3.14.29 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=390 \[ \frac {x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)-2 c^3 d^2 e (35 b d-18 a e)+b^4 e^4+30 c^4 d^4\right )}{e^7}-\frac {c x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )}{2 e^6}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)}-\frac {3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {c^2 x^3 \left (-c e (7 b d-2 a e)+3 b^2 e^2+4 c^2 d^2\right )}{e^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^3 x^4 (6 c d-7 b e)}{4 e^4}+\frac {2 c^4 x^5}{5 e^3} \]

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Rubi [A]  time = 0.51, antiderivative size = 390, 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 = {771} \begin {gather*} \frac {x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)-2 c^3 d^2 e (35 b d-18 a e)+b^4 e^4+30 c^4 d^4\right )}{e^7}+\frac {c^2 x^3 \left (-c e (7 b d-2 a e)+3 b^2 e^2+4 c^2 d^2\right )}{e^5}-\frac {c x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )}{2 e^6}-\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 )}{e^8 (d+e x)}-\frac {3 (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^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^3 x^4 (6 c d-7 b e)}{4 e^4}+\frac {2 c^4 x^5}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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)^3} \, dx &=\int \left (\frac {30 c^4 d^4+b^4 e^4-2 c^3 d^2 e (35 b d-18 a e)-3 b^2 c e^3 (5 b d-4 a e)+3 c^2 e^2 \left (18 b^2 d^2-15 a b d e+2 a^2 e^2\right )}{e^7}+\frac {c \left (-20 c^3 d^3+5 b^3 e^3-3 b c e^2 (9 b d-5 a e)+6 c^2 d e (7 b d-3 a e)\right ) x}{e^6}+\frac {3 c^2 \left (4 c^2 d^2+3 b^2 e^2-c e (7 b d-2 a e)\right ) x^2}{e^5}-\frac {c^3 (6 c d-7 b e) x^3}{e^4}+\frac {2 c^4 x^4}{e^3}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (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 )}{e^7 (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+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {\left (30 c^4 d^4+b^4 e^4-2 c^3 d^2 e (35 b d-18 a e)-3 b^2 c e^3 (5 b d-4 a e)+3 c^2 e^2 \left (18 b^2 d^2-15 a b d e+2 a^2 e^2\right )\right ) x}{e^7}-\frac {c \left (20 c^3 d^3-5 b^3 e^3+3 b c e^2 (9 b d-5 a e)-6 c^2 d e (7 b d-3 a e)\right ) x^2}{2 e^6}+\frac {c^2 \left (4 c^2 d^2+3 b^2 e^2-c e (7 b d-2 a e)\right ) x^3}{e^5}-\frac {c^3 (6 c d-7 b e) x^4}{4 e^4}+\frac {2 c^4 x^5}{5 e^3}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (d+e x)^2}-\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^8 (d+e x)}-\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 ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 403, normalized size = 1.03 \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 )+20 e x \left (3 c^2 e^2 \left (2 a^2 e^2-15 a b d e+18 b^2 d^2\right )-3 b^2 c e^3 (5 b d-4 a e)+2 c^3 d^2 e (18 a e-35 b d)+b^4 e^4+30 c^4 d^4\right )-10 c e^2 x^2 \left (-6 c^2 d e (7 b d-3 a e)+3 b c e^2 (9 b d-5 a e)-5 b^3 e^3+20 c^3 d^3\right )-\frac {20 \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}+20 c^2 e^3 x^3 \left (c e (2 a e-7 b d)+3 b^2 e^2+4 c^2 d^2\right )+\frac {10 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}-5 c^3 e^4 x^4 (6 c d-7 b e)+8 c^4 e^5 x^5}{20 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(30*c^4*d^4 + b^4*e^4 - 3*b^2*c*e^3*(5*b*d - 4*a*e) + 2*c^3*d^2*e*(-35*b*d + 18*a*e) + 3*c^2*e^2*(18*b^2
*d^2 - 15*a*b*d*e + 2*a^2*e^2))*x - 10*c*e^2*(20*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(9*b*d - 5*a*e) - 6*c^2*d*e*(
7*b*d - 3*a*e))*x^2 + 20*c^2*e^3*(4*c^2*d^2 + 3*b^2*e^2 + c*e*(-7*b*d + 2*a*e))*x^3 - 5*c^3*e^4*(6*c*d - 7*b*e
)*x^4 + 8*c^4*e^5*x^5 + (10*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (20*(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])/(20*e^8
)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 990, normalized size = 2.54 \begin {gather*} \frac {8 \, c^{4} e^{7} x^{7} - 260 \, c^{4} d^{7} + 770 \, b c^{3} d^{6} e - 10 \, a^{3} b e^{7} - 270 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 350 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 50 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 90 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 7 \, {\left (2 \, c^{4} d e^{6} - 5 \, b c^{3} e^{7}\right )} x^{6} + 2 \, {\left (14 \, c^{4} d^{2} e^{5} - 35 \, b c^{3} d e^{6} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 5 \, {\left (14 \, c^{4} d^{3} e^{4} - 35 \, b c^{3} d^{2} e^{5} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 20 \, {\left (14 \, c^{4} d^{4} e^{3} - 35 \, b c^{3} d^{3} e^{4} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 10 \, {\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^{3} + 10 \, {\left (100 \, c^{4} d^{5} e^{2} - 238 \, b c^{3} d^{4} e^{3} + 63 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 55 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6}\right )} x^{2} + 20 \, {\left (16 \, c^{4} d^{6} e - 28 \, b c^{3} d^{5} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} - 2 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x - 60 \, {\left (14 \, c^{4} d^{7} - 35 \, b c^{3} d^{6} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 10 \, {\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} - {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 10 \, {\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} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 2 \, {\left (14 \, c^{4} d^{6} e - 35 \, b c^{3} d^{5} e^{2} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 10 \, {\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} - {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/20*(8*c^4*e^7*x^7 - 260*c^4*d^7 + 770*b*c^3*d^6*e - 10*a^3*b*e^7 - 270*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 + 350*(
b^3*c + 3*a*b*c^2)*d^4*e^3 - 50*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 90*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - 10*(
3*a^2*b^2 + 2*a^3*c)*d*e^6 - 7*(2*c^4*d*e^6 - 5*b*c^3*e^7)*x^6 + 2*(14*c^4*d^2*e^5 - 35*b*c^3*d*e^6 + 10*(3*b^
2*c^2 + 2*a*c^3)*e^7)*x^5 - 5*(14*c^4*d^3*e^4 - 35*b*c^3*d^2*e^5 + 10*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 10*(b^3*c
+ 3*a*b*c^2)*e^7)*x^4 + 20*(14*c^4*d^4*e^3 - 35*b*c^3*d^3*e^4 + 10*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 10*(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^3 + 10*(100*c^4*d^5*e^2 - 238*b*c^3*d^4*e^3 + 63*(3*
b^2*c^2 + 2*a*c^3)*d^3*e^4 - 55*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6)*x^2 + 20
*(16*c^4*d^6*e - 28*b*c^3*d^5*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 + 5*(b^3*c + 3*a*b*c^2)*d^3*e^4 - 2*(b^4 +
12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 + 6*(a*b^3 + 3*a^2*b*c)*d*e^6 - (3*a^2*b^2 + 2*a^3*c)*e^7)*x - 60*(14*c^4*d^7
- 35*b*c^3*d^6*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 10*(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 - (a*b^3 + 3*a^2*b*c)*d^2*e^5 + (14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 10*(3*b^2*c^2 + 2*a*c^3)
*d^3*e^4 - 10*(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 - (a*b^3 + 3*a^2*b*c)*e^7)*x^
2 + 2*(14*c^4*d^6*e - 35*b*c^3*d^5*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 10*(b^3*c + 3*a*b*c^2)*d^3*e^4 + (
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - (a*b^3 + 3*a^2*b*c)*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^
2*e^8)

