3.3.93 \(\int \frac {(3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4)}{d+e x} \, dx\) [293]

Optimal. Leaf size=228 \[ -\frac {\left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right ) x}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x^2}{2 e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^3}{3 e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^4}{4 e^3}-\frac {(20 d+17 e) x^5}{5 e^2}+\frac {10 x^6}{3 e}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7} \]

[Out]

-(20*d^5+17*d^4*e+17*d^3*e^2+4*d^2*e^3+21*d*e^4-7*e^5)*x/e^6+1/2*(20*d^4+17*d^3*e+17*d^2*e^2+4*d*e^3+21*e^4)*x
^2/e^5-1/3*(20*d^3+17*d^2*e+17*d*e^2+4*e^3)*x^3/e^4+1/4*(20*d^2+17*d*e+17*e^2)*x^4/e^3-1/5*(20*d+17*e)*x^5/e^2
+10/3*x^6/e+(5*d^2-2*d*e+3*e^2)*(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)*ln(e*x+d)/e^7

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Rubi [A]
time = 0.12, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1642} \begin {gather*} \frac {x^4 \left (20 d^2+17 d e+17 e^2\right )}{4 e^3}-\frac {x^3 \left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right )}{3 e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7}+\frac {x^2 \left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right )}{2 e^5}-\frac {x \left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right )}{e^6}-\frac {x^5 (20 d+17 e)}{5 e^2}+\frac {10 x^6}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x),x]

[Out]

-(((20*d^5 + 17*d^4*e + 17*d^3*e^2 + 4*d^2*e^3 + 21*d*e^4 - 7*e^5)*x)/e^6) + ((20*d^4 + 17*d^3*e + 17*d^2*e^2
+ 4*d*e^3 + 21*e^4)*x^2)/(2*e^5) - ((20*d^3 + 17*d^2*e + 17*d*e^2 + 4*e^3)*x^3)/(3*e^4) + ((20*d^2 + 17*d*e +
17*e^2)*x^4)/(4*e^3) - ((20*d + 17*e)*x^5)/(5*e^2) + (10*x^6)/(3*e) + ((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*
e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/e^7

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx &=\int \left (\frac {-20 d^5-17 d^4 e-17 d^3 e^2-4 d^2 e^3-21 d e^4+7 e^5}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x}{e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^2}{e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^3}{e^3}-\frac {(20 d+17 e) x^4}{e^2}+\frac {20 x^5}{e}+\frac {20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {\left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right ) x}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x^2}{2 e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^3}{3 e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^4}{4 e^3}-\frac {(20 d+17 e) x^5}{5 e^2}+\frac {10 x^6}{3 e}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 179, normalized size = 0.79 \begin {gather*} \frac {e x \left (-1200 d^5+60 d^4 e (-17+10 x)-10 d^3 e^2 \left (102-51 x+40 x^2\right )+10 d^2 e^3 \left (-24+51 x-34 x^2+30 x^3\right )-5 d e^4 \left (252-24 x+68 x^2-51 x^3+48 x^4\right )+e^5 \left (420+630 x-80 x^2+255 x^3-204 x^4+200 x^5\right )\right )+60 \left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) \log (d+e x)}{60 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x),x]

[Out]

(e*x*(-1200*d^5 + 60*d^4*e*(-17 + 10*x) - 10*d^3*e^2*(102 - 51*x + 40*x^2) + 10*d^2*e^3*(-24 + 51*x - 34*x^2 +
30*x^3) - 5*d*e^4*(252 - 24*x + 68*x^2 - 51*x^3 + 48*x^4) + e^5*(420 + 630*x - 80*x^2 + 255*x^3 - 204*x^4 + 2
00*x^5)) + 60*(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)*Log[d + e*x])/(60*e^
7)

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Maple [A]
time = 0.12, size = 249, normalized size = 1.09

method	result	size
norman	\(\frac {10 x^{6}}{3 e}-\frac {\left (20 d +17 e \right ) x^{5}}{5 e^{2}}+\frac {\left (20 d^{2}+17 d e +17 e^{2}\right ) x^{4}}{4 e^{3}}-\frac {\left (20 d^{3}+17 d^{2} e +17 d \,e^{2}+4 e^{3}\right ) x^{3}}{3 e^{4}}+\frac {\left (20 d^{4}+17 d^{3} e +17 d^{2} e^{2}+4 d \,e^{3}+21 e^{4}\right ) x^{2}}{2 e^{5}}-\frac {\left (20 d^{5}+17 d^{4} e +17 d^{3} e^{2}+4 d^{2} e^{3}+21 d \,e^{4}-7 e^{5}\right ) x}{e^{6}}+\frac {\left (20 d^{6}+17 d^{5} e +17 d^{4} e^{2}+4 d^{3} e^{3}+21 d^{2} e^{4}-7 d \,e^{5}+6 e^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(220\)
default	\(\frac {\left (20 d^{6}+17 d^{5} e +17 d^{4} e^{2}+4 d^{3} e^{3}+21 d^{2} e^{4}-7 d \,e^{5}+6 e^{6}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {-\frac {10}{3} x^{6} e^{5}+4 x^{5} e^{4} d +\frac {17}{5} x^{5} e^{5}-5 d^{2} e^{3} x^{4}-\frac {17}{4} x^{4} e^{4} d -\frac {17}{4} x^{4} e^{5}+\frac {20}{3} d^{3} e^{2} x^{3}+\frac {17}{3} d^{2} e^{3} x^{3}+\frac {17}{3} d \,e^{4} x^{3}+\frac {4}{3} e^{5} x^{3}-10 d^{4} e \,x^{2}-\frac {17}{2} d^{3} e^{2} x^{2}-\frac {17}{2} d^{2} e^{3} x^{2}-2 d \,e^{4} x^{2}-\frac {21}{2} e^{5} x^{2}+20 d^{5} x +17 d^{4} e x +17 d^{3} e^{2} x +4 d^{2} e^{3} x +21 d \,e^{4} x -7 e^{5} x}{e^{6}}\)	\(249\)
risch	\(\frac {17 x^{4}}{4 e}-\frac {4 x^{3}}{3 e}+\frac {7 x}{e}+\frac {6 \ln \left (e x +d \right )}{e}-\frac {17 x^{5}}{5 e}+\frac {20 \ln \left (e x +d \right ) d^{6}}{e^{7}}+\frac {17 \ln \left (e x +d \right ) d^{5}}{e^{6}}+\frac {17 \ln \left (e x +d \right ) d^{4}}{e^{5}}+\frac {4 \ln \left (e x +d \right ) d^{3}}{e^{4}}+\frac {21 \ln \left (e x +d \right ) d^{2}}{e^{3}}-\frac {7 \ln \left (e x +d \right ) d}{e^{2}}-\frac {4 x^{5} d}{e^{2}}+\frac {5 d^{2} x^{4}}{e^{3}}+\frac {17 x^{4} d}{4 e^{2}}-\frac {20 d^{3} x^{3}}{3 e^{4}}-\frac {17 d^{2} x^{3}}{3 e^{3}}-\frac {17 d \,x^{3}}{3 e^{2}}+\frac {10 d^{4} x^{2}}{e^{5}}+\frac {17 d^{3} x^{2}}{2 e^{4}}+\frac {17 d^{2} x^{2}}{2 e^{3}}+\frac {2 d \,x^{2}}{e^{2}}-\frac {20 d^{5} x}{e^{6}}-\frac {17 d^{4} x}{e^{5}}-\frac {17 d^{3} x}{e^{4}}-\frac {4 d^{2} x}{e^{3}}-\frac {21 d x}{e^{2}}+\frac {21 x^{2}}{2 e}+\frac {10 x^{6}}{3 e}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

(20*d^6+17*d^5*e+17*d^4*e^2+4*d^3*e^3+21*d^2*e^4-7*d*e^5+6*e^6)/e^7*ln(e*x+d)-1/e^6*(-10/3*x^6*e^5+4*x^5*e^4*d
+17/5*x^5*e^5-5*d^2*e^3*x^4-17/4*x^4*e^4*d-17/4*x^4*e^5+20/3*d^3*e^2*x^3+17/3*d^2*e^3*x^3+17/3*d*e^4*x^3+4/3*e
^5*x^3-10*d^4*e*x^2-17/2*d^3*e^2*x^2-17/2*d^2*e^3*x^2-2*d*e^4*x^2-21/2*e^5*x^2+20*d^5*x+17*d^4*e*x+17*d^3*e^2*
x+4*d^2*e^3*x+21*d*e^4*x-7*e^5*x)

