Integrand size = 24, antiderivative size = 287 \[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}+\frac {2 A e x \sqrt {a+c x^2}}{\sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt [4]{a} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} B-3 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2*A*e*x*(c*x^2+a)^(1/2)/c^(1/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+2/3*B*(e*x )^(1/2)*(c*x^2+a)^(1/2)/c-2*a^(1/4)*A*e*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1 /4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arct an(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c* x^2+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(3/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/ 3*a^(1/4)*e*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan( c^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))) ,1/2*2^(1/2))*(B*a^(1/2)-3*A*c^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+ a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(5/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {e x} \left (B \left (a+c x^2\right )-a B \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+A c x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{3 c \sqrt {a+c x^2}} \]
(2*Sqrt[e*x]*(B*(a + c*x^2) - a*B*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/ 4, 1/2, 5/4, -((c*x^2)/a)] + A*c*x*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1 /2, 3/4, 7/4, -((c*x^2)/a)]))/(3*c*Sqrt[a + c*x^2])
Time = 0.39 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {552, 27, 556, 555, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \int \frac {a B-3 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {e \int \frac {a B-3 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{3 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {e \sqrt {x} \int \frac {a B-3 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{3 c \sqrt {e x}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \sqrt {x} \int \frac {a B-3 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{3 c \sqrt {e x}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+3 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{3 c \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{3 c \sqrt {e x}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \sqrt {x} \left (3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{3 c \sqrt {e x}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B \sqrt {e x} \sqrt {a+c x^2}}{3 c}-\frac {2 e \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+3 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{3 c \sqrt {e x}}\) |
(2*B*Sqrt[e*x]*Sqrt[a + c*x^2])/(3*c) - (2*e*Sqrt[x]*(3*A*Sqrt[c]*(-((Sqrt [x]*Sqrt[a + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]* x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*S qrt[x])/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^2])) + (a^(1/4)*(Sqrt[a]*B - 3*A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x) ^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[a + c*x^2])))/(3*c*Sqrt[e*x])
3.5.61.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free Q[{a, c, f, g}, x]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb ol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] , x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 0.22 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(-\frac {\sqrt {e x}\, \left (3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c -6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c +B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a -2 B \,c^{2} x^{3}-2 a B c x \right )}{3 \sqrt {c \,x^{2}+a}\, x \,c^{2}}\) | \(295\) |
| risch | \(\frac {2 B x \sqrt {c \,x^{2}+a}\, e}{3 c \sqrt {e x}}+\frac {\left (-\frac {B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {3 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) e \sqrt {\left (c \,x^{2}+a \right ) e x}}{3 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(327\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \sqrt {c e \,x^{3}+a e x}}{3 c}-\frac {B a e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 c^{2} \sqrt {c e \,x^{3}+a e x}}+\frac {A e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(328\) |
-1/3*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(3*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1 /2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^( 1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c-6* A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a *c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/( -a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c+B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1 /2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^( 1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*(-a*c) ^(1/2)*a-2*B*c^2*x^3-2*a*B*c*x)/x/c^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (\sqrt {c e} B a {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, \sqrt {c e} A c {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - \sqrt {c x^{2} + a} \sqrt {e x} B c\right )}}{3 \, c^{2}} \]
-2/3*(sqrt(c*e)*B*a*weierstrassPInverse(-4*a/c, 0, x) + 3*sqrt(c*e)*A*c*we ierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - sqrt(c*x^2 + a)*sqrt(e*x)*B*c)/c^2
Result contains complex when optimal does not.
Time = 2.84 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {A \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \]
A*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**2*exp_polar(I *pi)/a)/(2*sqrt(a)*gamma(7/4)) + B*sqrt(e)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(9/4))
\[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{\sqrt {c x^{2} + a}} \,d x } \]
\[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{\sqrt {c x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {e\,x}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \]