Integrand size = 24, antiderivative size = 404 \[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {16 a^3 A e x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {8 a^2 \sqrt {e x} (13 a B-77 A c x) \sqrt {a+c x^2}}{3003 c}-\frac {4 a \sqrt {e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac {2 \sqrt {e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {16 a^{13/4} A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{13/4} \left (13 \sqrt {a} B-77 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3003 c^{5/4} \sqrt {e x} \sqrt {a+c x^2}} \]
-4/9009*a*(-385*A*c*x+39*B*a)*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c-2/2145*(-165*A *c*x+13*B*a)*(c*x^2+a)^(5/2)*(e*x)^(1/2)/c+2/15*B*(c*x^2+a)^(7/2)*(e*x)^(1 /2)/c+16/39*a^3*A*e*x*(c*x^2+a)^(1/2)/c^(1/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1 /2)-8/3003*a^2*(-77*A*c*x+13*B*a)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c-16/39*a^(1 3/4)*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)-8/3003*a^(13/4)*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)))*E llipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(13*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^(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 = 10.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.29 \[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {e x} \sqrt {a+c x^2} \left (B \left (a+c x^2\right )^3 \sqrt {1+\frac {c x^2}{a}}-a^3 B \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )+5 a^2 A c x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{15 c \sqrt {1+\frac {c x^2}{a}}} \]
(2*Sqrt[e*x]*Sqrt[a + c*x^2]*(B*(a + c*x^2)^3*Sqrt[1 + (c*x^2)/a] - a^3*B* Hypergeometric2F1[-5/2, 1/4, 5/4, -((c*x^2)/a)] + 5*a^2*A*c*x*Hypergeometr ic2F1[-5/2, 3/4, 7/4, -((c*x^2)/a)]))/(15*c*Sqrt[1 + (c*x^2)/a])
Time = 0.53 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {552, 27, 548, 27, 548, 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.
| \(\displaystyle \int \sqrt {e x} \left (a+c x^2\right )^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {2 e \int \frac {(a B-15 A c x) \left (c x^2+a\right )^{5/2}}{2 \sqrt {e x}}dx}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \int \frac {(a B-15 A c x) \left (c x^2+a\right )^{5/2}}{\sqrt {e x}}dx}{15 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {20}{143} a \int \frac {(13 a B-165 A c x) \left (c x^2+a\right )^{3/2}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \int \frac {(13 a B-165 A c x) \left (c x^2+a\right )^{3/2}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {4}{21} a \int \frac {3 (39 a B-385 A c x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \int \frac {(39 a B-385 A c x) \sqrt {c x^2+a}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4}{15} a \int \frac {15 (13 a B-77 A c x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (2 a \int \frac {13 a B-77 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {2 a \sqrt {x} \int \frac {13 a B-77 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \int \frac {13 a B-77 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (13 \sqrt {a} B-77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+77 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (13 \sqrt {a} B-77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (13 \sqrt {a} B-77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac {e \left (\frac {10}{143} a \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (13 \sqrt {a} B-77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+77 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{\sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (13 a B-77 A c x)}{e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{21 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{143 e}\right )}{15 c}\) |
(2*B*Sqrt[e*x]*(a + c*x^2)^(7/2))/(15*c) - (e*((2*Sqrt[e*x]*(13*a*B - 165* A*c*x)*(a + c*x^2)^(5/2))/(143*e) + (10*a*((2*Sqrt[e*x]*(39*a*B - 385*A*c* x)*(a + c*x^2)^(3/2))/(21*e) + (2*a*((2*Sqrt[e*x]*(13*a*B - 77*A*c*x)*Sqrt [a + c*x^2])/e + (4*a*Sqrt[x]*(77*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)/(S qrt[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)*(13*Sqrt[a]*B - 77*A*Sqrt[c])*(Sqr t[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Ar cTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[a + c*x^2])))/Sqrt[ e*x]))/7))/143))/(15*c)