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Maxima [A]
time = 0.31, size = 207, normalized size = 0.91 \begin {gather*} {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (200 \, x^{6} e^{5} - 12 \, {\left (20 \, d e^{4} + 17 \, e^{5}\right )} x^{5} + 15 \, {\left (20 \, d^{2} e^{3} + 17 \, d e^{4} + 17 \, e^{5}\right )} x^{4} - 20 \, {\left (20 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 17 \, d e^{4} + 4 \, e^{5}\right )} x^{3} + 30 \, {\left (20 \, d^{4} e + 17 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 4 \, d e^{4} + 21 \, e^{5}\right )} x^{2} - 60 \, {\left (20 \, d^{5} + 17 \, d^{4} e + 17 \, d^{3} e^{2} + 4 \, d^{2} e^{3} + 21 \, d e^{4} - 7 \, e^{5}\right )} x\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="maxima")

[Out]

(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)*e^(-7)*log(x*e + d) + 1/60*(200*x^
6*e^5 - 12*(20*d*e^4 + 17*e^5)*x^5 + 15*(20*d^2*e^3 + 17*d*e^4 + 17*e^5)*x^4 - 20*(20*d^3*e^2 + 17*d^2*e^3 + 1
7*d*e^4 + 4*e^5)*x^3 + 30*(20*d^4*e + 17*d^3*e^2 + 17*d^2*e^3 + 4*d*e^4 + 21*e^5)*x^2 - 60*(20*d^5 + 17*d^4*e
+ 17*d^3*e^2 + 4*d^2*e^3 + 21*d*e^4 - 7*e^5)*x)*e^(-6)

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Fricas [A]
time = 0.33, size = 212, normalized size = 0.93 \begin {gather*} -\frac {1}{60} \, {\left (1200 \, d^{5} x e - {\left (200 \, x^{6} - 204 \, x^{5} + 255 \, x^{4} - 80 \, x^{3} + 630 \, x^{2} + 420 \, x\right )} e^{6} + 5 \, {\left (48 \, d x^{5} - 51 \, d x^{4} + 68 \, d x^{3} - 24 \, d x^{2} + 252 \, d x\right )} e^{5} - 10 \, {\left (30 \, d^{2} x^{4} - 34 \, d^{2} x^{3} + 51 \, d^{2} x^{2} - 24 \, d^{2} x\right )} e^{4} + 10 \, {\left (40 \, d^{3} x^{3} - 51 \, d^{3} x^{2} + 102 \, d^{3} x\right )} e^{3} - 60 \, {\left (10 \, d^{4} x^{2} - 17 \, d^{4} x\right )} e^{2} - 60 \, {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} \log \left (x e + d\right )\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="fricas")

[Out]

-1/60*(1200*d^5*x*e - (200*x^6 - 204*x^5 + 255*x^4 - 80*x^3 + 630*x^2 + 420*x)*e^6 + 5*(48*d*x^5 - 51*d*x^4 +
68*d*x^3 - 24*d*x^2 + 252*d*x)*e^5 - 10*(30*d^2*x^4 - 34*d^2*x^3 + 51*d^2*x^2 - 24*d^2*x)*e^4 + 10*(40*d^3*x^3
- 51*d^3*x^2 + 102*d^3*x)*e^3 - 60*(10*d^4*x^2 - 17*d^4*x)*e^2 - 60*(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e
^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)*log(x*e + d))*e^(-7)

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Sympy [A]
time = 0.22, size = 235, normalized size = 1.03 \begin {gather*} x^{5} \left (- \frac {4 d}{e^{2}} - \frac {17}{5 e}\right ) + x^{4} \cdot \left (\frac {5 d^{2}}{e^{3}} + \frac {17 d}{4 e^{2}} + \frac {17}{4 e}\right ) + x^{3} \left (- \frac {20 d^{3}}{3 e^{4}} - \frac {17 d^{2}}{3 e^{3}} - \frac {17 d}{3 e^{2}} - \frac {4}{3 e}\right ) + x^{2} \cdot \left (\frac {10 d^{4}}{e^{5}} + \frac {17 d^{3}}{2 e^{4}} + \frac {17 d^{2}}{2 e^{3}} + \frac {2 d}{e^{2}} + \frac {21}{2 e}\right ) + x \left (- \frac {20 d^{5}}{e^{6}} - \frac {17 d^{4}}{e^{5}} - \frac {17 d^{3}}{e^{4}} - \frac {4 d^{2}}{e^{3}} - \frac {21 d}{e^{2}} + \frac {7}{e}\right ) + \frac {10 x^{6}}{3 e} + \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right ) \left (4 d^{4} + 5 d^{3} e + 3 d^{2} e^{2} - d e^{3} + 2 e^{4}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+2*x+3)*(4*x**4-5*x**3+3*x**2+x+2)/(e*x+d),x)

