Integrand size = 24, antiderivative size = 363 \[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {6 A c^{3/2} x \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (5 \sqrt {a} B+9 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}} \]
-2/5*A*(c*x^2+a)^(1/2)/a/e/(e*x)^(5/2)-2/3*B*(c*x^2+a)^(1/2)/a/e^2/(e*x)^( 3/2)+6/5*A*c*(c*x^2+a)^(1/2)/a^2/e^3/(e*x)^(1/2)-6/5*A*c^(3/2)*x*(c*x^2+a) ^(1/2)/a^2/e^3/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+6/5*A*c^(5/4)*(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)/a^(7/4)/e^3/(e*x) ^(1/2)/(c*x^2+a)^(1/2)-1/15*c^(3/4)*(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)+9*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)/a^(7/4)/e^3/(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 = 83, normalized size of antiderivative = 0.23 \[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=-\frac {2 x \sqrt {1+\frac {c x^2}{a}} \left (3 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},-\frac {c x^2}{a}\right )+5 B x \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt {a+c x^2}} \]
(-2*x*Sqrt[1 + (c*x^2)/a]*(3*A*Hypergeometric2F1[-5/4, 1/2, -1/4, -((c*x^2 )/a)] + 5*B*x*Hypergeometric2F1[-3/4, 1/2, 1/4, -((c*x^2)/a)]))/(15*(e*x)^ (7/2)*Sqrt[a + c*x^2])
Time = 0.49 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {553, 27, 553, 27, 553, 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.
| \(\displaystyle \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {2 \int -\frac {5 a B-3 A c x}{2 (e x)^{5/2} \sqrt {c x^2+a}}dx}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a B-3 A c x}{(e x)^{5/2} \sqrt {c x^2+a}}dx}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a c (9 A+5 B x)}{2 (e x)^{3/2} \sqrt {c x^2+a}}dx}{3 a e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \int \frac {9 A+5 B x}{(e x)^{3/2} \sqrt {c x^2+a}}dx}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {c \left (-\frac {2 \int -\frac {5 a B+9 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \left (\frac {\int \frac {5 a B+9 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {-\frac {c \left (\frac {\sqrt {x} \int \frac {5 a B+9 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 \sqrt {x} \int \frac {5 a B+9 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-9 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-9 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-9 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-9 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{a e \sqrt {e x}}-\frac {18 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{3 e}-\frac {10 B \sqrt {a+c x^2}}{3 e (e x)^{3/2}}}{5 a e}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}\) |
(-2*A*Sqrt[a + c*x^2])/(5*a*e*(e*x)^(5/2)) + ((-10*B*Sqrt[a + c*x^2])/(3*e *(e*x)^(3/2)) - (c*((-18*A*Sqrt[a + c*x^2])/(a*e*Sqrt[e*x]) + (2*Sqrt[x]*( -9*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]*Elliptic E[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 + 9*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])))/(a*e*Sqrt[e*x])))/(3*e))/(5*a*e)
3.5.65.3.1 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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
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 = 0.56 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {9 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}-18 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}-5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,x^{2}+18 A \,c^{2} x^{4}-10 a B c \,x^{3}+12 a A c \,x^{2}-10 a^{2} B x -6 A \,a^{2}}{15 x^{2} \sqrt {c \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{2}}\) | \(331\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-9 A c \,x^{2}+5 a B x +3 a A \right )}{15 a^{2} x^{2} e^{3} \sqrt {e x}}-\frac {c \left (\frac {5 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {9 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{15 a^{2} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(350\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {c e \,x^{3}+a e x}}{5 e^{4} a \,x^{3}}-\frac {2 B \sqrt {c e \,x^{3}+a e x}}{3 e^{4} a \,x^{2}}+\frac {6 \left (c e \,x^{2}+a e \right ) A c}{5 a^{2} e^{4} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 a \,e^{3} \sqrt {c e \,x^{3}+a e x}}-\frac {3 A c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 a^{2} e^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(392\) |
1/15/x^2*(9*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*x^2-18*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*x^2-5*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*x^2+ 18*A*c^2*x^4-10*a*B*c*x^3+12*a*A*c*x^2-10*a^2*B*x-6*A*a^2)/(c*x^2+a)^(1/2) /e^3/(e*x)^(1/2)/a^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {c e} B a x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9 \, \sqrt {c e} A c x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (9 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{15 \, a^{2} e^{4} x^{3}} \]
-2/15*(5*sqrt(c*e)*B*a*x^3*weierstrassPInverse(-4*a/c, 0, x) - 9*sqrt(c*e) *A*c*x^3*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - ( 9*A*c*x^2 - 5*B*a*x - 3*A*a)*sqrt(c*x^2 + a)*sqrt(e*x))/(a^2*e^4*x^3)
Result contains complex when optimal does not.
Time = 15.75 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.29 \[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=\frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
A*gamma(-5/4)*hyper((-5/4, 1/2), (-1/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqr t(a)*e**(7/2)*x**(5/2)*gamma(-1/4)) + B*gamma(-3/4)*hyper((-3/4, 1/2), (1/ 4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*e**(7/2)*x**(3/2)*gamma(1/4))
\[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{7/2}\,\sqrt {c\,x^2+a}} \,d x \]