3.10.91 \(\int \frac {(A+B x) (b x+c x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=422 \[ -\frac {3 d (c d-b e) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{2 e^6}+\frac {\log (d+e x) \left (B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8}+\frac {x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )-B \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}+\frac {B c^3 x^4}{4 e^4} \]

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Rubi [A]  time = 0.64, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{2 e^6}+\frac {x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )-B \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}-\frac {3 d (c d-b e) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac {\log (d+e x) \left (B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}+\frac {B c^3 x^4}{4 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*c*e*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2) - B*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*x)
/e^7 - (c*(A*c*e*(4*c*d - 3*b*e) - B*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*x^2)/(2*e^6) - (c^2*(4*B*c*d - 3*b
*B*e - A*c*e)*x^3)/(3*e^5) + (B*c^3*x^4)/(4*e^4) + (d^3*(B*d - A*e)*(c*d - b*e)^3)/(3*e^8*(d + e*x)^3) - (d^2*
(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e)))/(2*e^8*(d + e*x)^2) - (3*d*(c*d - b*e)*(A*e*(5*c^2*
d^2 - 5*b*c*d*e + b^2*e^2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2)))/(e^8*(d + e*x)) + ((B*d*(35*c^3*d^3 - 6
0*b*c^2*d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3) - A*e*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*Lo
g[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 {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7}+\frac {c \left (-A c e (4 c d-3 b e)+B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x}{e^6}+\frac {c^2 (-4 B c d+3 b B e+A c e) x^2}{e^5}+\frac {B c^3 x^3}{e^4}-\frac {d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)^4}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^7 (d+e x)^3}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^7 (d+e x)^2}+\frac {B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {\left (A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}-\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^8 (d+e x)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 400, normalized size = 0.95 \begin {gather*} \frac {-6 c e^2 x^2 \left (A c e (4 c d-3 b e)+B \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )+\frac {36 d (c d-b e) \left (B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )-A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{d+e x}+12 e x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )+B \left (b^3 e^3-12 b^2 c d e^2+30 b c^2 d^2 e-20 c^3 d^3\right )\right )+12 \log (d+e x) \left (A e \left (b^3 e^3-12 b^2 c d e^2+30 b c^2 d^2 e-20 c^3 d^3\right )+B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )\right )+4 c^2 e^3 x^3 (A c e+3 b B e-4 B c d)+\frac {4 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^3}-\frac {6 d^2 (c d-b e)^2 (3 A e (b e-2 c d)+B d (7 c d-4 b e))}{(d+e x)^2}+3 B c^3 e^4 x^4}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*e*(A*c*e*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2) + B*(-20*c^3*d^3 + 30*b*c^2*d^2*e - 12*b^2*c*d*e^2 + b^3*e^
3))*x - 6*c*e^2*(A*c*e*(4*c*d - 3*b*e) + B*(-10*c^2*d^2 + 12*b*c*d*e - 3*b^2*e^2))*x^2 + 4*c^2*e^3*(-4*B*c*d +
3*b*B*e + A*c*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*d^3*(B*d - A*e)*(c*d - b*e)^3)/(d + e*x)^3 - (6*d^2*(c*d - b*e)^2
*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e)))/(d + e*x)^2 + (36*d*(c*d - b*e)*(-(A*e*(5*c^2*d^2 - 5*b*c*d*e +
b^2*e^2)) + B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2)))/(d + e*x) + 12*(B*d*(35*c^3*d^3 - 60*b*c^2*d^2*e + 30*b
^2*c*d*e^2 - 4*b^3*e^3) + A*e*(-20*c^3*d^3 + 30*b*c^2*d^2*e - 12*b^2*c*d*e^2 + b^3*e^3))*Log[d + e*x])/(12*e^8
)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^4, x]

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fricas [B]  time = 0.41, size = 910, normalized size = 2.16 \begin {gather*} \frac {3 \, B c^{3} e^{7} x^{7} + 214 \, B c^{3} d^{7} + 22 \, A b^{3} d^{3} e^{4} - 148 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 282 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 52 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} - {\left (7 \, B c^{3} d e^{6} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 3 \, {\left (7 \, B c^{3} d^{2} e^{5} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 3 \, {\left (35 \, B c^{3} d^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{6} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} - 2 \, {\left (278 \, B c^{3} d^{4} e^{3} - 146 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - 18 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, B c^{3} d^{5} e^{2} - 6 \, A b^{3} d e^{6} - 26 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} + 6 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 6 \, {\left (37 \, B c^{3} d^{6} e + 9 \, A b^{3} d^{2} e^{5} - 34 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 81 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 18 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x + 12 \, {\left (35 \, B c^{3} d^{7} + A b^{3} d^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + {\left (35 \, B c^{3} d^{4} e^{3} + A b^{3} e^{7} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} + 3 \, {\left (35 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{3} d^{6} e + A b^{3} d^{2} e^{5} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 + 22*A*b^3*d^3*e^4 - 148*(3*B*b*c^2 + A*c^3)*d^6*e + 282*(B*b^2*c + A*b*
c^2)*d^5*e^2 - 52*(B*b^3 + 3*A*b^2*c)*d^4*e^3 - (7*B*c^3*d*e^6 - 4*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7*B*c^3*d
^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d*e^6 + 6*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*(3*B*b*c^2 +
A*c^3)*d^2*e^5 + 30*(B*b^2*c + A*b*c^2)*d*e^6 - 4*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 - 2*(278*B*c^3*d^4*e^3 - 146*(3
*B*b*c^2 + A*c^3)*d^3*e^4 + 189*(B*b^2*c + A*b*c^2)*d^2*e^5 - 18*(B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 - 6*(68*B*c^3*
d^5*e^2 - 6*A*b^3*d*e^6 - 26*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 9*(B*b^2*c + A*b*c^2)*d^3*e^4 + 6*(B*b^3 + 3*A*b^2*
c)*d^2*e^5)*x^2 + 6*(37*B*c^3*d^6*e + 9*A*b^3*d^2*e^5 - 34*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 81*(B*b^2*c + A*b*c^2
)*d^4*e^3 - 18*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x + 12*(35*B*c^3*d^7 + A*b^3*d^3*e^4 - 20*(3*B*b*c^2 + A*c^3)*d^6*
e + 30*(B*b^2*c + A*b*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (35*B*c^3*d^4*e^3 + A*b^3*e^7 - 20*(3*B*b
*c^2 + A*c^3)*d^3*e^4 + 30*(B*b^2*c + A*b*c^2)*d^2*e^5 - 4*(B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 + 3*(35*B*c^3*d^5*e^
2 + A*b^3*d*e^6 - 20*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 30*(B*b^2*c + A*b*c^2)*d^3*e^4 - 4*(B*b^3 + 3*A*b^2*c)*d^2*
e^5)*x^2 + 3*(35*B*c^3*d^6*e + A*b^3*d^2*e^5 - 20*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 30*(B*b^2*c + A*b*c^2)*d^4*e^3
- 4*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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giac [A]  time = 0.16, size = 581, normalized size = 1.38 \begin {gather*} {\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 12 \, A b^{2} c d e^{3} + A b^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{12} - 16 \, B c^{3} d x^{3} e^{11} + 60 \, B c^{3} d^{2} x^{2} e^{10} - 240 \, B c^{3} d^{3} x e^{9} + 12 \, B b c^{2} x^{3} e^{12} + 4 \, A c^{3} x^{3} e^{12} - 72 \, B b c^{2} d x^{2} e^{11} - 24 \, A c^{3} d x^{2} e^{11} + 360 \, B b c^{2} d^{2} x e^{10} + 120 \, A c^{3} d^{2} x e^{10} + 18 \, B b^{2} c x^{2} e^{12} + 18 \, A b c^{2} x^{2} e^{12} - 144 \, B b^{2} c d x e^{11} - 144 \, A b c^{2} d x e^{11} + 12 \, B b^{3} x e^{12} + 36 \, A b^{2} c x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} + 11 \, A b^{3} d^{3} e^{4} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} + A b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} + 9 \, A b^{3} d^{2} e^{5}\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((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

(35*B*c^3*d^4 - 60*B*b*c^2*d^3*e - 20*A*c^3*d^3*e + 30*B*b^2*c*d^2*e^2 + 30*A*b*c^2*d^2*e^2 - 4*B*b^3*d*e^3 -
12*A*b^2*c*d*e^3 + A*b^3*e^4)*e^(-8)*log(abs(x*e + d)) + 1/12*(3*B*c^3*x^4*e^12 - 16*B*c^3*d*x^3*e^11 + 60*B*c
^3*d^2*x^2*e^10 - 240*B*c^3*d^3*x*e^9 + 12*B*b*c^2*x^3*e^12 + 4*A*c^3*x^3*e^12 - 72*B*b*c^2*d*x^2*e^11 - 24*A*
c^3*d*x^2*e^11 + 360*B*b*c^2*d^2*x*e^10 + 120*A*c^3*d^2*x*e^10 + 18*B*b^2*c*x^2*e^12 + 18*A*b*c^2*x^2*e^12 - 1
44*B*b^2*c*d*x*e^11 - 144*A*b*c^2*d*x*e^11 + 12*B*b^3*x*e^12 + 36*A*b^2*c*x*e^12)*e^(-16) + 1/6*(107*B*c^3*d^7
- 222*B*b*c^2*d^6*e - 74*A*c^3*d^6*e + 141*B*b^2*c*d^5*e^2 + 141*A*b*c^2*d^5*e^2 - 26*B*b^3*d^4*e^3 - 78*A*b^
2*c*d^4*e^3 + 11*A*b^3*d^3*e^4 + 18*(7*B*c^3*d^5*e^2 - 15*B*b*c^2*d^4*e^3 - 5*A*c^3*d^4*e^3 + 10*B*b^2*c*d^3*e
^4 + 10*A*b*c^2*d^3*e^4 - 2*B*b^3*d^2*e^5 - 6*A*b^2*c*d^2*e^5 + A*b^3*d*e^6)*x^2 + 3*(77*B*c^3*d^6*e - 162*B*b
*c^2*d^5*e^2 - 54*A*c^3*d^5*e^2 + 105*B*b^2*c*d^4*e^3 + 105*A*b*c^2*d^4*e^3 - 20*B*b^3*d^3*e^4 - 60*A*b^2*c*d^
3*e^4 + 9*A*b^3*d^2*e^5)*x)*e^(-8)/(x*e + d)^3

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maple [A]  time = 0.07, size = 807, normalized size = 1.91 \begin {gather*} \frac {B \,c^{3} x^{4}}{4 e^{4}}+\frac {A \,b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A \,b^{2} c \,d^{4}}{\left (e x +d \right )^{3} e^{5}}+\frac {A b \,c^{2} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {A \,c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {A \,c^{3} x^{3}}{3 e^{4}}-\frac {B \,b^{3} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,b^{2} c \,d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {B b \,c^{2} d^{6}}{\left (e x +d \right )^{3} e^{7}}+\frac {B b \,c^{2} x^{3}}{e^{4}}+\frac {B \,c^{3} d^{7}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {4 B \,c^{3} d \,x^{3}}{3 e^{5}}-\frac {3 A \,b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {6 A \,b^{2} c \,d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 A b \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {3 A b \,c^{2} x^{2}}{2 e^{4}}+\frac {3 A \,c^{3} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 A \,c^{3} d \,x^{2}}{e^{5}}+\frac {2 B \,b^{3} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 B \,b^{2} c \,d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {3 B \,b^{2} c \,x^{2}}{2 e^{4}}+\frac {9 B b \,c^{2} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {6 B b \,c^{2} d \,x^{2}}{e^{5}}-\frac {7 B \,c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {5 B \,c^{3} d^{2} x^{2}}{e^{6}}+\frac {3 A \,b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {A \,b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {18 A \,b^{2} c \,d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 A \,b^{2} c d \ln \left (e x +d \right )}{e^{5}}+\frac {3 A \,b^{2} c x}{e^{4}}+\frac {30 A b \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {30 A b \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {12 A b \,c^{2} d x}{e^{5}}-\frac {15 A \,c^{3} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 A \,c^{3} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 A \,c^{3} d^{2} x}{e^{6}}-\frac {6 B \,b^{3} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 B \,b^{3} d \ln \left (e x +d \right )}{e^{5}}+\frac {B \,b^{3} x}{e^{4}}+\frac {30 B \,b^{2} c \,d^{3}}{\left (e x +d \right ) e^{6}}+\frac {30 B \,b^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {12 B \,b^{2} c d x}{e^{5}}-\frac {45 B b \,c^{2} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {60 B b \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {30 B b \,c^{2} d^{2} x}{e^{6}}+\frac {21 B \,c^{3} d^{5}}{\left (e x +d \right ) e^{8}}+\frac {35 B \,c^{3} d^{4} \ln \left (e x +d \right )}{e^{8}}-\frac {20 B \,c^{3} d^{3} x}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^4*B*x^3*b*c^2-4/3/e^5*B*x^3*c^3*d+1/3/e^4*A*x^3*c^3+1/e^4*B*b^3*x+1/e^4*ln(e*x+d)*A*b^3-6/e^5*B*x^2*b*c^2*
d-12/e^5*A*b*c^2*d*x-12/e^5*B*b^2*c*d*x+30/e^6*B*b*c^2*d^2*x-18*d^2/e^5/(e*x+d)*A*b^2*c+30*d^3/e^6/(e*x+d)*A*b
*c^2+30*d^3/e^6/(e*x+d)*B*b^2*c-45*d^4/e^7/(e*x+d)*B*b*c^2+6*d^3/e^5/(e*x+d)^2*A*b^2*c-15/2*d^4/e^6/(e*x+d)^2*
A*b*c^2-15/2*d^4/e^6/(e*x+d)^2*B*b^2*c+9*d^5/e^7/(e*x+d)^2*B*b*c^2-d^4/e^5/(e*x+d)^3*A*b^2*c+d^5/e^6/(e*x+d)^3
*A*b*c^2+d^5/e^6/(e*x+d)^3*B*b^2*c-d^6/e^7/(e*x+d)^3*B*b*c^2-12/e^5*ln(e*x+d)*A*b^2*c*d+30/e^6*ln(e*x+d)*A*b*c
^2*d^2+30/e^6*ln(e*x+d)*B*b^2*c*d^2+1/3*d^3/e^4/(e*x+d)^3*A*b^3-60/e^7*ln(e*x+d)*B*b*c^2*d^3+5/e^6*B*x^2*c^3*d
^2+3/e^4*A*b^2*c*x+10/e^6*A*c^3*d^2*x-20/e^7*B*c^3*d^3*x+3*d/e^4/(e*x+d)*A*b^3-15*d^4/e^7/(e*x+d)*A*c^3-6*d^2/
e^5/(e*x+d)*B*b^3+21*d^5/e^8/(e*x+d)*B*c^3-3/2*d^2/e^4/(e*x+d)^2*A*b^3+3*d^5/e^7/(e*x+d)^2*A*c^3+3/2/e^4*A*x^2
*b*c^2-2/e^5*A*x^2*c^3*d+3/2/e^4*B*x^2*b^2*c+2*d^3/e^5/(e*x+d)^2*B*b^3-7/2*d^6/e^8/(e*x+d)^2*B*c^3-1/3*d^6/e^7
/(e*x+d)^3*A*c^3-1/3*d^4/e^5/(e*x+d)^3*B*b^3+1/3*d^7/e^8/(e*x+d)^3*B*c^3-20/e^7*ln(e*x+d)*A*c^3*d^3-4/e^5*ln(e
*x+d)*B*b^3*d+35/e^8*ln(e*x+d)*B*c^3*d^4+1/4*B*c^3*x^4/e^4

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maxima [A]  time = 0.57, size = 561, normalized size = 1.33 \begin {gather*} \frac {107 \, B c^{3} d^{7} + 11 \, A b^{3} d^{3} e^{4} - 74 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 141 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 26 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e + 9 \, A b^{3} d^{2} e^{5} - 54 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 105 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 20 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B c^{3} e^{3} x^{4} - 4 \, {\left (4 \, B c^{3} d e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{3}\right )} x}{12 \, e^{7}} + \frac {{\left (35 \, B c^{3} d^{4} + A b^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(107*B*c^3*d^7 + 11*A*b^3*d^3*e^4 - 74*(3*B*b*c^2 + A*c^3)*d^6*e + 141*(B*b^2*c + A*b*c^2)*d^5*e^2 - 26*(B
*b^3 + 3*A*b^2*c)*d^4*e^3 + 18*(7*B*c^3*d^5*e^2 + A*b^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c +
A*b*c^2)*d^3*e^4 - 2*(B*b^3 + 3*A*b^2*c)*d^2*e^5)*x^2 + 3*(77*B*c^3*d^6*e + 9*A*b^3*d^2*e^5 - 54*(3*B*b*c^2 +
