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

Optimal. Leaf size=213 \[ -\frac {3 d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (c d-b e)}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) \log (d+e x)}{e^7}+\frac {c x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac {d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac {3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac {c^3 x^3}{3 e^4} \]

[Out]

c*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2)*x/e^6-1/2*c^2*(-3*b*e+4*c*d)*x^2/e^5+1/3*c^3*x^3/e^4-1/3*d^3*(-b*e+c*d)^3/
e^7/(e*x+d)^3+3/2*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/(e*x+d)^2-3*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7
/(e*x+d)-(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*ln(e*x+d)/e^7

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Rubi [A]  time = 0.21, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {3 d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (c d-b e)}{e^7 (d+e x)}+\frac {c x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac {d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac {3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac {c^3 x^3}{3 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^4,x]

[Out]

(c*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x)/e^6 - (c^2*(4*c*d - 3*b*e)*x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (d^3
*(c*d - b*e)^3)/(3*e^7*(d + e*x)^3) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(2*e^7*(d + e*x)^2) - (3*d*(c*d - b*
e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*Log
[d + e*x])/e^7

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 (b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {c \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^6}-\frac {c^2 (4 c d-3 b e) x}{e^5}+\frac {c^3 x^2}{e^4}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^4}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^2}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac {c \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right ) x}{e^6}-\frac {c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac {c^3 x^3}{3 e^4}-\frac {d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 210, normalized size = 0.99 \[ \frac {6 c e x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )+\frac {18 d \left (b^3 e^3-6 b^2 c d e^2+10 b c^2 d^2 e-5 c^3 d^3\right )}{d+e x}+6 \left (b^3 e^3-12 b^2 c d e^2+30 b c^2 d^2 e-20 c^3 d^3\right ) \log (d+e x)-3 c^2 e^2 x^2 (4 c d-3 b e)-\frac {2 d^3 (c d-b e)^3}{(d+e x)^3}+\frac {9 d^2 (2 c d-b e) (c d-b e)^2}{(d+e x)^2}+2 c^3 e^3 x^3}{6 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*c*e*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x - 3*c^2*e^2*(4*c*d - 3*b*e)*x^2 + 2*c^3*e^3*x^3 - (2*d^3*(c*d -
b*e)^3)/(d + e*x)^3 + (9*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(d + e*x)^2 + (18*d*(-5*c^3*d^3 + 10*b*c^2*d^2*e -
6*b^2*c*d*e^2 + b^3*e^3))/(d + e*x) + 6*(-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]
)/(6*e^7)

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fricas [B]  time = 1.16, size = 480, normalized size = 2.25 \[ \frac {2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 78 \, b^{2} c d^{4} e^{2} + 11 \, b^{3} d^{3} e^{3} - 3 \, {\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} + {\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \, b^{2} c d e^{5}\right )} x^{3} + 3 \, {\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \, b^{2} c d^{2} e^{4} + 6 \, b^{3} d e^{5}\right )} x^{2} - 3 \, {\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 54 \, b^{2} c d^{3} e^{3} - 9 \, b^{3} d^{2} e^{4}\right )} x - 6 \, {\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 3 \, {\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c^3*e^6*x^6 - 74*c^3*d^6 + 141*b*c^2*d^5*e - 78*b^2*c*d^4*e^2 + 11*b^3*d^3*e^3 - 3*(2*c^3*d*e^5 - 3*b*c
^2*e^6)*x^5 + 3*(10*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 6*b^2*c*e^6)*x^4 + (146*c^3*d^3*e^3 - 189*b*c^2*d^2*e^4 + 5
4*b^2*c*d*e^5)*x^3 + 3*(26*c^3*d^4*e^2 - 9*b*c^2*d^3*e^3 - 18*b^2*c*d^2*e^4 + 6*b^3*d*e^5)*x^2 - 3*(34*c^3*d^5
*e - 81*b*c^2*d^4*e^2 + 54*b^2*c*d^3*e^3 - 9*b^3*d^2*e^4)*x - 6*(20*c^3*d^6 - 30*b*c^2*d^5*e + 12*b^2*c*d^4*e^
2 - b^3*d^3*e^3 + (20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 - b^3*e^6)*x^3 + 3*(20*c^3*d^4*e^2 - 30*
b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 3*(20*c^3*d^5*e - 30*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 - b^
3*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [A]  time = 0.18, size = 261, normalized size = 1.23 \[ -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, c^{3} x^{3} e^{8} - 12 \, c^{3} d x^{2} e^{7} + 60 \, c^{3} d^{2} x e^{6} + 9 \, b c^{2} x^{2} e^{8} - 72 \, b c^{2} d x e^{7} + 18 \, b^{2} c x e^{8}\right )} e^{\left (-12\right )} - \frac {{\left (74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{6 \, {\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*e^(-7)*log(abs(x*e + d)) + 1/6*(2*c^3*x^3*e^8 - 12*c
^3*d*x^2*e^7 + 60*c^3*d^2*x*e^6 + 9*b*c^2*x^2*e^8 - 72*b*c^2*d*x*e^7 + 18*b^2*c*x*e^8)*e^(-12) - 1/6*(74*c^3*d
^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 - 11*b^3*d^3*e^3 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*
e^4 - b^3*d*e^5)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2*d^4*e^2 + 20*b^2*c*d^3*e^3 - 3*b^3*d^2*e^4)*x)*e^(-7)/(x*e +
d)^3

