\(\int \frac {(b+2 c x) (d+e x)^{5/2}}{(a+b x+c x^2)^2} \, dx\) [1618]
Optimal result
Integrand size = 28, antiderivative size = 350 \[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {5 e^2 \sqrt {d+e x}}{c}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}-\frac {5 e \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {5 e \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}
\]
[Out]
-(e*x+d)^(5/2)/(c*x^2+b*x+a)+5*e^2*(e*x+d)^(1/2)/c-5/2*e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4
*a*c+b^2)^(1/2)))^(1/2))*(2*c^2*d^2+b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(b*d+a*e-d*(-4*a*c+b^2)^(1/2)))/c^(3/2)
*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+5/2*e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)
/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(2*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c*e*(b*d+a*e+d*(-4*a*c+b^2)
^(1/2)))/c^(3/2)*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Rubi [A] (verified)
Time = 0.77 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {782, 717, 840, 1180, 214}
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {5 e \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {5 e \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {5 e^2 \sqrt {d+e x}}{c}
\]
[In]
Int[((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^2,x]
[Out]
(5*e^2*Sqrt[d + e*x])/c - (d + e*x)^(5/2)/(a + b*x + c*x^2) - (5*e*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (5*e*(2*c^2*d^2 +
b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*
x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 717
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)),
x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 1]
Rule 782
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Dist[e*g*(m/(2*c*(p + 1))), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]
Rule 840
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {1}{2} (5 e) \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx \\ & = \frac {5 e^2 \sqrt {d+e x}}{c}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {(5 e) \int \frac {c d^2-a e^2+e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 c} \\ & = \frac {5 e^2 \sqrt {d+e x}}{c}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {(5 e) \text {Subst}\left (\int \frac {-d e (2 c d-b e)+e \left (c d^2-a e^2\right )+e (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c} \\ & = \frac {5 e^2 \sqrt {d+e x}}{c}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {\left (5 e \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c \sqrt {b^2-4 a c}}-\frac {\left (5 e \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c \sqrt {b^2-4 a c}} \\ & = \frac {5 e^2 \sqrt {d+e x}}{c}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}-\frac {5 e \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {5 e \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 5.35 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.04
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e \left (\frac {2 \sqrt {c} \sqrt {d+e x} \left (5 e^2 (a+b x)-c \left (d^2+2 d e x-4 e^2 x^2\right )\right )}{e (a+x (b+c x))}+\frac {2 \sqrt {2} \left (12 c^2 d^2+b \left (5 b-3 \sqrt {b^2-4 a c}\right ) e^2-2 c e \left (6 b d-3 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (14 i c^2 d^2+b \left (5 i b+\sqrt {-b^2+4 a c}\right ) e^2-2 c e \left (7 i b d+\sqrt {-b^2+4 a c} d+3 i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-14 i c^2 d^2+b \left (-5 i b+\sqrt {-b^2+4 a c}\right ) e^2+2 i c e \left (7 b d+i \sqrt {-b^2+4 a c} d+3 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}-\frac {2 \sqrt {2} \left (12 c^2 d^2+b \left (5 b+3 \sqrt {b^2-4 a c}\right ) e^2-2 c e \left (6 b d+3 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 c^{3/2}}
\]
[In]
Integrate[((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^2,x]
[Out]
(e*((2*Sqrt[c]*Sqrt[d + e*x]*(5*e^2*(a + b*x) - c*(d^2 + 2*d*e*x - 4*e^2*x^2)))/(e*(a + x*(b + c*x))) + (2*Sqr
t[2]*(12*c^2*d^2 + b*(5*b - 3*Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(6*b*d - 3*Sqrt[b^2 - 4*a*c]*d + 4*a*e))*ArcTan[(
Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b
- Sqrt[b^2 - 4*a*c])*e]) + (((14*I)*c^2*d^2 + b*((5*I)*b + Sqrt[-b^2 + 4*a*c])*e^2 - 2*c*e*((7*I)*b*d + Sqrt[-
b^2 + 4*a*c]*d + (3*I)*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e
]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (((-14*I)*c^2*d^2 + b*((-5*I)*b + S
qrt[-b^2 + 4*a*c])*e^2 + (2*I)*c*e*(7*b*d + I*Sqrt[-b^2 + 4*a*c]*d + 3*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d +
e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4
*a*c])*e]) - (2*Sqrt[2]*(12*c^2*d^2 + b*(5*b + 3*Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(6*b*d + 3*Sqrt[b^2 - 4*a*c]*d
+ 4*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c
]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e])))/(2*c^(3/2))
Maple [A] (verified)
Time = 0.66 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.29
| | |
method | result | size |
| | |
derivativedivides |
\(2 e^{2} \left (\frac {2 \sqrt {e x +d}}{c}-\frac {\frac {\left (-\frac {b e}{2}+c d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1}{2} e^{2} a +\frac {1}{2} b d e -\frac {1}{2} c \,d^{2}\right ) \sqrt {e x +d}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+10 c \left (\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\right )\) |