A*c^3)*d^5*e^2 + 105*(B*b^2*c + A*b*c^2)*d^4*e^3 - 20*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)/(e^11*x^3 + 3*d*e^10*x^2
+ 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*c^3*e^3*x^4 - 4*(4*B*c^3*d*e^2 - (3*B*b*c^2 + A*c^3)*e^3)*x^3 + 6*(10*B*
c^3*d^2*e - 4*(3*B*b*c^2 + A*c^3)*d*e^2 + 3*(B*b^2*c + A*b*c^2)*e^3)*x^2 - 12*(20*B*c^3*d^3 - 10*(3*B*b*c^2 +
A*c^3)*d^2*e + 12*(B*b^2*c + A*b*c^2)*d*e^2 - (B*b^3 + 3*A*b^2*c)*e^3)*x)/e^7 + (35*B*c^3*d^4 + A*b^3*e^4 - 20
*(3*B*b*c^2 + A*c^3)*d^3*e + 30*(B*b^2*c + A*b*c^2)*d^2*e^2 - 4*(B*b^3 + 3*A*b^2*c)*d*e^3)*log(e*x + d)/e^8

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mupad [B]  time = 0.20, size = 676, normalized size = 1.60 \begin {gather*} \frac {x\,\left (-10\,B\,b^3\,d^3\,e^3+\frac {9\,A\,b^3\,d^2\,e^4}{2}+\frac {105\,B\,b^2\,c\,d^4\,e^2}{2}-30\,A\,b^2\,c\,d^3\,e^3-81\,B\,b\,c^2\,d^5\,e+\frac {105\,A\,b\,c^2\,d^4\,e^2}{2}+\frac {77\,B\,c^3\,d^6}{2}-27\,A\,c^3\,d^5\,e\right )+x^2\,\left (-6\,B\,b^3\,d^2\,e^4+3\,A\,b^3\,d\,e^5+30\,B\,b^2\,c\,d^3\,e^3-18\,A\,b^2\,c\,d^2\,e^4-45\,B\,b\,c^2\,d^4\,e^2+30\,A\,b\,c^2\,d^3\,e^3+21\,B\,c^3\,d^5\,e-15\,A\,c^3\,d^4\,e^2\right )+\frac {-26\,B\,b^3\,d^4\,e^3+11\,A\,b^3\,d^3\,e^4+141\,B\,b^2\,c\,d^5\,e^2-78\,A\,b^2\,c\,d^4\,e^3-222\,B\,b\,c^2\,d^6\,e+141\,A\,b\,c^2\,d^5\,e^2+107\,B\,c^3\,d^7-74\,A\,c^3\,d^6\,e}{6\,e}}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e^4}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}-\frac {4\,B\,c^3\,d^3}{e^7}\right )+x^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-4\,B\,b^3\,d\,e^3+A\,b^3\,e^4+30\,B\,b^2\,c\,d^2\,e^2-12\,A\,b^2\,c\,d\,e^3-60\,B\,b\,c^2\,d^3\,e+30\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*((77*B*c^3*d^6)/2 - 27*A*c^3*d^5*e + (9*A*b^3*d^2*e^4)/2 - 10*B*b^3*d^3*e^3 + (105*A*b*c^2*d^4*e^2)/2 - 30*
A*b^2*c*d^3*e^3 + (105*B*b^2*c*d^4*e^2)/2 - 81*B*b*c^2*d^5*e) + x^2*(3*A*b^3*d*e^5 + 21*B*c^3*d^5*e - 15*A*c^3
*d^4*e^2 - 6*B*b^3*d^2*e^4 + 30*A*b*c^2*d^3*e^3 - 18*A*b^2*c*d^2*e^4 - 45*B*b*c^2*d^4*e^2 + 30*B*b^2*c*d^3*e^3
) + (107*B*c^3*d^7 - 74*A*c^3*d^6*e + 11*A*b^3*d^3*e^4 - 26*B*b^3*d^4*e^3 + 141*A*b*c^2*d^5*e^2 - 78*A*b^2*c*d
^4*e^3 + 141*B*b^2*c*d^5*e^2 - 222*B*b*c^2*d^6*e)/(6*e))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) + x*
((B*b^3 + 3*A*b^2*c)/e^4 + (4*d*((4*d*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e - (3*b*c*(A*c + B*b))/e^4
+ (6*B*c^3*d^2)/e^6))/e - (6*d^2*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e^2 - (4*B*c^3*d^3)/e^7) + x^3*
((A*c^3 + 3*B*b*c^2)/(3*e^4) - (4*B*c^3*d)/(3*e^5)) - x^2*((2*d*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e
- (3*b*c*(A*c + B*b))/(2*e^4) + (3*B*c^3*d^2)/e^6) + (log(d + e*x)*(A*b^3*e^4 + 35*B*c^3*d^4 - 20*A*c^3*d^3*e
- 4*B*b^3*d*e^3 + 30*A*b*c^2*d^2*e^2 + 30*B*b^2*c*d^2*e^2 - 12*A*b^2*c*d*e^3 - 60*B*b*c^2*d^3*e))/e^8 + (B*c^
3*x^4)/(4*e^4)

