Integrand size = 24, antiderivative size = 397 \[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=-\frac {4 a^2 A e^3 x \sqrt {a+c x^2}}{15 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 a e^2 \sqrt {e x} (25 a B-77 A c x) \sqrt {a+c x^2}}{1155 c^2}-\frac {10 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {4 a^{9/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a^{9/4} \left (25 \sqrt {a} B-77 A \sqrt {c}\right ) e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/9*A*e*(e*x)^(3/2)*(c*x^2+a)^(3/2)/c+2/11*B*(e*x)^(5/2)*(c*x^2+a)^(3/2)/c -10/77*a*B*e^2*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c^2-4/15*a^2*A*e^3*x*(c*x^2+a)^ (1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+2/1155*a*e^2*(-77*A*c*x+25*B *a)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c^2+4/15*a^(9/4)*A*e^3*(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)))*Ellip ticE(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^(7/4)/(e*x)^(1/2)/(c* x^2+a)^(1/2)+2/1155*a^(9/4)*e^3*(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))*(25*B*a^(1/2)-77*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^(9/4)/(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 = 10.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.34 \[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 e^2 \sqrt {e x} \sqrt {a+c x^2} \left (-\left (\left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} (45 a B-7 c x (11 A+9 B x))\right )+45 a^2 B \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )-77 a A c x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{693 c^2 \sqrt {1+\frac {c x^2}{a}}} \]
(2*e^2*Sqrt[e*x]*Sqrt[a + c*x^2]*(-((a + c*x^2)*Sqrt[1 + (c*x^2)/a]*(45*a* B - 7*c*x*(11*A + 9*B*x))) + 45*a^2*B*Hypergeometric2F1[-1/2, 1/4, 5/4, -( (c*x^2)/a)] - 77*a*A*c*x*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/a)])) /(693*c^2*Sqrt[1 + (c*x^2)/a])
Time = 0.52 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {552, 27, 552, 27, 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.
| \(\displaystyle \int (e x)^{5/2} \sqrt {a+c x^2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {2 e \int \frac {1}{2} (e x)^{3/2} (5 a B-11 A c x) \sqrt {c x^2+a}dx}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \int (e x)^{3/2} (5 a B-11 A c x) \sqrt {c x^2+a}dx}{11 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (-\frac {2 e \int -\frac {3}{2} a c \sqrt {e x} (11 A+15 B x) \sqrt {c x^2+a}dx}{9 c}-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \int \sqrt {e x} (11 A+15 B x) \sqrt {c x^2+a}dx-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {2 e \int \frac {(15 a B-77 A c x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \int \frac {(15 a B-77 A c x) \sqrt {c x^2+a}}{\sqrt {e x}}dx}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4}{15} a \int \frac {3 (25 a B-77 A c x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {2}{5} a \int \frac {25 a B-77 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {2 a \sqrt {x} \int \frac {25 a B-77 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \int \frac {25 a B-77 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B-77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+77 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B-77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B-77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {e \left (\frac {1}{3} a e \left (\frac {30 B \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {e \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B-77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+77 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B-77 A c x)}{5 e}\right )}{7 c}\right )-\frac {22}{9} A (e x)^{3/2} \left (a+c x^2\right )^{3/2}\right )}{11 c}\) |
(2*B*(e*x)^(5/2)*(a + c*x^2)^(3/2))/(11*c) - (e*((-22*A*(e*x)^(3/2)*(a + c *x^2)^(3/2))/9 + (a*e*((30*B*Sqrt[e*x]*(a + c*x^2)^(3/2))/(7*c) - (e*((2*S qrt[e*x]*(25*a*B - 77*A*c*x)*Sqrt[a + c*x^2])/(5*e) + (4*a*Sqrt[x]*(77*A*S qrt[c]*(-((Sqrt[x]*Sqrt[a + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqr t[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*Ar cTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^2])) + (a^(1/ 4)*(25*Sqrt[a]*B - 77*A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(S qrt[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])))/(5*Sqrt[e*x])))/(7*c)))/3))/(11*c)
3.5.34.3.1 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 = 0.53 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {2 e^{2} \sqrt {e x}\, \left (315 B \,c^{4} x^{7}+385 A \,c^{4} x^{6}+231 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3} c -462 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3} c +75 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+405 a B \,c^{3} x^{5}+539 a A \,c^{3} x^{4}-60 a^{2} B \,c^{2} x^{3}+154 a^{2} A \,c^{2} x^{2}-150 a^{3} B c x \right )}{3465 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(360\) |
| risch | \(\frac {2 \left (315 B \,c^{2} x^{4}+385 A \,c^{2} x^{3}+90 a B c \,x^{2}+154 a A c x -150 B \,a^{2}\right ) x \sqrt {c \,x^{2}+a}\, e^{3}}{3465 c^{2} \sqrt {e x}}-\frac {2 a^{2} \left (-\frac {25 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {77 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) e^{3} \sqrt {\left (c \,x^{2}+a \right ) e x}}{1155 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(372\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{2} x^{4} \sqrt {c e \,x^{3}+a e x}}{11}+\frac {2 A \,e^{2} x^{3} \sqrt {c e \,x^{3}+a e x}}{9}+\frac {4 B a \,e^{2} x^{2} \sqrt {c e \,x^{3}+a e x}}{77 c}+\frac {4 A a \,e^{2} x \sqrt {c e \,x^{3}+a e x}}{45 c}-\frac {20 B \,a^{2} e^{2} \sqrt {c e \,x^{3}+a e x}}{231 c^{2}}+\frac {10 B \,a^{3} e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{3} \sqrt {c e \,x^{3}+a e x}}-\frac {2 A \,a^{2} e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{15 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(438\) |
2/3465*e^2/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(315*B*c^4*x^7+385*A*c^4*x^6+231* 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^3*c-462*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^3*c+75*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^3+405*a*B*c^3*x^5+ 539*a*A*c^3*x^4-60*a^2*B*c^2*x^3+154*a^2*A*c^2*x^2-150*a^3*B*c*x)/c^3
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.34 \[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (150 \, \sqrt {c e} B a^{3} e^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 462 \, \sqrt {c e} A a^{2} c e^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (315 \, B c^{3} e^{2} x^{4} + 385 \, A c^{3} e^{2} x^{3} + 90 \, B a c^{2} e^{2} x^{2} + 154 \, A a c^{2} e^{2} x - 150 \, B a^{2} c e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{3465 \, c^{3}} \]
2/3465*(150*sqrt(c*e)*B*a^3*e^2*weierstrassPInverse(-4*a/c, 0, x) + 462*sq rt(c*e)*A*a^2*c*e^2*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (315*B*c^3*e^2*x^4 + 385*A*c^3*e^2*x^3 + 90*B*a*c^2*e^2*x^2 + 15 4*A*a*c^2*e^2*x - 150*B*a^2*c*e^2)*sqrt(c*x^2 + a)*sqrt(e*x))/c^3
Result contains complex when optimal does not.
Time = 15.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.24 \[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {A \sqrt {a} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B \sqrt {a} e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \]
A*sqrt(a)*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2* exp_polar(I*pi)/a)/(2*gamma(11/4)) + B*sqrt(a)*e**(5/2)*x**(9/2)*gamma(9/4 )*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(13/4))
\[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}} \,d x } \]
\[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (e x)^{5/2} (A+B x) \sqrt {a+c x^2} \, dx=\int {\left (e\,x\right )}^{5/2}\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \]