3.3.55 \(\int \frac {(d+e x)^5}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=171 \[ \frac {(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac {d^4 (3 c d-5 b e)}{b^4 x}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}-\frac {d^5}{2 b^3 x^2}+\frac {d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3} \]

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Rubi [A]  time = 0.18, antiderivative size = 171, 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} \begin {gather*} \frac {d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3}+\frac {(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac {d^4 (3 c d-5 b e)}{b^4 x}-\frac {d^5}{2 b^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-d^5/(2*b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b + c*x)^2) + ((c*d - b*e)^4*(3*c
*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(6*
c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/(b^5*c^3)

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 {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d^5}{b^3 x^3}+\frac {d^4 (-3 c d+5 b e)}{b^4 x^2}+\frac {d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 x}+\frac {(-c d+b e)^5}{b^3 c^2 (b+c x)^3}-\frac {(-c d+b e)^4 (3 c d+2 b e)}{b^4 c^2 (b+c x)^2}+\frac {(-c d+b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right )}{b^5 c^2 (b+c x)}\right ) \, dx\\ &=-\frac {d^5}{2 b^3 x^2}+\frac {d^4 (3 c d-5 b e)}{b^4 x}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac {(c d-b e)^4 (3 c d+2 b e)}{b^4 c^3 (b+c x)}+\frac {d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (x)}{b^5}-\frac {(c d-b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 165, normalized size = 0.96 \begin {gather*} -\frac {\frac {b^2 (b e-c d)^5}{c^3 (b+c x)^2}-2 d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )+\frac {2 (c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{c^3}+\frac {b^2 d^5}{x^2}-\frac {2 b (c d-b e)^4 (2 b e+3 c d)}{c^3 (b+c x)}+\frac {2 b d^4 (5 b e-3 c d)}{x}}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((b^2*d^5)/x^2 + (2*b*d^4*(-3*c*d + 5*b*e))/x + (b^2*(-(c*d) + b*e)^5)/(c^3*(b + c*x)^2) - (2*b*(c*d - b*
e)^4*(3*c*d + 2*b*e))/(c^3*(b + c*x)) - 2*d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x] + (2*(c*d - b*e)^3*
(6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/c^3)/b^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^5/(b*x + c*x^2)^3, x]

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fricas [B]  time = 0.43, size = 493, normalized size = 2.88 \begin {gather*} -\frac {b^{4} c^{3} d^{5} - 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - 5 \, b^{5} c^{2} d e^{4} + 2 \, b^{6} c e^{5}\right )} x^{3} - {\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} - 10 \, b^{5} c^{2} d^{2} e^{3} - 5 \, b^{6} c d e^{4} + 3 \, b^{7} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e\right )} x + 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^4*c^3*d^5 - 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - 5*b^5*c^2*d*e^4 + 2*b^6*c*e^5)*x^
3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^3*d^3*e^2 - 10*b^5*c^2*d^2*e^3 - 5*b^6*c*d*e^4 + 3*b^7*e^5)*
x^2 - 2*(2*b^3*c^4*d^5 - 5*b^4*c^3*d^4*e)*x + 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 - b^5*c^2*e^
5)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*c*e^5)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4
*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x^2)*log(c*x + b) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2
)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b
^4*c^3*d^3*e^2)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)

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giac [A]  time = 0.20, size = 275, normalized size = 1.61 \begin {gather*} \frac {{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac {b^{3} c^{3} d^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*c^2*d^5 - 15*b*c*d^4*e + 10*b^2*d^3*e^2)*log(abs(x))/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2
- b^5*e^5)*log(abs(c*x + b))/(b^5*c^3) - 1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^
2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^
2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/((c*x + b)^2*b^4*c^3*x^2)

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maple [A]  time = 0.06, size = 329, normalized size = 1.92 \begin {gather*} -\frac {b^{2} e^{5}}{2 \left (c x +b \right )^{2} c^{3}}+\frac {5 b d \,e^{4}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {5 d^{3} e^{2}}{\left (c x +b \right )^{2} b}-\frac {5 c \,d^{4} e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{5}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {5 d^{2} e^{3}}{\left (c x +b \right )^{2} c}+\frac {2 b \,e^{5}}{\left (c x +b \right ) c^{3}}+\frac {10 d^{3} e^{2}}{\left (c x +b \right ) b^{2}}-\frac {10 c \,d^{4} e}{\left (c x +b \right ) b^{3}}+\frac {10 d^{3} e^{2} \ln \relax (x )}{b^{3}}-\frac {10 d^{3} e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{5}}{\left (c x +b \right ) b^{4}}-\frac {15 c \,d^{4} e \ln \relax (x )}{b^{4}}+\frac {15 c \,d^{4} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{5} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{5} \ln \left (c x +b \right )}{b^{5}}-\frac {5 d \,e^{4}}{\left (c x +b \right ) c^{2}}+\frac {e^{5} \ln \left (c x +b \right )}{c^{3}}-\frac {5 d^{4} e}{b^{3} x}+\frac {3 c \,d^{5}}{b^{4} x}-\frac {d^{5}}{2 b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x)^3,x)