\(452\) |
default |
\(2 e^{2} \left (\frac {2 \sqrt {e x +d}}{c}-\frac {\frac {\left (-\frac {b e}{2}+c d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1}{2} e^{2} a +\frac {1}{2} b d e -\frac {1}{2} c \,d^{2}\right ) \sqrt {e x +d}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+10 c \left (\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\right )\) |
\(452\) |
risch |
\(\frac {4 e^{2} \sqrt {e x +d}}{c}-\frac {2 e^{2} \left (\frac {\left (-\frac {b e}{2}+c d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1}{2} e^{2} a +\frac {1}{2} b d e -\frac {1}{2} c \,d^{2}\right ) \sqrt {e x +d}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+10 c \left (\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )}{c}\) |
\(453\) |
pseudoelliptic |
\(\frac {5 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \left (\left (\frac {b e}{2}-c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (a c -\frac {b^{2}}{2}\right ) e^{2}+b c d e -c^{2} d^{2}\right ) e^{2} \left (c \,x^{2}+b x +a \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+5 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (-\frac {b e}{2}+c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (a c -\frac {b^{2}}{2}\right ) e^{2}+b c d e -c^{2} d^{2}\right ) \sqrt {2}\, e^{2} \left (c \,x^{2}+b x +a \right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (a +\frac {4}{5} c \,x^{2}+b x \right ) e^{2}-\frac {2 c d e x}{5}-\frac {c \,d^{2}}{5}\right ) \sqrt {e x +d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \left (c \,x^{2}+b x +a \right )}\) |
\(453\) |
| | |
|
|
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[In]
int((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
2*e^2*(2/c*(e*x+d)^(1/2)-1/c*(((-1/2*b*e+c*d)*(e*x+d)^(3/2)+(-1/2*e^2*a+1/2*b*d*e-1/2*c*d^2)*(e*x+d)^(1/2))/(c
*(e*x+d)^2+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a-b*d*e+c*d^2)+10*c*(1/8*(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+(-e^
2*(4*a*c-b^2))^(1/2)*b*e-2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*
(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/
8*(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2+(-e^2*(4*a*c-b^2))^(1/2)*b*e-2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/c/(-e^2*
(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-
b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2928 vs. \(2 (298) = 596\).
Time = 0.39 (sec) , antiderivative size = 2928, normalized size of antiderivative = 8.37
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
-1/2*(5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 -
(b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2
*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)
)*log(125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^
2)*e^6 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2
*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*
sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*
sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 -
2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 +
2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*(c^2*
x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (
b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c
^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-125*sqrt(1/2)*(3
*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 - (2*(b^2*c^4 -
4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^
8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 -
3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 -
18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e
^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d
^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) + 5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqr
t((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqr
t((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a
*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^
2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3
- 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)
*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^
2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5
*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7
)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c
)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*
c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b
*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)
/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a
*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((
9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^
2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 -
(b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^
2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4
)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2
*c)*e^8)*sqrt(e*x + d)) - 2*(4*c*e^2*x^2 - c*d^2 + 5*a*e^2 - (2*c*d*e - 5*b*e^2)*x)*sqrt(e*x + d))/(c^2*x^2 +
b*c*x + a*c)
Sympy [F(-1)]
Timed out. \[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate((2*c*x+b)*(e*x+d)**(5/2)/(c*x**2+b*x+a)**2,x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x }
\]
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
integrate((2*c*x + b)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^2, x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (298) = 596\).