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maple [A]  time = 0.05, size = 353, normalized size = 1.66 \[ \frac {b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {b^{2} c \,d^{4}}{\left (e x +d \right )^{3} e^{5}}+\frac {b \,c^{2} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {c^{3} x^{3}}{3 e^{4}}-\frac {3 b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {6 b^{2} c \,d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 b \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {3 b \,c^{2} x^{2}}{2 e^{4}}+\frac {3 c^{3} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}+\frac {3 b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {18 b^{2} c \,d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 b^{2} c d \ln \left (e x +d \right )}{e^{5}}+\frac {3 b^{2} c x}{e^{4}}+\frac {30 b \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {30 b \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {12 b \,c^{2} d x}{e^{5}}-\frac {15 c^{3} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 c^{3} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 c^{3} d^{2} x}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c^3*x^3/e^4+3/2*c^2/e^4*x^2*b-2*c^3*d*x^2/e^5+3*c/e^4*b^2*x-12*c^2/e^5*b*d*x+10*c^3/e^6*d^2*x+1/3*d^3/e^4/
(e*x+d)^3*b^3-d^4/e^5/(e*x+d)^3*b^2*c+d^5/e^6/(e*x+d)^3*b*c^2-1/3*d^6/e^7/(e*x+d)^3*c^3-3/2*d^2/e^4/(e*x+d)^2*
b^3+6*d^3/e^5/(e*x+d)^2*b^2*c-15/2*d^4/e^6/(e*x+d)^2*b*c^2+3*d^5/e^7/(e*x+d)^2*c^3+3*d/e^4/(e*x+d)*b^3-18*d^2/
e^5/(e*x+d)*b^2*c+30*d^3/e^6/(e*x+d)*b*c^2-15*d^4/e^7/(e*x+d)*c^3+1/e^4*ln(e*x+d)*b^3-12/e^5*ln(e*x+d)*b^2*c*d
+30/e^6*ln(e*x+d)*b*c^2*d^2-20/e^7*ln(e*x+d)*c^3*d^3

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maxima [A]  time = 1.49, size = 294, normalized size = 1.38 \[ -\frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4}\right )} x}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {2 \, c^{3} e^{2} x^{3} - 3 \, {\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x}{6 \, e^{6}} - \frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 - 11*b^3*d^3*e^3 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3
+ 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2*d^4*e^2 + 20*b^2*c*d^3*e^3 - 3*b^3*d^2*e^4)*x)
/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/6*(2*c^3*e^2*x^3 - 3*(4*c^3*d*e - 3*b*c^2*e^2)*x^2 + 6*(
10*c^3*d^2 - 12*b*c^2*d*e + 3*b^2*c*e^2)*x)/e^6 - (20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*log
(e*x + d)/e^7