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sympy [A]  time = 23.06, size = 700, normalized size = 1.66 \begin {gather*} \frac {B c^{3} x^{4}}{4 e^{4}} + x^{3} \left (\frac {A c^{3}}{3 e^{4}} + \frac {B b c^{2}}{e^{4}} - \frac {4 B c^{3} d}{3 e^{5}}\right ) + x^{2} \left (\frac {3 A b c^{2}}{2 e^{4}} - \frac {2 A c^{3} d}{e^{5}} + \frac {3 B b^{2} c}{2 e^{4}} - \frac {6 B b c^{2} d}{e^{5}} + \frac {5 B c^{3} d^{2}}{e^{6}}\right ) + x \left (\frac {3 A b^{2} c}{e^{4}} - \frac {12 A b c^{2} d}{e^{5}} + \frac {10 A c^{3} d^{2}}{e^{6}} + \frac {B b^{3}}{e^{4}} - \frac {12 B b^{2} c d}{e^{5}} + \frac {30 B b c^{2} d^{2}}{e^{6}} - \frac {20 B c^{3} d^{3}}{e^{7}}\right ) + \frac {11 A b^{3} d^{3} e^{4} - 78 A b^{2} c d^{4} e^{3} + 141 A b c^{2} d^{5} e^{2} - 74 A c^{3} d^{6} e - 26 B b^{3} d^{4} e^{3} + 141 B b^{2} c d^{5} e^{2} - 222 B b c^{2} d^{6} e + 107 B c^{3} d^{7} + x^{2} \left (18 A b^{3} d e^{6} - 108 A b^{2} c d^{2} e^{5} + 180 A b c^{2} d^{3} e^{4} - 90 A c^{3} d^{4} e^{3} - 36 B b^{3} d^{2} e^{5} + 180 B b^{2} c d^{3} e^{4} - 270 B b c^{2} d^{4} e^{3} + 126 B c^{3} d^{5} e^{2}\right ) + x \left (27 A b^{3} d^{2} e^{5} - 180 A b^{2} c d^{3} e^{4} + 315 A b c^{2} d^{4} e^{3} - 162 A c^{3} d^{5} e^{2} - 60 B b^{3} d^{3} e^{4} + 315 B b^{2} c d^{4} e^{3} - 486 B b c^{2} d^{5} e^{2} + 231 B c^{3} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} - \frac {\left (- A b^{3} e^{4} + 12 A b^{2} c d e^{3} - 30 A b c^{2} d^{2} e^{2} + 20 A c^{3} d^{3} e + 4 B b^{3} d e^{3} - 30 B b^{2} c d^{2} e^{2} + 60 B b c^{2} d^{3} e - 35 B c^{3} d^{4}\right ) \log {\left (d + e x \right )}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d)**4,x)

[Out]

B*c**3*x**4/(4*e**4) + x**3*(A*c**3/(3*e**4) + B*b*c**2/e**4 - 4*B*c**3*d/(3*e**5)) + x**2*(3*A*b*c**2/(2*e**4
) - 2*A*c**3*d/e**5 + 3*B*b**2*c/(2*e**4) - 6*B*b*c**2*d/e**5 + 5*B*c**3*d**2/e**6) + x*(3*A*b**2*c/e**4 - 12*
A*b*c**2*d/e**5 + 10*A*c**3*d**2/e**6 + B*b**3/e**4 - 12*B*b**2*c*d/e**5 + 30*B*b*c**2*d**2/e**6 - 20*B*c**3*d
**3/e**7) + (11*A*b**3*d**3*e**4 - 78*A*b**2*c*d**4*e**3 + 141*A*b*c**2*d**5*e**2 - 74*A*c**3*d**6*e - 26*B*b*
*3*d**4*e**3 + 141*B*b**2*c*d**5*e**2 - 222*B*b*c**2*d**6*e + 107*B*c**3*d**7 + x**2*(18*A*b**3*d*e**6 - 108*A
*b**2*c*d**2*e**5 + 180*A*b*c**2*d**3*e**4 - 90*A*c**3*d**4*e**3 - 36*B*b**3*d**2*e**5 + 180*B*b**2*c*d**3*e**
4 - 270*B*b*c**2*d**4*e**3 + 126*B*c**3*d**5*e**2) + x*(27*A*b**3*d**2*e**5 - 180*A*b**2*c*d**3*e**4 + 315*A*b
*c**2*d**4*e**3 - 162*A*c**3*d**5*e**2 - 60*B*b**3*d**3*e**4 + 315*B*b**2*c*d**4*e**3 - 486*B*b*c**2*d**5*e**2
+ 231*B*c**3*d**6*e))/(6*d**3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3) - (-A*b**3*e**4 + 12*A*
b**2*c*d*e**3 - 30*A*b*c**2*d**2*e**2 + 20*A*c**3*d**3*e + 4*B*b**3*d*e**3 - 30*B*b**2*c*d**2*e**2 + 60*B*b*c*
*2*d**3*e - 35*B*c**3*d**4)*log(d + e*x)/e**8

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