Time = 0.54 (sec) , antiderivative size = 930, normalized size of antiderivative = 2.66
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {4 \, \sqrt {e x + d} e^{2}}{c} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c d e^{2} - \sqrt {e x + d} c d^{2} e^{2} - {\left (e x + d\right )}^{\frac {3}{2}} b e^{3} + \sqrt {e x + d} b d e^{3} - \sqrt {e x + d} a e^{4}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e + a e^{2}\right )} c} + \frac {5 \, {\left ({\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 4 \, a b c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{3} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} {\left | e \right |} - {\left (4 \, c^{5} d^{3} e^{2} - 6 \, b c^{4} d^{2} e^{3} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{4} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{5}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e + \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{3} d e + \sqrt {b^{2} - 4 \, a c} a c^{3} e^{2}\right )} c^{2} {\left | e \right |}} - \frac {5 \, {\left ({\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 4 \, a b c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{3} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} {\left | e \right |} - {\left (4 \, c^{5} d^{3} e^{2} - 6 \, b c^{4} d^{2} e^{3} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{4} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{5}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e - \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{3} d e + \sqrt {b^{2} - 4 \, a c} a c^{3} e^{2}\right )} c^{2} {\left | e \right |}}
\]
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
4*sqrt(e*x + d)*e^2/c - (2*(e*x + d)^(3/2)*c*d*e^2 - sqrt(e*x + d)*c*d^2*e^2 - (e*x + d)^(3/2)*b*e^3 + sqrt(e*
x + d)*b*d*e^3 - sqrt(e*x + d)*a*e^4)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e + a*e^
2)*c) + 5/8*((2*(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e
)*c^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c^2*d*e^3 + sqrt(b^2 - 4*a*c)*a*c^2*e^4)*sq
rt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) - (4*c^5*d^3*e^2 - 6*b*c^4*d^2*e^3 + 4*(b^2*c^3 -
a*c^4)*d*e^4 - (b^3*c^2 - 2*a*b*c^3)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2
)*sqrt(e*x + d)/sqrt(-(2*c^2*d - b*c*e + sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2
))/((sqrt(b^2 - 4*a*c)*c^4*d^2 - sqrt(b^2 - 4*a*c)*b*c^3*d*e + sqrt(b^2 - 4*a*c)*a*c^3*e^2)*c^2*abs(e)) - 5/8*
((2*(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2*e^2 +
2*(sqrt(b^2 - 4*a*c)*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c^2*d*e^3 + sqrt(b^2 - 4*a*c)*a*c^2*e^4)*sqrt(-4*c^2*d
+ 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) - (4*c^5*d^3*e^2 - 6*b*c^4*d^2*e^3 + 4*(b^2*c^3 - a*c^4)*d*e^
4 - (b^3*c^2 - 2*a*b*c^3)*e^5)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x +
d)/sqrt(-(2*c^2*d - b*c*e - sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((sqrt(b^
2 - 4*a*c)*c^4*d^2 - sqrt(b^2 - 4*a*c)*b*c^3*d*e + sqrt(b^2 - 4*a*c)*a*c^3*e^2)*c^2*abs(e))
Mupad [B] (verification not implemented)
Time = 1.49 (sec) , antiderivative size = 8776, normalized size of antiderivative = 25.07
\[
\int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int(((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^2,x)
[Out]
((b*e^3 - 2*c*d*e^2)*(d + e*x)^(3/2) + (d + e*x)^(1/2)*(a*e^4 + c*d^2*e^2 - b*d*e^3))/(c^2*(d + e*x)^2 - (2*c^
2*d - b*c*e)*(d + e*x) + c^2*d^2 + a*c*e^2 - b*c*d*e) - atan(((((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^
4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b
^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 +
3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2)
- 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*
c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(
-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b
^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c
- b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*
e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c
^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 +
300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2
- 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3
*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*
d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((5*(16*a^2*c^3*e^6 - 4*a
*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1
/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4
- 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(
4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c
^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 +
32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 -
24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*
e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2
*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 +
50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e
^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2
*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b
^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^
3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((5
*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5
))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e
^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b
^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^
3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*
e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^
5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)
^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^
(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*
x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d
^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3
)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^
2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*
d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*
c^4)))^(1/2) + (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4
- 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*
e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^
2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3
)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3
*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2
) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*
e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*
(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))
^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*
e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^
5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3
*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b
^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 +
b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (10*(50*c^3*d^5*e^6 - 25*b^3*d^2*e^9 - 25*a^2*b*e^11 + 100*a*c^2*d^3*e^8 - 1
25*b*c^2*d^4*e^7 + 100*b^2*c*d^3*e^8 + 50*a*b^2*d*e^10 + 50*a^2*c*d*e^10 - 150*a*b*c*d^2*e^9))/c))*(-(25*(b^5*
e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3
*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)
^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(
16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i - atan(((((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e
^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*
(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c
^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4
*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b
^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b
^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*
d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)
^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(
8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*
e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*
b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a
^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5
+ a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3
+ 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^
2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(
25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2
*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c -
b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^
4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c
^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*
c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a
*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 +
18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2
*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 10
0*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 +
8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2
) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((5*(16*a^
2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c -
(2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24
*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^
5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e
^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 1
6*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a
*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) -
12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2
)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6
- 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(
-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^
(1/2) + (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a
*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8
*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(
1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2)
- 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 -
8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(
4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*
c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)
+ (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 1
50*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^
2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*
e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^
3 - 8*a*b^2*c^4)))^(1/2) - (10*(50*c^3*d^5*e^6 - 25*b^3*d^2*e^9 - 25*a^2*b*e^11 + 100*a*c^2*d^3*e^8 - 125*b*c^
2*d^4*e^7 + 100*b^2*c*d^3*e^8 + 50*a*b^2*d*e^10 + 50*a^2*c*d*e^10 - 150*a*b*c*d^2*e^9))/c))*(-(25*(b^5*e^5 - b
^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 +
3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2)
- 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*
c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i + (4*e^2*(d + e*x)^(1/2))/c