (9*b^4*d^2*e^4)
/2 - 105*a*c^3*d^4*e^2 + 50*b^3*c*d^3*e^3 - 27*a^2*c^2*d^2*e^4 - (315*b^2*c^2*d^4*e^2)/2 + 3*a*b^3*d*e^5 + 189
*b*c^3*d^5*e + 9*a^2*b*c*d*e^5 + 150*a*b*c^2*d^3*e^3 - 54*a*b^2*c*d^2*e^4) - x^2*(3*b^4*d*e^5 - 3*a*b^3*e^6 +
42*c^4*d^5*e + 60*a*c^3*d^3*e^3 + 18*a^2*c^2*d*e^5 - 105*b*c^3*d^4*e^2 - 30*b^3*c*d^2*e^4 + 90*b^2*c^2*d^3*e^3
- 9*a^2*b*c*e^6 + 36*a*b^2*c*d*e^5 - 90*a*b*c^2*d^2*e^4) + (2*a^3*b*e^7 - 214*c^4*d^7 - 11*b^4*d^3*e^4 + 6*a*
b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 - 282*a*c^3*d^5*e^2 + 130*b^3*c*d^4*e^3 - 66*a^2*c^2*d^3*e^4 - 423*b^2*c^2*d^5*e
^2 + 2*a^3*c*d*e^6 + 518*b*c^3*d^6*e + 390*a*b*c^2*d^4*e^3 - 132*a*b^2*c*d^3*e^4 + 18*a^2*b*c*d^2*e^5)/(6*e))/
(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) - x^2*((2*d*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e - (6*a*c^3 + 9
*b^2*c^2)/(2*e^4) + (6*c^4*d^2)/e^6) - x*((8*c^4*d^3)/e^7 + (6*d^2*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e^2 - (4*d
*((4*d*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e - (6*a*c^3 + 9*b^2*c^2)/e^4 + (12*c^4*d^2)/e^6))/e - (5*b*c*(3*a*c +
b^2))/e^4) + (c^4*x^4)/(2*e^4) + (log(d + e*x)*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*
b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/e^8

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