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mupad [B]  time = 0.12, size = 307, normalized size = 1.44 \[ x^2\,\left (\frac {3\,b\,c^2}{2\,e^4}-\frac {2\,c^3\,d}{e^5}\right )-x\,\left (\frac {4\,d\,\left (\frac {3\,b\,c^2}{e^4}-\frac {4\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b^2\,c}{e^4}+\frac {6\,c^3\,d^2}{e^6}\right )-\frac {x\,\left (-\frac {9\,b^3\,d^2\,e^3}{2}+30\,b^2\,c\,d^3\,e^2-\frac {105\,b\,c^2\,d^4\,e}{2}+27\,c^3\,d^5\right )-x^2\,\left (3\,b^3\,d\,e^4-18\,b^2\,c\,d^2\,e^3+30\,b\,c^2\,d^3\,e^2-15\,c^3\,d^4\,e\right )+\frac {-11\,b^3\,d^3\,e^3+78\,b^2\,c\,d^4\,e^2-141\,b\,c^2\,d^5\,e+74\,c^3\,d^6}{6\,e}}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e-20\,c^3\,d^3\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x)^4,x)

[Out]

x^2*((3*b*c^2)/(2*e^4) - (2*c^3*d)/e^5) - x*((4*d*((3*b*c^2)/e^4 - (4*c^3*d)/e^5))/e - (3*b^2*c)/e^4 + (6*c^3*
d^2)/e^6) - (x*(27*c^3*d^5 - (9*b^3*d^2*e^3)/2 + 30*b^2*c*d^3*e^2 - (105*b*c^2*d^4*e)/2) - x^2*(3*b^3*d*e^4 -
15*c^3*d^4*e + 30*b*c^2*d^3*e^2 - 18*b^2*c*d^2*e^3) + (74*c^3*d^6 - 11*b^3*d^3*e^3 + 78*b^2*c*d^4*e^2 - 141*b*
c^2*d^5*e)/(6*e))/(d^3*e^6 + e^9*x^3 + 3*d^2*e^7*x + 3*d*e^8*x^2) + (log(d + e*x)*(b^3*e^3 - 20*c^3*d^3 + 30*b
*c^2*d^2*e - 12*b^2*c*d*e^2))/e^7 + (c^3*x^3)/(3*e^4)

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sympy [A]  time = 2.73, size = 301, normalized size = 1.41 \[ \frac {c^{3} x^{3}}{3 e^{4}} + x^{2} \left (\frac {3 b c^{2}}{2 e^{4}} - \frac {2 c^{3} d}{e^{5}}\right ) + x \left (\frac {3 b^{2} c}{e^{4}} - \frac {12 b c^{2} d}{e^{5}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {11 b^{3} d^{3} e^{3} - 78 b^{2} c d^{4} e^{2} + 141 b c^{2} d^{5} e - 74 c^{3} d^{6} + x^{2} \left (18 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 180 b c^{2} d^{3} e^{3} - 90 c^{3} d^{4} e^{2}\right ) + x \left (27 b^{3} d^{2} e^{4} - 180 b^{2} c d^{3} e^{3} + 315 b c^{2} d^{4} e^{2} - 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac {\left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*x**3/(3*e**4) + x**2*(3*b*c**2/(2*e**4) - 2*c**3*d/e**5) + x*(3*b**2*c/e**4 - 12*b*c**2*d/e**5 + 10*c**3*
d**2/e**6) + (11*b**3*d**3*e**3 - 78*b**2*c*d**4*e**2 + 141*b*c**2*d**5*e - 74*c**3*d**6 + x**2*(18*b**3*d*e**
5 - 108*b**2*c*d**2*e**4 + 180*b*c**2*d**3*e**3 - 90*c**3*d**4*e**2) + x*(27*b**3*d**2*e**4 - 180*b**2*c*d**3*
e**3 + 315*b*c**2*d**4*e**2 - 162*c**3*d**5*e))/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3)
+ (b*e - 2*c*d)*(b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*log(d + e*x)/e**7

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