Integrand size = 24, antiderivative size = 328 \[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\frac {4 a A e x \sqrt {a+c x^2}}{5 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt {e x} (5 a B-21 A c x) \sqrt {a+c x^2}}{105 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {4 a^{5/4} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 a^{5/4} \left (5 \sqrt {a} B-21 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/7*B*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c+4/5*a*A*e*x*(c*x^2+a)^(1/2)/c^(1/2)/(a ^(1/2)+x*c^(1/2))/(e*x)^(1/2)-2/105*(-21*A*c*x+5*B*a)*(e*x)^(1/2)*(c*x^2+a )^(1/2)/c-4/5*a^(5/4)*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*arctan(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)-2/105*a^(5/4)*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) )*(5*B*a^(1/2)-21*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 = 10.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.34 \[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 \sqrt {e x} \sqrt {a+c x^2} \left (3 B \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}}-3 a B \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )+7 A c x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{21 c \sqrt {1+\frac {c x^2}{a}}} \]
(2*Sqrt[e*x]*Sqrt[a + c*x^2]*(3*B*(a + c*x^2)*Sqrt[1 + (c*x^2)/a] - 3*a*B* Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/a)] + 7*A*c*x*Hypergeometric2F 1[-1/2, 3/4, 7/4, -((c*x^2)/a)]))/(21*c*Sqrt[1 + (c*x^2)/a])
Time = 0.44 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {552, 27, 548, 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.
| \(\displaystyle \int \sqrt {e x} \sqrt {a+c x^2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {2 e \int \frac {(a B-7 A c x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \int \frac {(a B-7 A c x) \sqrt {c x^2+a}}{\sqrt {e x}}dx}{7 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4}{15} a \int \frac {5 a B-21 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {2}{15} a \int \frac {5 a B-21 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {2 a \sqrt {x} \int \frac {5 a B-21 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \int \frac {5 a B-21 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+21 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+21 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (21 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+21 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 a B-21 A c x)}{15 e}\right )}{7 c}\) |
(2*B*Sqrt[e*x]*(a + c*x^2)^(3/2))/(7*c) - (e*((2*Sqrt[e*x]*(5*a*B - 21*A*c *x)*Sqrt[a + c*x^2])/(15*e) + (4*a*Sqrt[x]*(21*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)*Sqrt[x])/a ^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^2])) + (a^(1/4)*(5*Sqrt[a]*B - 21*A*S qrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*El lipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[a + c*x ^2])))/(15*Sqrt[e*x])))/(7*c)
3.5.36.3.1 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[(e*x)^(m + 1)*(c*(m + 2*p + 2) + d*(m + 2*p + 1)*x)*((a + b*x^ 2)^p/(e*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*a*(p/((m + 2*p + 1)*(m + 2*p + 2))) Int[(e*x)^m*(a + b*x^2)^(p - 1)*(c*(m + 2*p + 2) + d*(m + 2*p + 1)*x), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[ p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.38 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(\frac {2 \sqrt {e x}\, \left (42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+15 B \,c^{3} x^{5}+21 A \,c^{3} x^{4}+25 a B \,c^{2} x^{3}+21 a A \,c^{2} x^{2}+10 a^{2} B c x \right )}{105 \sqrt {c \,x^{2}+a}\, x \,c^{2}}\) | \(333\) |
| risch | \(\frac {2 \left (15 B c \,x^{2}+21 A c x +10 B a \right ) x \sqrt {c \,x^{2}+a}\, e}{105 c \sqrt {e x}}+\frac {2 a \left (-\frac {5 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {21 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) e \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(344\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,x^{2} \sqrt {c e \,x^{3}+a e x}}{7}+\frac {2 A x \sqrt {c e \,x^{3}+a e x}}{5}+\frac {4 B a \sqrt {c e \,x^{3}+a e x}}{21 c}-\frac {2 B \,a^{2} e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 c^{2} \sqrt {c e \,x^{3}+a e x}}+\frac {2 a A e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 c \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(369\) |
2/105*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(42*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^2* c-21*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^2*c-5*B*(-a*c)^(1/2)*((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^2+15*B*c^3*x^5+21*A*c^3*x^4+25*a*B*c^2*x^3+21*a*A*c^2*x^ 2+10*a^2*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 = 91, normalized size of antiderivative = 0.28 \[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=-\frac {2 \, {\left (10 \, \sqrt {c e} B a^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 42 \, \sqrt {c e} A a c {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (15 \, B c^{2} x^{2} + 21 \, A c^{2} x + 10 \, B a c\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{105 \, c^{2}} \]
-2/105*(10*sqrt(c*e)*B*a^2*weierstrassPInverse(-4*a/c, 0, x) + 42*sqrt(c*e )*A*a*c*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (1 5*B*c^2*x^2 + 21*A*c^2*x + 10*B*a*c)*sqrt(c*x^2 + a)*sqrt(e*x))/c^2
Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.30 \[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\frac {A \sqrt {a} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
A*sqrt(a)*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*ex p_polar(I*pi)/a)/(2*gamma(7/4)) + B*sqrt(a)*sqrt(e)*x**(5/2)*gamma(5/4)*hy per((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(9/4))
\[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x} \,d x } \]
\[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x} \,d x } \]
Timed out. \[ \int \sqrt {e x} (A+B x) \sqrt {a+c x^2} \, dx=\int \sqrt {e\,x}\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \]