\(\int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx\) [619]

Optimal result

Integrand size = 19, antiderivative size = 689 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 e \sqrt {d+e x}}{c}-\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

2*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))*(c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/c^(5/4)*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))*(c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/c^(5/4)
*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))*(c*d^2+a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^
(1/2))/c^(5/4)*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))*(c*d^2+a*e^2+2*d*c^(1/2)*
(a*e^2+c*d^2)^(1/2))/c^(5/4)*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 = 1.00 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {718, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=-\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {2 e \sqrt {d+e x}}{c} \]

[In]

Int[(d + e*x)^(3/2)/(a + c*x^2),x]

[Out]

(2*e*Sqrt[d + e*x])/c - (e*(c*d^2 + a*e^2 - 2*Sqrt[c]*d*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^(5/4)*Sqrt[c
*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Ar
cTanh[(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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^2 + 2*Sq
rt[c]*d*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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]
]) - (e*(c*d^2 + a*e^2 + 2*Sqrt[c]*d*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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sq
rt[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 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 \sqrt {d+e x}}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{c} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {2 \text {Subst}\left (\int \frac {-2 c d^2 e+e \left (c d^2-a e^2\right )+2 c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \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^{5/4} \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 (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {2 e \sqrt {d+e x}}{c}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \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^{3/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \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^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {2 e \sqrt {d+e x}}{c}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \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^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \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^{3/2} \sqrt {c d^2+a e^2}} \\ & = \frac {2 e \sqrt {d+e x}}{c}-\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \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 = 0.61 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.34 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 e \sqrt {d+e x}+\frac {i \left (\sqrt {c} d+i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{c} \]

[In]

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

[Out]

(2*e*Sqrt[d + e*x] + (I*(Sqrt[c]*d + I*Sqrt[a]*e)^2*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]) - (I*(Sqrt[c]*d - I*Sqrt[a]*e)^2*ArcT
an[(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*Sqr
t[a]*Sqrt[c]*e]))/c

Maple [A] (verified)

Time = 2.75 (sec) , antiderivative size = 685, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (\sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}+2 c d \right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c^{2} d^{2}-c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (\sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}+2 c d \right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c^{2} d^{2}-c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+e^{2} \left (2 \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, c^{\frac {3}{2}} \sqrt {e x +d}+\left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (-2 c^{2} d +\sqrt {e^{2} a +c \,d^{2}}\, c^{\frac {3}{2}}\right )\right ) a}{c^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, a e}\)	\(685\)
derivativedivides	\(\text {Expression too large to display}\)	\(1244\)
default	\(\text {Expression too large to display}\)	\(1244\)
risch	\(\text {Expression too large to display}\)	\(1245\)

[In]

int((e*x+d)^(3/2)/(c*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + 4 \, \sqrt {e x + d} e}{2 \, c} \]

[In]

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

[Out]

1/2*(c*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log
(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) + (3*a*c^2*d^2*e^2 - a^2*c*e^4 + a*c^4*d*sqrt(-(9*c^2*
d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2
*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4
+ a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 -
a^2*c*e^4 + a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2
*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) + c*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt
(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sq
rt(e*x + d) + (3*a*c^2*d^2*e^2 - a^2*c*e^4 - a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))
*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt
(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^
4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 - a^2*c*e^4 - a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6
*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*
e^6)/(a*c^5)))/(a*c^2))) + 4*sqrt(e*x + d)*e)/c

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{a + c x^{2}}\, dx \]

[In]

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

[Out]

Integral((d + e*x)**(3/2)/(a + c*x**2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} + a} \,d x } \]

[In]

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

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + a), x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\frac {2 \, \sqrt {e x + d} e}{c} - \frac {{\left (\sqrt {-a c} c^{3} d^{3} e + \sqrt {-a c} a c^{2} d e^{3} - {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {-a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {-a c} c^{3} d^{3} e + \sqrt {-a c} a c^{2} d e^{3} + {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {-a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 1625, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(3/2)/(a + c*x^2),x)

[Out]

2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3
*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^2*d^5*e^3 - 16*a^2*d*e^7 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3
+ 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)) + (32*d*e^5*(-a^3*
c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^
3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 + 32*a*c^2*d^3*e^5 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^2 - 16*a^
2*c*d*e^7 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/a + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c) - (96*d^3*e^3*(-a^3*c^5)^(1/2)*
(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^3*c^5)^(1/2
))/(4*a^2*c^4))^(1/2))/(48*a*c^2*d^5*e^3 - 16*a^3*d*e^7 + 32*a^2*c*d^3*e^5 - (16*a^2*e^8*(-a^3*c^5)^(1/2))/c^3
+ (48*d^4*e^4*(-a^3*c^5)^(1/2))/c + (32*a*d^2*e^6*(-a^3*c^5)^(1/2))/c^2) - (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*
((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^
(1/2))/(48*c^2*d^5*e^3 - 16*a^2*d*e^7 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3 + 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c
^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)))*(-(a*c^4*d^3 - a*e^3*(-a^3*c^5)^(1/2) - 3*a^2*c^3*d*e^2
+ 3*c*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^5))^(1/2) - 2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) -
d^3/(4*a*c) - (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7
- 48*c^2*d^5*e^3 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3 - 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48
*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)) + (32*d*e^5*(-a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c
) - (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 + 32*a*c
^2*d^3*e^5 + (16*a*e^8*(-a^3*c^5)^(1/2))/c^2 - 16*a^2*c*d*e^7 - (48*d^4*e^4*(-a^3*c^5)^(1/2))/a - (32*d^2*e^6*
(-a^3*c^5)^(1/2))/c) + (96*d^3*e^3*(-a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) - (e^3*(-
a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^3*d*e^7 - 48*a*c^2*d^5*e^3 -
32*a^2*c*d^3*e^5 - (16*a^2*e^8*(-a^3*c^5)^(1/2))/c^3 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/c + (32*a*d^2*e^6*(-a^3*c
^5)^(1/2))/c^2) - (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) - (e^3*(-a^3*c^5)^(1/2))/
(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 - (16*a*e^8*(-a^3*c^
5)^(1/2))/c^3 - 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)))*(-(
a*c^4*d^3 + a*e^3*(-a^3*c^5)^(1/2) - 3*a^2*c^3*d*e^2 - 3*c*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^5))^(1/2) + (2*e*(
d + e*x)^(1/2))/c