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giac [A]  time = 0.17, size = 694, normalized size = 1.78 \begin {gather*} -3 \, {\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 (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (8 \, c^{4} x^{5} e^{12} - 30 \, c^{4} d x^{4} e^{11} + 80 \, c^{4} d^{2} x^{3} e^{10} - 200 \, c^{4} d^{3} x^{2} e^{9} + 600 \, c^{4} d^{4} x e^{8} + 35 \, b c^{3} x^{4} e^{12} - 140 \, b c^{3} d x^{3} e^{11} + 420 \, b c^{3} d^{2} x^{2} e^{10} - 1400 \, b c^{3} d^{3} x e^{9} + 60 \, b^{2} c^{2} x^{3} e^{12} + 40 \, a c^{3} x^{3} e^{12} - 270 \, b^{2} c^{2} d x^{2} e^{11} - 180 \, a c^{3} d x^{2} e^{11} + 1080 \, b^{2} c^{2} d^{2} x e^{10} + 720 \, a c^{3} d^{2} x e^{10} + 50 \, b^{3} c x^{2} e^{12} + 150 \, a b c^{2} x^{2} e^{12} - 300 \, b^{3} c d x e^{11} - 900 \, a b c^{2} d x e^{11} + 20 \, b^{4} x e^{12} + 240 \, a b^{2} c x e^{12} + 120 \, a^{2} c^{2} x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (26 \, c^{4} d^{7} - 77 \, b c^{3} d^{6} e + 81 \, b^{2} c^{2} d^{5} e^{2} + 54 \, a c^{3} d^{5} e^{2} - 35 \, b^{3} c d^{4} e^{3} - 105 \, a b c^{2} d^{4} e^{3} + 5 \, b^{4} d^{3} e^{4} + 60 \, a b^{2} c d^{3} e^{4} + 30 \, a^{2} c^{2} d^{3} e^{4} - 9 \, a b^{3} d^{2} e^{5} - 27 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} + a^{3} b e^{7} + 2 \, {\left (14 \, c^{4} d^{6} e - 42 \, b c^{3} d^{5} e^{2} + 45 \, b^{2} c^{2} d^{4} e^{3} + 30 \, a c^{3} d^{4} e^{3} - 20 \, b^{3} c d^{3} e^{4} - 60 \, a b c^{2} d^{3} e^{4} + 3 \, b^{4} d^{2} e^{5} + 36 \, a b^{2} c d^{2} e^{5} + 18 \, 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 )}}{2 \, {\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-3*(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^(-8)*log(abs(x*e + d)) + 1/2
0*(8*c^4*x^5*e^12 - 30*c^4*d*x^4*e^11 + 80*c^4*d^2*x^3*e^10 - 200*c^4*d^3*x^2*e^9 + 600*c^4*d^4*x*e^8 + 35*b*c
^3*x^4*e^12 - 140*b*c^3*d*x^3*e^11 + 420*b*c^3*d^2*x^2*e^10 - 1400*b*c^3*d^3*x*e^9 + 60*b^2*c^2*x^3*e^12 + 40*
a*c^3*x^3*e^12 - 270*b^2*c^2*d*x^2*e^11 - 180*a*c^3*d*x^2*e^11 + 1080*b^2*c^2*d^2*x*e^10 + 720*a*c^3*d^2*x*e^1
0 + 50*b^3*c*x^2*e^12 + 150*a*b*c^2*x^2*e^12 - 300*b^3*c*d*x*e^11 - 900*a*b*c^2*d*x*e^11 + 20*b^4*x*e^12 + 240
*a*b^2*c*x*e^12 + 120*a^2*c^2*x*e^12)*e^(-15) - 1/2*(26*c^4*d^7 - 77*b*c^3*d^6*e + 81*b^2*c^2*d^5*e^2 + 54*a*c
^3*d^5*e^2 - 35*b^3*c*d^4*e^3 - 105*a*b*c^2*d^4*e^3 + 5*b^4*d^3*e^4 + 60*a*b^2*c*d^3*e^4 + 30*a^2*c^2*d^3*e^4
- 9*a*b^3*d^2*e^5 - 27*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e^6 + 2*a^3*c*d*e^6 + a^3*b*e^7 + 2*(14*c^4*d^6*e - 42*b*
c^3*d^5*e^2 + 45*b^2*c^2*d^4*e^3 + 30*a*c^3*d^4*e^3 - 20*b^3*c*d^3*e^4 - 60*a*b*c^2*d^3*e^4 + 3*b^4*d^2*e^5 +
36*a*b^2*c*d^2*e^5 + 18*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)^2

