Integrand size = 19, antiderivative size = 478 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {-i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )+i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} c} \]
((-I)*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*S qrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)] + I*Sqrt[-(c*d) + I*Sq rt[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)
Time = 0.74 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {483, 1449, 1142, 25, 27, 1083, 219, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx\) |
| \(\Big \downarrow \) 483 |
| \(\displaystyle 2 e \int \frac {d+e x}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 1449 |
| \(\displaystyle 2 e \left (\frac {\int \frac {\sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\int \frac {\sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 e \left (\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {1}{2} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 e \left (\frac {-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}-\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt [4]{c}}+\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 e \left (\frac {-\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {\int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 e \left (\frac {\frac {1}{2} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {1}{2} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\) |
2*e*((-((Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*ArcTanh[(c^(1/4)*(-((Sqrt[2 ]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])/c^(1/4)) + 2*Sqrt[d + e*x]))/(Sqr t[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])])/Sqrt[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 + Sq rt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)]/2)/(2*Sqrt[2]*c^(3/4 )*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - ((Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*ArcTanh[(c^(1/4)*((Sqrt[2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) /c^(1/4) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2] ])])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]] + Log[Sqrt[c*d^2 + a*e^2] + Sqr t[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)/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]))
3.7.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x ] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]}, Simp[1/(2*c*r) Int[x^(m - 1)/(q - r*x + x^2), x], x] - Simp[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]
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 | \(2 e \left (-\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}+\frac {\sqrt {2 \sqrt {a c \,e^{2}+c^{2} d^{2}}+2 c d}\, \left (-c d +\sqrt {a c \,e^{2}+c^{2} d^{2}}\right ) \left (\frac {\ln \left (-\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}-\sqrt {e^{2} a +c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )}{4 a c \,e^{2}}\right )\) | \(550\) |
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))-ar ctan((-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
Time = 0.41 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=-\frac {1}{2} \, \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt {-\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt {\frac {a c \sqrt {-\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {-\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) \]
-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)
\[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\int \frac {\sqrt {d + e x}}{a + c x^{2}}\, dx \]
\[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\int { \frac {\sqrt {e x + d}}{c x^{2} + a} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=\frac {{\left (c d^{2} e {\left | c \right |} + a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d - \sqrt {-a c} c e} {\left (a c e + \sqrt {-a c} c d\right )} {\left | e \right |}} + \frac {{\left (c d^{2} e {\left | c \right |} + a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d + \sqrt {-a c} c e} {\left (a c e - \sqrt {-a c} c d\right )} {\left | e \right |}} \]
(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 + sqr t(-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))
Time = 9.75 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d+e x}}{a+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,a\,c^2\,e^4-16\,c^3\,d^2\,e^2\right )\,\sqrt {d+e\,x}+\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {-a^3\,c^3}+a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3+16\,a\,c\,e^5}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,a\,c^2\,e^4-16\,c^3\,d^2\,e^2\right )\,\sqrt {d+e\,x}-\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {-a^3\,c^3}-a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3+16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}} \]
- 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))/(1 6*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)