Integrand size = 24, antiderivative size = 368 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=-\frac {4 A c \sqrt {a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac {4 B c \sqrt {a+c x^2}}{5 a e^4 \sqrt {e x}}-\frac {2 (5 A+7 B x) \sqrt {a+c x^2}}{35 e (e x)^{7/2}}+\frac {4 B c^{3/2} x \sqrt {a+c x^2}}{5 a e^4 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 B c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^4 \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 \left (21 \sqrt {a} B-5 A \sqrt {c}\right ) c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{5/4} e^4 \sqrt {e x} \sqrt {a+c x^2}} \]
-4/21*A*c*(c*x^2+a)^(1/2)/a/e^3/(e*x)^(3/2)-2/35*(7*B*x+5*A)*(c*x^2+a)^(1/ 2)/e/(e*x)^(7/2)-4/5*B*c*(c*x^2+a)^(1/2)/a/e^4/(e*x)^(1/2)+4/5*B*c^(3/2)*x *(c*x^2+a)^(1/2)/a/e^4/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-4/5*B*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^(3/4)/e ^4/(e*x)^(1/2)/(c*x^2+a)^(1/2)+2/105*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)))*EllipticF(sin(2 *arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(21*B*a^(1/2)-5*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^(5/4 )/e^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 = 88, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=-\frac {2 \sqrt {e x} \sqrt {a+c x^2} \left (5 A \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},-\frac {c x^2}{a}\right )+7 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{35 e^5 x^4 \sqrt {1+\frac {c x^2}{a}}} \]
(-2*Sqrt[e*x]*Sqrt[a + c*x^2]*(5*A*Hypergeometric2F1[-7/4, -1/2, -3/4, -(( c*x^2)/a)] + 7*B*x*Hypergeometric2F1[-5/4, -1/2, -1/4, -((c*x^2)/a)]))/(35 *e^5*x^4*Sqrt[1 + (c*x^2)/a])
Time = 0.49 (sec) , antiderivative size = 358, 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 = {546, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (A+B x)}{(e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {4 c \int -\frac {5 A+7 B x}{2 (e x)^{5/2} \sqrt {c x^2+a}}dx}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \int \frac {5 A+7 B x}{(e x)^{5/2} \sqrt {c x^2+a}}dx}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 c \left (-\frac {2 \int -\frac {21 a B-5 A c x}{2 (e x)^{3/2} \sqrt {c x^2+a}}dx}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {\int \frac {21 a B-5 A c x}{(e x)^{3/2} \sqrt {c x^2+a}}dx}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 \int \frac {a c (5 A-21 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {c \int \frac {5 A-21 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{e}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {c \sqrt {x} \int \frac {5 A-21 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 c \sqrt {x} \int \frac {5 A-21 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 c \sqrt {x} \left (\left (5 A-\frac {21 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {21 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 c \sqrt {x} \left (\left (5 A-\frac {21 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {21 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 c \sqrt {x} \left (\frac {21 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}+\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 A-\frac {21 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 c \left (\frac {-\frac {2 c \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 A-\frac {21 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}+\frac {21 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{e \sqrt {e x}}-\frac {42 B \sqrt {a+c x^2}}{e \sqrt {e x}}}{3 a e}-\frac {10 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}\right )}{35 e^2}-\frac {2 \sqrt {a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
(-2*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(35*e*(e*x)^(7/2)) + (2*c*((-10*A*Sqrt[ a + c*x^2])/(3*a*e*(e*x)^(3/2)) + ((-42*B*Sqrt[a + c*x^2])/(e*Sqrt[e*x]) - (2*c*Sqrt[x]*((21*B*(-((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])))/Sqrt[c] + ((5*A - (21*Sqrt[a]*B)/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqr t[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x] )/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^2])))/(e*Sqrt[e*x]))/(3* a*e)))/(35*e^2)
3.5.41.3.1 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) + d*(m + 1)*x)*((a + b*x^2)^p/(e*(m + 1)*(m + 2))), x] - Simp[2*b*(p/(e^2*(m + 1)*(m + 2))) Int[(e*x)^(m + 2)* (a + b*x^2)^(p - 1)*(c*(m + 2) + d*(m + 1)*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[p, 0] && LtQ[m, -2] && !ILtQ[m + 2*p + 3, 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 = 0.88 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {2 \left (5 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) c \,x^{3}+21 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{3}-42 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{3}+42 B \,c^{2} x^{5}+10 A \,c^{2} x^{4}+63 a B c \,x^{3}+25 a A c \,x^{2}+21 a^{2} B x +15 A \,a^{2}\right )}{105 x^{3} \sqrt {c \,x^{2}+a}\, e^{4} \sqrt {e x}\, a}\) | \(340\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (42 B c \,x^{3}+10 A c \,x^{2}+21 a B x +15 a A \right )}{105 x^{3} a \,e^{4} \sqrt {e x}}-\frac {2 c^{2} \left (\frac {5 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}-\frac {21 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 a \,e^{4} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(361\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {c e \,x^{3}+a e x}}{7 e^{5} x^{4}}-\frac {2 B \sqrt {c e \,x^{3}+a e x}}{5 e^{5} x^{3}}-\frac {4 c A \sqrt {c e \,x^{3}+a e x}}{21 a \,e^{5} x^{2}}-\frac {4 \left (c e \,x^{2}+a e \right ) B c}{5 a \,e^{5} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {2 c A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 a \,e^{4} \sqrt {c e \,x^{3}+a e x}}+\frac {2 B c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 a \,e^{4} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(413\) |
-2/105/x^3*(5*A*(-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)*Elli pticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*c*x^3+21*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*x^3-42*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)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c*x ^3+42*B*c^2*x^5+10*A*c^2*x^4+63*a*B*c*x^3+25*a*A*c*x^2+21*a^2*B*x+15*A*a^2 )/(c*x^2+a)^(1/2)/e^4/(e*x)^(1/2)/a
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.27 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=-\frac {2 \, {\left (10 \, \sqrt {c e} A c x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 42 \, \sqrt {c e} B c x^{4} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (42 \, B c x^{3} + 10 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{105 \, a e^{5} x^{4}} \]
-2/105*(10*sqrt(c*e)*A*c*x^4*weierstrassPInverse(-4*a/c, 0, x) + 42*sqrt(c *e)*B*c*x^4*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (42*B*c*x^3 + 10*A*c*x^2 + 21*B*a*x + 15*A*a)*sqrt(c*x^2 + a)*sqrt(e*x)) /(a*e^5*x^4)
Result contains complex when optimal does not.
Time = 43.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.30 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \]
A*sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), c*x**2*exp_polar(I*pi)/ a)/(2*e**(9/2)*x**(7/2)*gamma(-3/4)) + B*sqrt(a)*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*x**(5/2)*gamma(-1/4) )
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\left (e x\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\left (e x\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{9/2}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{9/2}} \,d x \]