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maple [B]  time = 0.07, size = 978, normalized size = 2.51 \begin {gather*} \frac {2 c^{4} x^{5}}{5 e^{3}}+\frac {7 b \,c^{3} x^{4}}{4 e^{3}}-\frac {3 c^{4} d \,x^{4}}{2 e^{4}}+\frac {2 a \,c^{3} x^{3}}{e^{3}}+\frac {3 b^{2} c^{2} x^{3}}{e^{3}}-\frac {7 b \,c^{3} d \,x^{3}}{e^{4}}+\frac {4 c^{4} d^{2} x^{3}}{e^{5}}-\frac {a^{3} b}{2 \left (e x +d \right )^{2} e}+\frac {a^{3} c d}{\left (e x +d \right )^{2} e^{2}}+\frac {3 a^{2} b^{2} d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {9 a^{2} b c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 a^{2} c^{2} d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {3 a \,b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {6 a \,b^{2} c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {15 a b \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {15 a b \,c^{2} x^{2}}{2 e^{3}}+\frac {3 a \,c^{3} d^{5}}{\left (e x +d \right )^{2} e^{6}}-\frac {9 a \,c^{3} d \,x^{2}}{e^{4}}+\frac {b^{4} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {5 b^{3} c \,d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {5 b^{3} c \,x^{2}}{2 e^{3}}+\frac {9 b^{2} c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {27 b^{2} c^{2} d \,x^{2}}{2 e^{4}}-\frac {7 b \,c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {21 b \,c^{3} d^{2} x^{2}}{e^{5}}+\frac {c^{4} d^{7}}{\left (e x +d \right )^{2} e^{8}}-\frac {10 c^{4} d^{3} x^{2}}{e^{6}}-\frac {2 a^{3} c}{\left (e x +d \right ) e^{2}}-\frac {3 a^{2} b^{2}}{\left (e x +d \right ) e^{2}}+\frac {18 a^{2} b c d}{\left (e x +d \right ) e^{3}}+\frac {9 a^{2} b c \ln \left (e x +d \right )}{e^{3}}-\frac {18 a^{2} c^{2} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {18 a^{2} c^{2} d \ln \left (e x +d \right )}{e^{4}}+\frac {6 a^{2} c^{2} x}{e^{3}}+\frac {6 a \,b^{3} d}{\left (e x +d \right ) e^{3}}+\frac {3 a \,b^{3} \ln \left (e x +d \right )}{e^{3}}-\frac {36 a \,b^{2} c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {36 a \,b^{2} c d \ln \left (e x +d \right )}{e^{4}}+\frac {12 a \,b^{2} c x}{e^{3}}+\frac {60 a b \,c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {90 a b \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {45 a b \,c^{2} d x}{e^{4}}-\frac {30 a \,c^{3} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {60 a \,c^{3} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {36 a \,c^{3} d^{2} x}{e^{5}}-\frac {3 b^{4} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 b^{4} d \ln \left (e x +d \right )}{e^{4}}+\frac {b^{4} x}{e^{3}}+\frac {20 b^{3} c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {30 b^{3} c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {15 b^{3} c d x}{e^{4}}-\frac {45 b^{2} c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {90 b^{2} c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {54 b^{2} c^{2} d^{2} x}{e^{5}}+\frac {42 b \,c^{3} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {105 b \,c^{3} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {70 b \,c^{3} d^{3} x}{e^{6}}-\frac {14 c^{4} d^{6}}{\left (e x +d \right ) e^{8}}-\frac {42 c^{4} d^{5} \ln \left (e x +d \right )}{e^{8}}+\frac {30 c^{4} d^{4} x}{e^{7}} \end {gather*}

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)^3,x)

[Out]

6/e^4/(e*x+d)^2*a*b^2*c*d^3+60/e^5/(e*x+d)*a*b*c^2*d^3-9/2/e^3/(e*x+d)^2*d^2*a^2*b*c-15/2/e^5/(e*x+d)^2*a*b*c^
2*d^4-45/e^4*a*b*c^2*d*x+18/e^3/(e*x+d)*a^2*b*c*d-36/e^4/(e*x+d)*a*b^2*c*d^2+3/2/e^2/(e*x+d)^2*d*a^2*b^2+3/e^4
/(e*x+d)^2*a^2*c^2*d^3-3/2/e^3/(e*x+d)^2*d^2*a*b^3+3/e^6/(e*x+d)^2*a*c^3*d^5-5/2/e^5/(e*x+d)^2*b^3*c*d^4+9/2/e
^6/(e*x+d)^2*b^2*c^2*d^5+30/e^5*ln(e*x+d)*b^3*c*d^2-18/e^4/(e*x+d)*a^2*c^2*d^2+6/e^3/(e*x+d)*a*b^3*d-30/e^6/(e
*x+d)*a*c^3*d^4+20/e^5/(e*x+d)*b^3*c*d^3-45/e^6/(e*x+d)*b^2*c^2*d^4+42/e^7/(e*x+d)*b*c^3*d^5+1/e^2/(e*x+d)^2*a
^3*c*d+7/4/e^3*x^4*b*c^3+2/e^3*x^3*a*c^3-3/2/e^4*x^4*c^4*d-2/e^2/(e*x+d)*a^3*c-3/e^2/(e*x+d)*a^2*b^2-3/e^4/(e*
x+d)*b^4*d^2-14/e^8/(e*x+d)*c^4*d^6-1/2/e/(e*x+d)^2*a^3*b+1/2/e^4/(e*x+d)^2*b^4*d^3+1/e^8/(e*x+d)^2*c^4*d^7+3/
e^3*ln(e*x+d)*a*b^3-3/e^4*ln(e*x+d)*b^4*d-42/e^8*ln(e*x+d)*c^4*d^5+3/e^3*x^3*b^2*c^2+4/e^5*x^3*c^4*d^2+5/2/e^3
*x^2*b^3*c-10/e^6*x^2*c^4*d^3-36/e^4*ln(e*x+d)*a*b^2*c*d+90/e^5*ln(e*x+d)*a*b*c^2*d^2+1/e^3*b^4*x-7/e^4*x^3*b*
c^3*d+6/e^3*c^2*a^2*x+30/e^7*c^4*d^4*x-7/2/e^7/(e*x+d)^2*b*c^3*d^6+2/5*c^4*x^5/e^3+15/2/e^3*x^2*a*b*c^2-9/e^4*
x^2*a*c^3*d-27/2/e^4*x^2*b^2*c^2*d+21/e^5*x^2*b*c^3*d^2+12/e^3*a*b^2*c*x+36/e^5*a*c^3*d^2*x-15/e^4*b^3*c*d*x+5
4/e^5*b^2*c^2*d^2*x-70/e^6*b*c^3*d^3*x-90/e^6*ln(e*x+d)*b^2*c^2*d^3+105/e^7*ln(e*x+d)*b*c^3*d^4+9/e^3*ln(e*x+d
)*a^2*b*c-18/e^4*ln(e*x+d)*a^2*c^2*d-60/e^6*ln(e*x+d)*a*c^3*d^3

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maxima [A]  time = 0.69, size = 656, normalized size = 1.68 \begin {gather*} -\frac {26 \, c^{4} d^{7} - 77 \, b c^{3} d^{6} e + a^{3} b e^{7} + 27 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 35 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 5 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 9 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} + 2 \, {\left (14 \, c^{4} d^{6} e - 42 \, b c^{3} d^{5} e^{2} + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {8 \, c^{4} e^{4} x^{5} - 5 \, {\left (6 \, c^{4} d e^{3} - 7 \, b c^{3} e^{4}\right )} x^{4} + 20 \, {\left (4 \, c^{4} d^{2} e^{2} - 7 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{3} - 10 \, {\left (20 \, c^{4} d^{3} e - 42 \, b c^{3} d^{2} e^{2} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{2} + 20 \, {\left (30 \, c^{4} d^{4} - 70 \, b c^{3} d^{3} e + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 15 \, {\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}{20 \, e^{7}} - \frac {3 \, {\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^{8}} \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)^3,x, algorithm="maxima")

[Out]

-1/2*(26*c^4*d^7 - 77*b*c^3*d^6*e + a^3*b*e^7 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 35*(b^3*c + 3*a*b*c^2)*d^4*
e^3 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 9*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6 +
2*(14*c^4*d^6*e - 42*b*c^3*d^5*e^2 + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 3*(b
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2*b^2 + 2*a^3*c)*e^7)*x)/(e^10*x^2 +
2*d*e^9*x + d^2*e^8) + 1/20*(8*c^4*e^4*x^5 - 5*(6*c^4*d*e^3 - 7*b*c^3*e^4)*x^4 + 20*(4*c^4*d^2*e^2 - 7*b*c^3*
d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 10*(20*c^4*d^3*e - 42*b*c^3*d^2*e^2 + 9*(3*b^2*c^2 + 2*a*c^3)*d*e^3 -
5*(b^3*c + 3*a*b*c^2)*e^4)*x^2 + 20*(30*c^4*d^4 - 70*b*c^3*d^3*e + 18*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 15*(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^7 - 3*(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 + 12*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^8

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mupad [B]  time = 1.88, size = 936, normalized size = 2.40 \begin {gather*} x^4\,\left (\frac {7\,b\,c^3}{4\,e^3}-\frac {3\,c^4\,d}{2\,e^4}\right )-\frac {x\,\left (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\right )+\frac {a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6-27\,a^2\,b\,c\,d^2\,e^5+30\,a^2\,c^2\,d^3\,e^4-9\,a\,b^3\,d^2\,e^5+60\,a\,b^2\,c\,d^3\,e^4-105\,a\,b\,c^2\,d^4\,e^3+54\,a\,c^3\,d^5\,e^2+5\,b^4\,d^3\,e^4-35\,b^3\,c\,d^4\,e^3+81\,b^2\,c^2\,d^5\,e^2-77\,b\,c^3\,d^6\,e+26\,c^4\,d^7}{2\,e}}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-x^2\,\left (\frac {c^4\,d^3}{e^6}+\frac {3\,d^2\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{2\,e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^3}+\frac {6\,c^4\,d^2}{e^5}\right )}{2\,e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{2\,e^3}\right )-x^3\,\left (\frac {d\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{3\,e^3}+\frac {2\,c^4\,d^2}{e^5}\right )+x\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^3}+\frac {6\,c^4\,d^2}{e^5}\right )}{e^2}+\frac {3\,d\,\left (\frac {2\,c^4\,d^3}{e^6}+\frac {3\,d^2\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^3}+\frac {6\,c^4\,d^2}{e^5}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e^3}\right )}{e}-\frac {d^3\,\left (\frac {7\,b\,c^3}{e^3}-\frac {6\,c^4\,d}{e^4}\right )}{e^3}\right )+\frac {2\,c^4\,x^5}{5\,e^3}-\frac {\ln \left (d+e\,x\right )\,\left (-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\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)^3,x)

[Out]

x^4*((7*b*c^3)/(4*e^3) - (3*c^4*d)/(2*e^4)) - (x*(14*c^4*d^6 + 2*a^3*c*e^6 + 3*a^2*b^2*e^6 + 3*b^4*d^2*e^4 + 3
0*a*c^3*d^4*e^2 - 20*b^3*c*d^3*e^3 + 18*a^2*c^2*d^2*e^4 + 45*b^2*c^2*d^4*e^2 - 6*a*b^3*d*e^5 - 42*b*c^3*d^5*e
- 18*a^2*b*c*d*e^5 - 60*a*b*c^2*d^3*e^3 + 36*a*b^2*c*d^2*e^4) + (26*c^4*d^7 + a^3*b*e^7 + 5*b^4*d^3*e^4 - 9*a*
b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 + 54*a*c^3*d^5*e^2 - 35*b^3*c*d^4*e^3 + 30*a^2*c^2*d^3*e^4 + 81*b^2*c^2*d^5*e^2
+ 2*a^3*c*d*e^6 - 77*b*c^3*d^6*e - 105*a*b*c^2*d^4*e^3 + 60*a*b^2*c*d^3*e^4 - 27*a^2*b*c*d^2*e^5)/(2*e))/(d^2*
e^7 + e^9*x^2 + 2*d*e^8*x) - x^2*((c^4*d^3)/e^6 + (3*d^2*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/(2*e^2) - (3*d*((3*d
*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e - (6*a*c^3 + 9*b^2*c^2)/e^3 + (6*c^4*d^2)/e^5))/(2*e) - (5*b*c*(3*a*c + b^
2))/(2*e^3)) - x^3*((d*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e - (6*a*c^3 + 9*b^2*c^2)/(3*e^3) + (2*c^4*d^2)/e^5) +
x*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^3 + (3*d^2*((3*d*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e - (6*a*c^3 + 9*b^2*c^
2)/e^3 + (6*c^4*d^2)/e^5))/e^2 + (3*d*((2*c^4*d^3)/e^6 + (3*d^2*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e^2 - (3*d*((
3*d*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e - (6*a*c^3 + 9*b^2*c^2)/e^3 + (6*c^4*d^2)/e^5))/e - (5*b*c*(3*a*c + b^2
))/e^3))/e - (d^3*((7*b*c^3)/e^3 - (6*c^4*d)/e^4))/e^3) + (2*c^4*x^5)/(5*e^3) - (log(d + e*x)*(42*c^4*d^5 - 3*
a*b^3*e^5 + 3*b^4*d*e^4 + 60*a*c^3*d^3*e^2 + 18*a^2*c^2*d*e^4 - 30*b^3*c*d^2*e^3 + 90*b^2*c^2*d^3*e^2 - 9*a^2*
b*c*e^5 - 105*b*c^3*d^4*e + 36*a*b^2*c*d*e^4 - 90*a*b*c^2*d^2*e^3))/e^8