[Out]

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

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maxima [A]  time = 1.46, size = 296, normalized size = 1.73 \begin {gather*} -\frac {b^{3} c^{3} d^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \relax (x)}{b^{5}} - \frac {{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 -
(18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 -
2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2) + (6*c^2*d^5 - 15*b*c*d^4*e
+ 10*b^2*d^3*e^2)*log(x)/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - b^5*e^5)*log(c*x + b)/(b^5*
c^3)

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mupad [B]  time = 0.43, size = 268, normalized size = 1.57 \begin {gather*} \frac {d^3\,\ln \relax (x)\,\left (10\,b^2\,e^2-15\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5}-\frac {\frac {d^5}{2\,b}+\frac {d^4\,x\,\left (5\,b\,e-2\,c\,d\right )}{b^2}-\frac {x^2\,\left (3\,b^5\,e^5-5\,b^4\,c\,d\,e^4-10\,b^3\,c^2\,d^2\,e^3+30\,b^2\,c^3\,d^3\,e^2-45\,b\,c^4\,d^4\,e+18\,c^5\,d^5\right )}{2\,b^3\,c^3}-\frac {x^3\,\left (2\,b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^2\,c^3\,d^3\,e^2-15\,b\,c^4\,d^4\,e+6\,c^5\,d^5\right )}{b^4\,c^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^3\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(b*x + c*x^2)^3,x)

[Out]

(d^3*log(x)*(10*b^2*e^2 + 6*c^2*d^2 - 15*b*c*d*e))/b^5 - (d^5/(2*b) + (d^4*x*(5*b*e - 2*c*d))/b^2 - (x^2*(3*b^
5*e^5 + 18*c^5*d^5 + 30*b^2*c^3*d^3*e^2 - 10*b^3*c^2*d^2*e^3 - 45*b*c^4*d^4*e - 5*b^4*c*d*e^4))/(2*b^3*c^3) -
(x^3*(2*b^5*e^5 + 6*c^5*d^5 + 10*b^2*c^3*d^3*e^2 - 15*b*c^4*d^4*e - 5*b^4*c*d*e^4))/(b^4*c^2))/(b^2*x^2 + c^2*
x^4 + 2*b*c*x^3) + (log(b + c*x)*(b*e - c*d)^3*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(b^5*c^3)

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sympy [B]  time = 5.81, size = 524, normalized size = 3.06 \begin {gather*} \frac {- b^{3} c^{3} d^{5} + x^{3} \left (4 b^{5} c e^{5} - 10 b^{4} c^{2} d e^{4} + 20 b^{2} c^{4} d^{3} e^{2} - 30 b c^{5} d^{4} e + 12 c^{6} d^{5}\right ) + x^{2} \left (3 b^{6} e^{5} - 5 b^{5} c d e^{4} - 10 b^{4} c^{2} d^{2} e^{3} + 30 b^{3} c^{3} d^{3} e^{2} - 45 b^{2} c^{4} d^{4} e + 18 b c^{5} d^{5}\right ) + x \left (- 10 b^{3} c^{3} d^{4} e + 4 b^{2} c^{4} d^{5}\right )}{2 b^{6} c^{3} x^{2} + 4 b^{5} c^{4} x^{3} + 2 b^{4} c^{5} x^{4}} + \frac {d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + b c^{2} d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right )}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5}} + \frac {\left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + \frac {b \left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right )}{c}}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+b*x)**3,x)

[Out]

(-b**3*c**3*d**5 + x**3*(4*b**5*c*e**5 - 10*b**4*c**2*d*e**4 + 20*b**2*c**4*d**3*e**2 - 30*b*c**5*d**4*e + 12*
c**6*d**5) + x**2*(3*b**6*e**5 - 5*b**5*c*d*e**4 - 10*b**4*c**2*d**2*e**3 + 30*b**3*c**3*d**3*e**2 - 45*b**2*c
**4*d**4*e + 18*b*c**5*d**5) + x*(-10*b**3*c**3*d**4*e + 4*b**2*c**4*d**5))/(2*b**6*c**3*x**2 + 4*b**5*c**4*x*
*3 + 2*b**4*c**5*x**4) + d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*
b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*c**2*d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2))/(b**5*e**5 - 20*b**2
*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/b**5 + (b*e - c*d)**3*(b**2*e**2 + 3*b*c*d*e + 6*c**2*d**2
)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*(b*e - c*d)**3*(b**2*e**2 + 3*b*c
*d*e + 6*c**2*d**2)/c)/(b**5*e**5 - 20*b**2*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/(b**5*c**3)

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