Integrand size = 24, antiderivative size = 400 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=-\frac {4 a^2 e \sqrt {e x} (65 A+77 B x) \sqrt {a+c x^2}}{5005 c}-\frac {8 a^3 B e^2 x \sqrt {a+c x^2}}{65 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 a e \sqrt {e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {8 a^{13/4} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 a^{11/4} \left (77 \sqrt {a} B+65 A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5005 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/13*B*(e*x)^(3/2)*(c*x^2+a)^(5/2)/c-2/3003*a*e*(77*B*x+39*A)*(c*x^2+a)^(3 /2)*(e*x)^(1/2)/c+2/11*A*e*(c*x^2+a)^(5/2)*(e*x)^(1/2)/c-8/65*a^3*B*e^2*x* (c*x^2+a)^(1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-4/5005*a^2*e*(77*B *x+65*A)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c+8/65*a^(13/4)*B*e^2*(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 llipticE(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)-4/5005*a^(11/4)*e^2*(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))*(77*B*a^(1/2)+65*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^(7/4)/(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 = 10.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.31 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {2 e \sqrt {e x} \sqrt {a+c x^2} \left ((13 A+11 B x) \left (a+c x^2\right )^2 \sqrt {1+\frac {c x^2}{a}}-13 a^2 A \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )-11 a^2 B x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{143 c \sqrt {1+\frac {c x^2}{a}}} \]
(2*e*Sqrt[e*x]*Sqrt[a + c*x^2]*((13*A + 11*B*x)*(a + c*x^2)^2*Sqrt[1 + (c* x^2)/a] - 13*a^2*A*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/a)] - 11*a^ 2*B*x*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/a)]))/(143*c*Sqrt[1 + (c *x^2)/a])
Time = 0.53 (sec) , antiderivative size = 388, 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, 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.
| \(\displaystyle \int (e x)^{3/2} \left (a+c x^2\right )^{3/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {2 e \int \frac {1}{2} \sqrt {e x} (3 a B-13 A c x) \left (c x^2+a\right )^{3/2}dx}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \int \sqrt {e x} (3 a B-13 A c x) \left (c x^2+a\right )^{3/2}dx}{13 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (-\frac {2 e \int -\frac {a c (13 A+33 B x) \left (c x^2+a\right )^{3/2}}{2 \sqrt {e x}}dx}{11 c}-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \int \frac {(13 A+33 B x) \left (c x^2+a\right )^{3/2}}{\sqrt {e x}}dx-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {4}{21} a \int \frac {3 (39 A+77 B x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \int \frac {(39 A+77 B x) \sqrt {c x^2+a}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4}{15} a \int \frac {3 (65 A+77 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {2}{5} a \int \frac {65 A+77 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {2 a \sqrt {x} \int \frac {65 A+77 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \int \frac {65 A+77 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\left (\frac {77 \sqrt {a} B}{\sqrt {c}}+65 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {77 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\left (\frac {77 \sqrt {a} B}{\sqrt {c}}+65 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {77 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {77 \sqrt {a} B}{\sqrt {c}}+65 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {77 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {2}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {77 \sqrt {a} B}{\sqrt {c}}+65 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {77 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 A+77 B x)}{5 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{21 e}\right )-\frac {26}{11} A \sqrt {e x} \left (a+c x^2\right )^{5/2}\right )}{13 c}\) |
(2*B*(e*x)^(3/2)*(a + c*x^2)^(5/2))/(13*c) - (e*((-26*A*Sqrt[e*x]*(a + c*x ^2)^(5/2))/11 + (a*e*((2*Sqrt[e*x]*(39*A + 77*B*x)*(a + c*x^2)^(3/2))/(21* e) + (2*a*((2*Sqrt[e*x]*(65*A + 77*B*x)*Sqrt[a + c*x^2])/(5*e) + (4*a*Sqrt [x]*((-77*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])))/Sq rt[c] + ((65*A + (77*Sqrt[a]*B)/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*a^(1/4)*c^(1/4)*Sqrt[a + c*x^2])))/(5*Sqrt[e*x])))/7))/11))/( 13*c)
3.5.43.3.1 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 = 0.50 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-1155 B \,c^{4} x^{8}-1365 A \,c^{4} x^{7}-3080 a B \,c^{3} x^{6}+390 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{3}-3900 a A \,c^{3} x^{5}+924 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-462 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-2233 a^{2} B \,c^{2} x^{4}-3315 a^{2} A \,c^{2} x^{3}-308 a^{3} B c \,x^{2}-780 a^{3} A c x \right )}{15015 x \sqrt {c \,x^{2}+a}\, c^{2}}\) | \(366\) |
| risch | \(\frac {2 \left (1155 B \,c^{2} x^{5}+1365 A \,c^{2} x^{4}+1925 a B c \,x^{3}+2535 a A c \,x^{2}+308 a^{2} B x +780 A \,a^{2}\right ) x \sqrt {c \,x^{2}+a}\, e^{2}}{15015 c \sqrt {e x}}-\frac {4 a^{3} \left (\frac {65 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {77 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{2} \sqrt {\left (c \,x^{2}+a \right ) e x}}{5005 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(383\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B c e \,x^{5} \sqrt {c e \,x^{3}+a e x}}{13}+\frac {2 A c e \,x^{4} \sqrt {c e \,x^{3}+a e x}}{11}+\frac {10 B a e \,x^{3} \sqrt {c e \,x^{3}+a e x}}{39}+\frac {26 a A e \,x^{2} \sqrt {c e \,x^{3}+a e x}}{77}+\frac {8 B \,a^{2} e x \sqrt {c e \,x^{3}+a e x}}{195 c}+\frac {8 A \,a^{2} e \sqrt {c e \,x^{3}+a e x}}{77 c}-\frac {4 A \,a^{3} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{77 c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {4 B \,a^{3} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{65 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(450\) |
-2/15015*e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^2*(-1155*B*c^4*x^8-1365*A*c^4*x ^7-3080*a*B*c^3*x^6+390*A*(-a*c)^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^( 1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*El lipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3-3 900*a*A*c^3*x^5+924*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*a^4-462*B*((c*x+(-a*c) ^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*a^4-2233*a^2*B*c^2*x^4-3315*a^2*A*c^2*x^3-308*a^3*B*c*x^ 2-780*a^3*A*c*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.33 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (780 \, \sqrt {c e} A a^{3} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 924 \, \sqrt {c e} B a^{3} e {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (1155 \, B c^{3} e x^{5} + 1365 \, A c^{3} e x^{4} + 1925 \, B a c^{2} e x^{3} + 2535 \, A a c^{2} e x^{2} + 308 \, B a^{2} c e x + 780 \, A a^{2} c e\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{15015 \, c^{2}} \]
-2/15015*(780*sqrt(c*e)*A*a^3*e*weierstrassPInverse(-4*a/c, 0, x) - 924*sq rt(c*e)*B*a^3*e*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (1155*B*c^3*e*x^5 + 1365*A*c^3*e*x^4 + 1925*B*a*c^2*e*x^3 + 2535*A*a *c^2*e*x^2 + 308*B*a^2*c*e*x + 780*A*a^2*c*e)*sqrt(c*x^2 + a)*sqrt(e*x))/c ^2
Result contains complex when optimal does not.
Time = 12.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.50 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {A a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {A \sqrt {a} c e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B \sqrt {a} c e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \]
A*a**(3/2)*e**(3/2)*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)) + A*sqrt(a)*c*e**(3/2)*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)) + B*a**(3/2)*e**(3/2)*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)*c*e**(3/2)*x**(11/2)*gamm a(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(15 /4))
\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int {\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]