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sympy [A]  time = 17.28, size = 733, normalized size = 1.88 \begin {gather*} \frac {2 c^{4} x^{5}}{5 e^{3}} + x^{4} \left (\frac {7 b c^{3}}{4 e^{3}} - \frac {3 c^{4} d}{2 e^{4}}\right ) + x^{3} \left (\frac {2 a c^{3}}{e^{3}} + \frac {3 b^{2} c^{2}}{e^{3}} - \frac {7 b c^{3} d}{e^{4}} + \frac {4 c^{4} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {15 a b c^{2}}{2 e^{3}} - \frac {9 a c^{3} d}{e^{4}} + \frac {5 b^{3} c}{2 e^{3}} - \frac {27 b^{2} c^{2} d}{2 e^{4}} + \frac {21 b c^{3} d^{2}}{e^{5}} - \frac {10 c^{4} d^{3}}{e^{6}}\right ) + x \left (\frac {6 a^{2} c^{2}}{e^{3}} + \frac {12 a b^{2} c}{e^{3}} - \frac {45 a b c^{2} d}{e^{4}} + \frac {36 a c^{3} d^{2}}{e^{5}} + \frac {b^{4}}{e^{3}} - \frac {15 b^{3} c d}{e^{4}} + \frac {54 b^{2} c^{2} d^{2}}{e^{5}} - \frac {70 b c^{3} d^{3}}{e^{6}} + \frac {30 c^{4} d^{4}}{e^{7}}\right ) + \frac {- a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 27 a^{2} b c d^{2} e^{5} - 30 a^{2} c^{2} d^{3} e^{4} + 9 a b^{3} d^{2} e^{5} - 60 a b^{2} c d^{3} e^{4} + 105 a b c^{2} d^{4} e^{3} - 54 a c^{3} d^{5} e^{2} - 5 b^{4} d^{3} e^{4} + 35 b^{3} c d^{4} e^{3} - 81 b^{2} c^{2} d^{5} e^{2} + 77 b c^{3} d^{6} e - 26 c^{4} d^{7} + x \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 )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac {3 \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^{8}} \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)**3,x)

[Out]

2*c**4*x**5/(5*e**3) + x**4*(7*b*c**3/(4*e**3) - 3*c**4*d/(2*e**4)) + x**3*(2*a*c**3/e**3 + 3*b**2*c**2/e**3 -
7*b*c**3*d/e**4 + 4*c**4*d**2/e**5) + x**2*(15*a*b*c**2/(2*e**3) - 9*a*c**3*d/e**4 + 5*b**3*c/(2*e**3) - 27*b
**2*c**2*d/(2*e**4) + 21*b*c**3*d**2/e**5 - 10*c**4*d**3/e**6) + x*(6*a**2*c**2/e**3 + 12*a*b**2*c/e**3 - 45*a
*b*c**2*d/e**4 + 36*a*c**3*d**2/e**5 + b**4/e**3 - 15*b**3*c*d/e**4 + 54*b**2*c**2*d**2/e**5 - 70*b*c**3*d**3/
e**6 + 30*c**4*d**4/e**7) + (-a**3*b*e**7 - 2*a**3*c*d*e**6 - 3*a**2*b**2*d*e**6 + 27*a**2*b*c*d**2*e**5 - 30*
a**2*c**2*d**3*e**4 + 9*a*b**3*d**2*e**5 - 60*a*b**2*c*d**3*e**4 + 105*a*b*c**2*d**4*e**3 - 54*a*c**3*d**5*e**
2 - 5*b**4*d**3*e**4 + 35*b**3*c*d**4*e**3 - 81*b**2*c**2*d**5*e**2 + 77*b*c**3*d**6*e - 26*c**4*d**7 + x*(-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))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2) + 3*(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**8

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