Integrand size = 20, antiderivative size = 134 \[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}} \]
-arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d *c^(1/2))^(1/2)/c^(3/4)/a^(1/2)+arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d *c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))^(1/2)/c^(3/4)/a^(1/2)
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=\frac {-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c} \]
(-(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]* e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)]) + Sqrt[-(c*d) + Sqrt[a]*Sqrt[c ]*e]*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*c)
Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {483, 25, 1450, 221}
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 |
\(\displaystyle 2 e \int -\frac {d+e x}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 e \int \frac {d+e x}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}\) |
\(\Big \downarrow \) 1450 |
\(\displaystyle 2 e \left (-\frac {1}{2} \left (1-\frac {\sqrt {c} d}{\sqrt {a} e}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}-\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 e}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 e \left (\frac {\left (1-\frac {\sqrt {c} d}{\sqrt {a} e}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 c^{3/4} e \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\) |
2*e*(((1 - (Sqrt[c]*d)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S qrt[c]*d - Sqrt[a]*e]])/(2*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (((Sqrt[ c]*d)/Sqrt[a] + e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a ]*e]])/(2*c^(3/4)*e*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))
3.7.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Time = 2.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {e \left (\frac {\left (-c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (c d +\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}}\) | \(127\) |
derivativedivides | \(-2 e c \left (\frac {\left (-c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (c d +\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(143\) |
default | \(-2 e c \left (\frac {\left (-c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (c d +\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(143\) |
-e/(a*c*e^2)^(1/2)*((-c*d+(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2 )*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))-(c*d+(a*c*e^2)^ (1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c* e^2)^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=\frac {1}{2} \, \sqrt {\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt {\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt {\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} + d}{a c}} \sqrt {\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt {-\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt {-\frac {a c \sqrt {\frac {e^{2}}{a c^{3}}} - d}{a c}} \sqrt {\frac {e^{2}}{a c^{3}}} + \sqrt {e x + d} e\right ) \]
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*s qrt(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)
\[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=- \int \frac {\sqrt {d + e x}}{- a + c x^{2}}\, dx \]
\[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=\int { -\frac {\sqrt {e x + d}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (94) = 188\).
Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=\frac {{\left (c d^{2} e {\left | c \right |} - a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d - \sqrt {a c} c e} {\left (a c e - \sqrt {a c} c d\right )} {\left | e \right |}} + \frac {{\left (c d^{2} e {\left | c \right |} - a e^{3} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{\sqrt {-c^{2} d + \sqrt {a c} c e} {\left (a c e + \sqrt {a c} c d\right )} {\left | e \right |}} \]
(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 - 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))
Time = 0.34 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )\,\sqrt {d+e\,x}-\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {a^3\,c^3}+a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {\frac {e\,\sqrt {a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3-16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )\,\sqrt {d+e\,x}+\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {a^3\,c^3}-a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {-\frac {e\,\sqrt {a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3-16\,a\,c\,e^5}\right )\,\sqrt {-\frac {e\,\sqrt {a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}} \]
- 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))/(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)