3.5.51.3.1 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 = 0.69 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(\frac {2 \sqrt {e x}\, \left (3003 B \,c^{5} x^{9}+3465 A \,c^{5} x^{8}+11739 B a \,c^{4} x^{7}+14245 A a \,c^{4} x^{6}+9240 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4} c -4620 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4} c -780 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4}+16809 B \,a^{2} c^{3} x^{5}+22715 A \,a^{2} c^{3} x^{4}+9633 B \,a^{3} c^{2} x^{3}+11935 A \,a^{3} c^{2} x^{2}+1560 B \,a^{4} c x \right )}{45045 \sqrt {c \,x^{2}+a}\, c^{2} x}\) | \(381\) |
| risch | \(\frac {2 \left (3003 B \,c^{3} x^{6}+3465 A \,c^{3} x^{5}+8736 a B \,c^{2} x^{4}+10780 a A \,c^{2} x^{3}+8073 a^{2} B c \,x^{2}+11935 a^{2} A c x +1560 B \,a^{3}\right ) x \sqrt {c \,x^{2}+a}\, e}{45045 c \sqrt {e x}}+\frac {8 a^{3} \left (-\frac {13 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {77 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) e \sqrt {\left (c \,x^{2}+a \right ) e x}}{3003 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(392\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,c^{2} x^{6} \sqrt {c e \,x^{3}+a e x}}{15}+\frac {2 A \,c^{2} x^{5} \sqrt {c e \,x^{3}+a e x}}{13}+\frac {64 B a c \,x^{4} \sqrt {c e \,x^{3}+a e x}}{165}+\frac {56 A a c \,x^{3} \sqrt {c e \,x^{3}+a e x}}{117}+\frac {138 B \,a^{2} x^{2} \sqrt {c e \,x^{3}+a e x}}{385}+\frac {62 A \,a^{2} x \sqrt {c e \,x^{3}+a e x}}{117}+\frac {16 B \,a^{3} \sqrt {c e \,x^{3}+a e x}}{231 c}-\frac {8 B \,a^{4} e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{2} \sqrt {c e \,x^{3}+a e x}}+\frac {8 A \,a^{3} e \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{39 c \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(465\) |
2/45045*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(3003*B*c^5*x^9+3465*A*c^5*x^8+11739*B *a*c^4*x^7+14245*A*a*c^4*x^6+9240*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^4*c-46 20*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^4*c-780*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^4+16809*B*a^2*c^3*x^5+22715*A*a^2*c^3*x^4+9633*B*a^3*c^2 *x^3+11935*A*a^3*c^2*x^2+1560*B*a^4*c*x)/c^2/x
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.34 \[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=-\frac {2 \, {\left (1560 \, \sqrt {c e} B a^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 9240 \, \sqrt {c e} A a^{3} c {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3003 \, B c^{4} x^{6} + 3465 \, A c^{4} x^{5} + 8736 \, B a c^{3} x^{4} + 10780 \, A a c^{3} x^{3} + 8073 \, B a^{2} c^{2} x^{2} + 11935 \, A a^{2} c^{2} x + 1560 \, B a^{3} c\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{45045 \, c^{2}} \]
-2/45045*(1560*sqrt(c*e)*B*a^4*weierstrassPInverse(-4*a/c, 0, x) + 9240*sq rt(c*e)*A*a^3*c*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (3003*B*c^4*x^6 + 3465*A*c^4*x^5 + 8736*B*a*c^3*x^4 + 10780*A*a*c^3* x^3 + 8073*B*a^2*c^2*x^2 + 11935*A*a^2*c^2*x + 1560*B*a^3*c)*sqrt(c*x^2 + a)*sqrt(e*x))/c^2
Result contains complex when optimal does not.
Time = 11.31 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.75 \[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {A a^{\frac {5}{2}} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {A a^{\frac {3}{2}} c \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {11}{4}\right )} + \frac {A \sqrt {a} c^{2} \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} + \frac {B a^{\frac {5}{2}} \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {B a^{\frac {3}{2}} c \sqrt {e} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} c^{2} \sqrt {e} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \]
A*a**(5/2)*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*e xp_polar(I*pi)/a)/(2*gamma(7/4)) + A*a**(3/2)*c*sqrt(e)*x**(7/2)*gamma(7/4 )*hyper((-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/gamma(11/4) + A*sq rt(a)*c**2*sqrt(e)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x* *2*exp_polar(I*pi)/a)/(2*gamma(15/4)) + B*a**(5/2)*sqrt(e)*x**(5/2)*gamma( 5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(9/4)) + B*a**(3/2)*c*sqrt(e)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x* *2*exp_polar(I*pi)/a)/gamma(13/4) + B*sqrt(a)*c**2*sqrt(e)*x**(13/2)*gamma (13/4)*hyper((-1/2, 13/4), (17/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(17/ 4))
\[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \sqrt {e x} \,d x } \]
\[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \sqrt {e x} \,d x } \]
Timed out. \[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int \sqrt {e\,x}\,{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]