3.7.21 \(\int \frac {1}{\sqrt {d+e x} (a+c x^2)} \, dx\) [621]

Optimal. Leaf size=538 \[ \frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

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^(1/4)*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^(1/4)*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^(1/4)*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^(1/4)*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)

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Rubi [A]
time = 0.32, antiderivative size = 538, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {722, 1108, 648, 632, 212, 642} \begin {gather*} -\frac {e \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} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \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} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \tanh ^{-1}\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} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \tanh ^{-1}\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} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(a + c*x^2)),x]

[Out]

(e*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^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*ArcTanh[(Sqrt[Sq
rt[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^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*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^(1/4)*Sqrt[c*d^2
+ a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*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^(1/4)*Sqrt[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 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx &=(2 e) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=\frac {e \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {e \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 \sqrt {c} \sqrt {c d^2+a e^2}}+\frac {e \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 \sqrt {c} \sqrt {c d^2+a e^2}}-\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \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 )}{\sqrt {c} \sqrt {c d^2+a e^2}}-\frac {e \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 )}{\sqrt {c} \sqrt {c d^2+a e^2}}\\ &=\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 171, normalized size = 0.32 \begin {gather*} \frac {i \left (\frac {\tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {\tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)),x]

[Out]

(I*(ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)]/Sqrt[-(c*d) - I*Sqrt[
a]*Sqrt[c]*e] - ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)]/Sqrt[-(c*
d) + I*Sqrt[a]*Sqrt[c]*e]))/Sqrt[a]

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Maple [A]
time = 0.45, size = 796, normalized size = 1.48

method	result	size
derivativedivides	\(2 e \left (\frac {-\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \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 )}{2 \sqrt {c}}+\frac {2 \left (-2 \sqrt {c}\, a \,e^{2}+\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{2 \sqrt {c}}\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 )}{\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}}}{4 \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}\, a \,e^{2}}+\frac {\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \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 )}{2 \sqrt {c}}+\frac {2 \left (2 \sqrt {c}\, a \,e^{2}-\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{2 \sqrt {c}}\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 )}{\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}}}{4 \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}\, a \,e^{2}}\right )\)	\(796\)
default	\(2 e \left (\frac {-\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \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 )}{2 \sqrt {c}}+\frac {2 \left (-2 \sqrt {c}\, a \,e^{2}+\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{2 \sqrt {c}}\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 )}{\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}}}{4 \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}\, a \,e^{2}}+\frac {\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \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 )}{2 \sqrt {c}}+\frac {2 \left (2 \sqrt {c}\, a \,e^{2}-\frac {\left (-\sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\, c d +\sqrt {a c \,e^{2}+d^{2} c^{2}}\, \sqrt {2 \sqrt {a c \,e^{2}+d^{2} c^{2}}+2 c d}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{2 \sqrt {c}}\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 )}{\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}}}{4 \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}\, a \,e^{2}}\right )\)	\(796\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+a)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*sqrt(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (415) = 830\).
time = 1.45, size = 889, normalized size = 1.65 \begin {gather*} \frac {1}{2} \, \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} \log \left ({\left (a e^{2} + {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} + \sqrt {x e + d} e\right ) - \frac {1}{2} \, \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} \log \left (-{\left (a e^{2} + {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} + \sqrt {x e + d} e\right ) + \frac {1}{2} \, \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} \log \left ({\left (a e^{2} - {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} + \sqrt {x e + d} e\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} \log \left (-{\left (a e^{2} - {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} + \sqrt {x e + d} e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right ) \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+a)/(e*x+d)**(1/2),x)

[Out]

Integral(1/((a + c*x**2)*sqrt(d + e*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [B]
time = 0.65, size = 1366, normalized size = 2.54 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {32\,a^2\,c^5\,d^2\,e^2\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}-\frac {32\,c^3\,e^2\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^2\,c^4\,d\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a\,c^3\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}+\frac {32\,a\,c^4\,d\,e^3\,\sqrt {-a^3\,c}\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}+a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}-2\,\mathrm {atanh}\left (\frac {32\,c^3\,e^2\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^2\,c^4\,d\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a\,c^3\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}-\frac {32\,a^2\,c^5\,d^2\,e^2\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}+\frac {32\,a\,c^4\,d\,e^3\,\sqrt {-a^3\,c}\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c}-a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)*(d + e*x)^(1/2)),x)

[Out]

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

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