Integrand size = 24, antiderivative size = 373 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]
1/3*(B*x+A)/a/e/(c*x^2+a)^(3/2)/(e*x)^(1/2)+1/6*(5*B*x+7*A)/a^2/e/(e*x)^(1 /2)/(c*x^2+a)^(1/2)-7/2*A*(c*x^2+a)^(1/2)/a^3/e/(e*x)^(1/2)+7/2*A*x*c^(1/2 )*(c*x^2+a)^(1/2)/a^3/e/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-7/2*A*c^(1/4)*(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^(11/4) /e/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/12*(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)/a^(11/4)/c^(1/4 )/e/(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 = 10.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {x \left (9 a A+7 a B x+7 A c x^2+5 B c x^3-21 A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c x^2}{a}\right )+5 B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )\right )}{6 a^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]
(x*(9*a*A + 7*a*B*x + 7*A*c*x^2 + 5*B*c*x^3 - 21*A*(a + c*x^2)*Sqrt[1 + (c *x^2)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((c*x^2)/a)] + 5*B*x*(a + c*x^ 2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)]))/(6 *a^2*(e*x)^(3/2)*(a + c*x^2)^(3/2))
Time = 0.50 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {551, 27, 551, 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.
| \(\displaystyle \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}-\frac {\int -\frac {7 A+5 B x}{2 (e x)^{3/2} \left (c x^2+a\right )^{3/2}}dx}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 A+5 B x}{(e x)^{3/2} \left (c x^2+a\right )^{3/2}}dx}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {\int -\frac {21 A+5 B x}{2 (e x)^{3/2} \sqrt {c x^2+a}}dx}{a}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {21 A+5 B x}{(e x)^{3/2} \sqrt {c x^2+a}}dx}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {5 a B+21 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {5 a B+21 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {x} \int \frac {5 a B+21 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {x} \int \frac {5 a B+21 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-21 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-21 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-21 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-21 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{a e \sqrt {e x}}-\frac {42 A \sqrt {a+c x^2}}{a e \sqrt {e x}}}{2 a}+\frac {7 A+5 B x}{a e \sqrt {e x} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}\) |
(A + B*x)/(3*a*e*Sqrt[e*x]*(a + c*x^2)^(3/2)) + ((7*A + 5*B*x)/(a*e*Sqrt[e *x]*Sqrt[a + c*x^2]) + ((-42*A*Sqrt[a + c*x^2])/(a*e*Sqrt[e*x]) + (2*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]*Ell ipticE[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*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]))/(2*a))/(6*a)
3.5.82.3.1 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 + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
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 = 2.32 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.17
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (-\frac {A x}{3 a^{2} e^{2} c}+\frac {B}{3 a \,e^{2} c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 A x}{4 a^{3} e^{2}}-\frac {5 B}{12 a^{2} e^{2} c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {2 \left (c e \,x^{2}+a e \right ) A}{a^{3} e^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} e c \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a^{3} e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(437\) |
| default | \(-\frac {21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+42 A \,c^{3} x^{4}-10 a B \,c^{2} x^{3}+70 a A \,c^{2} x^{2}-14 a^{2} B c x +24 A \,a^{2} c}{12 a^{3} e \sqrt {e x}\, c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(605\) |
| risch | \(-\frac {2 A \sqrt {c \,x^{2}+a}}{a^{3} e \sqrt {e x}}+\frac {\left (\frac {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}}-A a c \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {\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 )}{2 a c \sqrt {c e \,x^{3}+a e x}}\right )-a^{2} \left (\frac {\left (\frac {A x}{3 a e c}-\frac {B}{3 e \,c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 c e x \left (-\frac {A x}{4 a^{2} e}+\frac {5 B}{12 a e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {5 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 )}{12 a c \sqrt {c e \,x^{3}+a e x}}-\frac {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 )}{4 a^{2} \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{a^{3} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(779\) |
((c*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(c*x^2+a)^(1/2)*((-1/3/a^2/e^2/c*A*x+1/3 /a/e^2/c^2*B)*(c*e*x^3+a*e*x)^(1/2)/(x^2+a/c)^2-2*x*e*c*(3/4/a^3/e^2*A*x-5 /12/a^2/e^2*B/c)/((x^2+a/c)*x*e*c)^(1/2)-2*(c*e*x^2+a*e)/a^3/e^2*A/(x*(c*e *x^2+a*e))^(1/2)+5/12/a^2*B/e*(-a*c)^(1/2)/c*((x+(-a*c)^(1/2)/c)/(-a*c)^(1 /2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-x/(-a*c)^(1/2) *c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2) *c)^(1/2),1/2*2^(1/2))+7/4/a^3*A/e*(-a*c)^(1/2)*((x+(-a*c)^(1/2)/c)/(-a*c) ^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-x/(-a*c)^(1 /2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*(-2*(-a*c)^(1/2)/c*EllipticE(((x+(-a*c) ^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+ (-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (B a c^{2} x^{5} + 2 \, B a^{2} c x^{3} + B a^{3} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 21 \, {\left (A c^{3} x^{5} + 2 \, A a c^{2} x^{3} + A a^{2} c x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (21 \, A c^{3} x^{4} - 5 \, B a c^{2} x^{3} + 35 \, A a c^{2} x^{2} - 7 \, B a^{2} c x + 12 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{3} c^{3} e^{2} x^{5} + 2 \, a^{4} c^{2} e^{2} x^{3} + a^{5} c e^{2} x\right )}} \]
1/6*(5*(B*a*c^2*x^5 + 2*B*a^2*c*x^3 + B*a^3*x)*sqrt(c*e)*weierstrassPInver se(-4*a/c, 0, x) - 21*(A*c^3*x^5 + 2*A*a*c^2*x^3 + A*a^2*c*x)*sqrt(c*e)*we ierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (21*A*c^3*x^ 4 - 5*B*a*c^2*x^3 + 35*A*a*c^2*x^2 - 7*B*a^2*c*x + 12*A*a^2*c)*sqrt(c*x^2 + a)*sqrt(e*x))/(a^3*c^3*e^2*x^5 + 2*a^4*c^2*e^2*x^3 + a^5*c*e^2*x)
Result contains complex when optimal does not.
Time = 105.88 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
A*gamma(-1/4)*hyper((-1/4, 5/2), (3/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**( 5/2)*e**(3/2)*sqrt(x)*gamma(3/4)) + B*sqrt(x)*gamma(1/4)*hyper((1/4, 5/2), (5/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*e**(3/2)*gamma(5/4))
\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{5/2}} \,d x \]