\(\int \frac {\sqrt {d+e x}}{a+c x^2} \, dx\) [620]

Optimal result

Integrand size = 19, antiderivative size = 478 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[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^(3/4)*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^(3/4)*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^(3/4)*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^(3/4)*2^(1/2)/(d*c^(1
/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {714, 1143, 648, 632, 212, 642} \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}} \]

[In]

Int[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^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (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^(3/4)*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^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - (e*Log[Sq
rt[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^(3/4)*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 714

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

Rule 1143

Int[(x_)^(m_.)/((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*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = (2 e) \text {Subst}\left (\int \frac {x^2}{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 {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^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \text {Subst}\left (\int \frac {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^{3/4} \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 c}+\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 c}+\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} c^{3/4} \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} c^{3/4} \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} c^{3/4} \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} c^{3/4} \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 )}{c}-\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 )}{c} \\ & = \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} c^{3/4} \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} c^{3/4} \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} c^{3/4} \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} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {-i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )+i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} c} \]

[In]

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

[Out]

((-I)*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)] + I*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[a]*c)

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {\left (\frac {\ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{4}-\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right ) \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{4}+\left (c d +\sqrt {\left (e^{2} a +c \,d^{2}\right ) c}\right ) \left (\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right )\right ) \left (-c d +\sqrt {\left (e^{2} a +c \,d^{2}\right ) c}\right )}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, c^{\frac {3}{2}} a e}\)	\(508\)
derivativedivides	\(2 e \left (-\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}+\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (-\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}-\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}\right )\)	\(550\)
default	\(2 e \left (-\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}+\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (-\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}-\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}\right )\)	\(550\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=-\frac {1}{2} \, \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) \]

[In]

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

[Out]

-1/2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*log(a*c^2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*sqrt(-e^2/(
a*c^3)) + sqrt(e*x + d)*e) + 1/2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*log(-a*c^2*sqrt(-(a*c*sqrt(-e^2/(a*
c^3)) + d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e) + 1/2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*log(a*c
^2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e) - 1/2*sqrt((a*c*sqrt(-e^2/(a
*c^3)) - d)/(a*c))*log(-a*c^2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e)

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\int \frac {\sqrt {d + e x}}{a + c x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\int { \frac {\sqrt {e x + d}}{c x^{2} + a} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {{\left (c d^{2} e {\left | c \right |} + a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d - \sqrt {-a c} c e} {\left (a c e + \sqrt {-a c} c d\right )} {\left | e \right |}} + \frac {{\left (c d^{2} e {\left | c \right |} + a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d + \sqrt {-a c} c e} {\left (a c e - \sqrt {-a c} c d\right )} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.75 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,a\,c^2\,e^4-16\,c^3\,d^2\,e^2\right )\,\sqrt {d+e\,x}+\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {-a^3\,c^3}+a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3+16\,a\,c\,e^5}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,a\,c^2\,e^4-16\,c^3\,d^2\,e^2\right )\,\sqrt {d+e\,x}-\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {-a^3\,c^3}-a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3+16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}} \]

[In]

int((d + e*x)^(1/2)/(a + c*x^2),x)

[Out]

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