[Out]

x**5*(-4*d/e**2 - 17/(5*e)) + x**4*(5*d**2/e**3 + 17*d/(4*e**2) + 17/(4*e)) + x**3*(-20*d**3/(3*e**4) - 17*d**
2/(3*e**3) - 17*d/(3*e**2) - 4/(3*e)) + x**2*(10*d**4/e**5 + 17*d**3/(2*e**4) + 17*d**2/(2*e**3) + 2*d/e**2 +
21/(2*e)) + x*(-20*d**5/e**6 - 17*d**4/e**5 - 17*d**3/e**4 - 4*d**2/e**3 - 21*d/e**2 + 7/e) + 10*x**6/(3*e) +
(5*d**2 - 2*d*e + 3*e**2)*(4*d**4 + 5*d**3*e + 3*d**2*e**2 - d*e**3 + 2*e**4)*log(d + e*x)/e**7

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Giac [A]
time = 3.73, size = 228, normalized size = 1.00 \begin {gather*} {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (200 \, x^{6} e^{5} - 240 \, d x^{5} e^{4} + 300 \, d^{2} x^{4} e^{3} - 400 \, d^{3} x^{3} e^{2} + 600 \, d^{4} x^{2} e - 1200 \, d^{5} x - 204 \, x^{5} e^{5} + 255 \, d x^{4} e^{4} - 340 \, d^{2} x^{3} e^{3} + 510 \, d^{3} x^{2} e^{2} - 1020 \, d^{4} x e + 255 \, x^{4} e^{5} - 340 \, d x^{3} e^{4} + 510 \, d^{2} x^{2} e^{3} - 1020 \, d^{3} x e^{2} - 80 \, x^{3} e^{5} + 120 \, d x^{2} e^{4} - 240 \, d^{2} x e^{3} + 630 \, x^{2} e^{5} - 1260 \, d x e^{4} + 420 \, x e^{5}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d),x, algorithm="giac")

[Out]

(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(2
00*x^6*e^5 - 240*d*x^5*e^4 + 300*d^2*x^4*e^3 - 400*d^3*x^3*e^2 + 600*d^4*x^2*e - 1200*d^5*x - 204*x^5*e^5 + 25
5*d*x^4*e^4 - 340*d^2*x^3*e^3 + 510*d^3*x^2*e^2 - 1020*d^4*x*e + 255*x^4*e^5 - 340*d*x^3*e^4 + 510*d^2*x^2*e^3
- 1020*d^3*x*e^2 - 80*x^3*e^5 + 120*d*x^2*e^4 - 240*d^2*x*e^3 + 630*x^2*e^5 - 1260*d*x*e^4 + 420*x*e^5)*e^(-6
)

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Mupad [B]
time = 4.14, size = 260, normalized size = 1.14 \begin {gather*} x\,\left (\frac {7}{e}-\frac {d\,\left (\frac {21}{e}+\frac {d\,\left (\frac {4}{e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )-x^5\,\left (\frac {4\,d}{e^2}+\frac {17}{5\,e}\right )+x^4\,\left (\frac {17}{4\,e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{4\,e}\right )-x^3\,\left (\frac {4}{3\,e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{3\,e}\right )+x^2\,\left (\frac {21}{2\,e}+\frac {d\,\left (\frac {4}{e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+\frac {10\,x^6}{3\,e}+\frac {\ln \left (d+e\,x\right )\,\left (20\,d^6+17\,d^5\,e+17\,d^4\,e^2+4\,d^3\,e^3+21\,d^2\,e^4-7\,d\,e^5+6\,e^6\right )}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + 5*x^2 + 3)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2))/(d + e*x),x)

[Out]

x*(7/e - (d*(21/e + (d*(4/e + (d*(17/e + (d*((20*d)/e^2 + 17/e))/e))/e))/e))/e) - x^5*((4*d)/e^2 + 17/(5*e)) +
x^4*(17/(4*e) + (d*((20*d)/e^2 + 17/e))/(4*e)) - x^3*(4/(3*e) + (d*(17/e + (d*((20*d)/e^2 + 17/e))/e))/(3*e))
+ x^2*(21/(2*e) + (d*(4/e + (d*(17/e + (d*((20*d)/e^2 + 17/e))/e))/e))/(2*e)) + (10*x^6)/(3*e) + (log(d + e*x
)*(17*d^5*e - 7*d*e^5 + 20*d^6 + 6*e^6 + 21*d^2*e^4 + 4*d^3*e^3 + 17*d^4*e^2))/e^7

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