\(\int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx\) [618]

Optimal result

Integrand size = 19, antiderivative size = 781 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

2/3*e*(e*x+d)^(3/2)/c+4*d*e*(e*x+d)^(1/2)/c-1/2*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*
d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a
*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)+1/2*e*arctanh((c^
(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c
^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1
/2)-(a*e^2+c*d^2)^(1/2))^(1/2)+1/4*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c
^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7
/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)-1/4*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(
1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(-a
*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {718, 839, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=-\frac {e \left (-\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}}{c} \]

[In]

Int[(d + e*x)^(5/2)/(a + c*x^2),x]

[Out]

(4*d*e*Sqrt[d + e*x])/c + (2*e*(d + e*x)^(3/2))/(3*c) - (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - (3*c*d^2 - a*e
^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[
Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])
+ (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sq
rt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*S
qrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 -
a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*S
qrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]
]) - (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] +
Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*S
qrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 718

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (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 + 2*c*d*e*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e}
, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 1]

Rule 839

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(
c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + 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] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {d+e x} \left (c d^2-a e^2+2 c d e x\right )}{a+c x^2} \, dx}{c} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {c d \left (c d^2-3 a e^2\right )+c e \left (3 c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{c^2} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {2 \text {Subst}\left (\int \frac {c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )+c e \left (3 c d^2-a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )-\sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )-\sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \sqrt {c d^2+a e^2}}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{c^2 \sqrt {c d^2+a e^2}} \\ & = \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.32 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x} (7 d+e x)+\frac {3 i \left (\sqrt {c} d+i \sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (\sqrt {c} d-i \sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{3 c^{3/2}} \]

[In]

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

[Out]

(2*Sqrt[c]*e*Sqrt[d + e*x]*(7*d + e*x) + ((3*I)*(Sqrt[c]*d + I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sq
rt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - ((3*I)*(Sqr
t[c]*d - I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/
(Sqrt[a]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(3*c^(3/2))

Maple [A] (verified)

Time = 5.66 (sec) , antiderivative size = 739, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+e^{2} \left (\frac {14 \sqrt {e x +d}\, \left (\frac {e x}{7}+d \right ) c^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{3}+\left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (a c \,e^{2}-3 c^{2} d^{2}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right )\right ) a}{c^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, a e}\)	\(739\)
risch	\(\text {Expression too large to display}\)	\(1594\)
derivativedivides	\(\text {Expression too large to display}\)	\(1599\)
default	\(\text {Expression too large to display}\)	\(1599\)

[In]

int((e*x+d)^(5/2)/(c*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/c^(5/2)*(-1/4*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*((e^2*a-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2)*((a*e
^2+c*d^2)*c)^(1/2)-d*e^2*a*c+3*c^2*d^3+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a
*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*
e^2+c*d^2)^(1/2))+1/4*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*((e^2*a-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2)
*((a*e^2+c*d^2)*c)^(1/2)-d*e^2*a*c+3*c^2*d^3+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)
-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/
2)+(a*e^2+c*d^2)^(1/2))+e^2*(14/3*(e*x+d)^(1/2)*(1/7*e*x+d)*c^(3/2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c
*d^2)*c)^(1/2)-2*c*d)^(1/2)+(arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e
^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2))-arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d
^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))*(a*c*e^2-3*
c^2*d^2+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d))*a)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(
1/2)/a/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1641 vs. \(2 (634) = 1268\).

Time = 0.61 (sec) , antiderivative size = 1641, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 11
0*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*
a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2
- a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)
/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 1
10*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 +
5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e
^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10
*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*
a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 +
5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*
e^10)/(a*c^7)))/(a*c^3))) + 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 -
100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 1
4*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*
c^2*d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3
*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 -
100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt(-(c^2*d
^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20
*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^
9)*sqrt(e*x + d) - (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-
(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*
d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 2
0*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 4*(e^2*x + 7*d*e)*sqrt(e*x + d))/c

Sympy [F]

\[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{a + c x^{2}}\, dx \]

[In]

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

[Out]

Integral((d + e*x)**(5/2)/(a + c*x**2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} + a} \,d x } \]

[In]

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

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + a), x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.54 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=-\frac {{\left (\sqrt {-a c} c^{4} d^{4} e - 3 \, \sqrt {-a c} a c^{3} d^{2} e^{3} + {\left (3 \, \sqrt {-a c} a c d^{2} e - \sqrt {-a c} a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {-a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {-a c} c^{4} d^{4} e - 3 \, \sqrt {-a c} a c^{3} d^{2} e^{3} + {\left (3 \, \sqrt {-a c} a c d^{2} e - \sqrt {-a c} a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {-a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e\right )}}{3 \, c^{3}} \]

[In]

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

[Out]

-(sqrt(-a*c)*c^4*d^4*e - 3*sqrt(-a*c)*a*c^3*d^2*e^3 + (3*sqrt(-a*c)*a*c*d^2*e - sqrt(-a*c)*a^2*e^3)*c^2*e^2 -
2*(a*c^3*d^3*e + a^2*c^2*d*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 + a
*c^3*e^2)*c^4))/c^4))/((a*c^4*d - sqrt(-a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(e)) + (sqrt(-a*c)*c^4*
d^4*e - 3*sqrt(-a*c)*a*c^3*d^2*e^3 + (3*sqrt(-a*c)*a*c*d^2*e - sqrt(-a*c)*a^2*e^3)*c^2*e^2 + 2*(a*c^3*d^3*e +
a^2*c^2*d*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^2 + a*c^3*e^2)*c^4))/c
^4))/((a*c^4*d + sqrt(-a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(e)) + 2/3*((e*x + d)^(3/2)*c^2*e + 6*sq
rt(e*x + d)*c^2*d*e)/c^3

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 3481, normalized size of antiderivative = 4.46 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(5/2)/(a + c*x^2),x)

[Out]

(2*e*(d + e*x)^(3/2))/(3*c) - atan((a^3*e^8*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5
*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2)
)/(2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d
^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(-a^3*c^7)
^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (d^5*e^3*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*
c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*
a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c - 160*a^4*d^2*e^9 - 80*a*c^3*d
^8*e^3 + 64*a^3*c*d^4*e^7 + 160*a^2*c^2*d^6*e^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/c - (160*a*d^5*e^6*(-a^3*c^7)
^(1/2))/c^2 + (32*a^3*d*e^10*(-a^3*c^7)^(1/2))/c^4 - (288*a^2*d^3*e^8*(-a^3*c^7)^(1/2))/c^3) + (d^3*e^5*(-a^3*
c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(
4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/(16*a^4*
e^11 - 80*c^4*d^8*e^3 + 160*a*c^3*d^6*e^5 - 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 + (160*d^7*e^4*(-a^3*c^7)^(
1/2))/a - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^2 + (32*a^2*d*e^10*(-a^3*c^7)^
(1/2))/c^3) - (a*d*e^7*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3
*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2
*a*c^6))^(1/2)*32i)/(16*a^4*c*e^11 - 160*d^5*e^6*(-a^3*c^7)^(1/2) - 80*c^5*d^8*e^3 + 160*a*c^4*d^6*e^5 + 64*a^
2*c^3*d^4*e^7 - 160*a^3*c^2*d^2*e^9 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c + (160*c*d^7*e^4*(-a^3*c^7)^(1/2))/a
+ (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5
/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a
^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)
/c + 160*a*c*d^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e
^10*(-a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (a^2*c*d^2*e^6*(d + e*x)^(1/2)*((e^5*(-a
^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/
(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2
*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c
^7)^(1/2))/c^4 + (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)))*((a^2*e^5*(-a
^3*c^7)^(1/2) - a*c^6*d^5 - 5*a^3*c^4*d*e^4 + 10*a^2*c^5*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^7)^(1/2) - 10*a*c*d^2*e
^3*(-a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i - atan((a^3*e^8*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c)
- (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(
-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9
)/c + 160*a*c*d^6*e^5 + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 - (32*a^2*d*
e^10*(-a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) + (d^5*e^3*(-a^3*c^7)^(1/2)*(d + e*x)^(1/
2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*
c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c - 160*a^4*d^2*e
^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 + 160*a^2*c^2*d^6*e^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/c + (160*a*d^5
*e^6*(-a^3*c^7)^(1/2))/c^2 - (32*a^3*d*e^10*(-a^3*c^7)^(1/2))/c^4 + (288*a^2*d^3*e^8*(-a^3*c^7)^(1/2))/c^3) -
(d^3*e^5*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7)
- (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)
*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 + 160*a*c^3*d^6*e^5 - 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 - (160*d^7*e
^4*(-a^3*c^7)^(1/2))/a + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^2 - (32*a^2*d*e
^10*(-a^3*c^7)^(1/2))/c^3) + (a*d*e^7*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e
^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3
*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/(160*d^5*e^6*(-a^3*c^7)^(1/2) + 16*a^4*c*e^11 - 80*c^5*d^8*e^3 + 160*a*c^4*
d^6*e^5 + 64*a^2*c^3*d^4*e^7 - 160*a^3*c^2*d^2*e^9 + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c - (160*c*d^7*e^4*(-a^3
*c^7)^(1/2))/a - (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) -
d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) +
(5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (1
60*a^3*d^2*e^9)/c + 160*a*c*d^6*e^5 + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^
4 - (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (a^2*c*d^2*e^6*(d + e*x)^
(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a
^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/((16*a^4*e^11)/c^2 + 64*a^2*d
^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d^6*e^5 + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 + (288*a*
d^3*e^8*(-a^3*c^7)^(1/2))/c^4 - (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2))
)*(-(a^2*e^5*(-a^3*c^7)^(1/2) + a*c^6*d^5 + 5*a^3*c^4*d*e^4 - 10*a^2*c^5*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^7)^(1/2
) - 10*a*c*d^2*e^3*(-a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i + (4*d*e*(d + e*x)^(1/2))/c