Integrand size = 24, antiderivative size = 363 \[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=-\frac {2 a e \sqrt {e x} (5 A+7 B x) \sqrt {a+c x^2}}{105 c}-\frac {4 a^2 B e^2 x \sqrt {a+c x^2}}{15 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}+\frac {4 a^{9/4} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 a^{7/4} \left (7 \sqrt {a} B+5 A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/9*B*(e*x)^(3/2)*(c*x^2+a)^(3/2)/c+2/7*A*e*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c- 4/15*a^2*B*e^2*x*(c*x^2+a)^(1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-2 /105*a*e*(7*B*x+5*A)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c+4/15*a^(9/4)*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)))*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^(7/4)/ (e*x)^(1/2)/(c*x^2+a)^(1/2)-2/105*a^(7/4)*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))*(7*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)/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 = 10.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.33 \[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 e \sqrt {e x} \sqrt {a+c x^2} \left ((9 A+7 B x) \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}}-9 a A \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )-7 a B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{63 c \sqrt {1+\frac {c x^2}{a}}} \]
(2*e*Sqrt[e*x]*Sqrt[a + c*x^2]*((9*A + 7*B*x)*(a + c*x^2)*Sqrt[1 + (c*x^2) /a] - 9*a*A*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/a)] - 7*a*B*x*Hype rgeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/a)]))/(63*c*Sqrt[1 + (c*x^2)/a])
Time = 0.47 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.
| \(\displaystyle \int (e x)^{3/2} \sqrt {a+c x^2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {2 e \int \frac {3}{2} \sqrt {e x} (a B-3 A c x) \sqrt {c x^2+a}dx}{9 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \int \sqrt {e x} (a B-3 A c x) \sqrt {c x^2+a}dx}{3 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (-\frac {2 e \int -\frac {a c (3 A+7 B x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{7 c}-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \int \frac {(3 A+7 B x) \sqrt {c x^2+a}}{\sqrt {e x}}dx-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4}{15} a \int \frac {3 (5 A+7 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {2}{5} a \int \frac {5 A+7 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {2 a \sqrt {x} \int \frac {5 A+7 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \int \frac {5 A+7 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\left (\frac {7 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {7 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\left (\frac {7 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {7 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {7 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {7 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+7 B x)}{5 e}\right )-\frac {6}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\) |
(2*B*(e*x)^(3/2)*(a + c*x^2)^(3/2))/(9*c) - (e*((-6*A*Sqrt[e*x]*(a + c*x^2 )^(3/2))/7 + (a*e*((2*Sqrt[e*x]*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(5*e) + (4* a*Sqrt[x]*((-7*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]*Elli pticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^2]) ))/Sqrt[c] + ((5*A + (7*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))/(3* c)
3.5.35.3.1 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 = 0.44 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-35 B \,c^{3} x^{6}+15 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-45 A \,c^{3} x^{5}+42 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-21 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-49 a B \,c^{2} x^{4}-75 a A \,c^{2} x^{3}-14 a^{2} B c \,x^{2}-30 a^{2} A c x \right )}{315 x \sqrt {c \,x^{2}+a}\, c^{2}}\) | \(342\) |
| risch | \(\frac {2 \left (35 B c \,x^{3}+45 A c \,x^{2}+14 a B x +30 a A \right ) x \sqrt {c \,x^{2}+a}\, e^{2}}{315 c \sqrt {e x}}-\frac {2 a^{2} \left (\frac {5 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {7 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{2} \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(359\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{3} \sqrt {c e \,x^{3}+a e x}}{9}+\frac {2 A e \,x^{2} \sqrt {c e \,x^{3}+a e x}}{7}+\frac {4 B a e x \sqrt {c e \,x^{3}+a e x}}{45 c}+\frac {4 a A e \sqrt {c e \,x^{3}+a e x}}{21 c}-\frac {2 a^{2} A \,e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {2 B \,a^{2} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{15 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(402\) |
-2/315*e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^2*(-35*B*c^3*x^6+15*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)*EllipticF(((c*x+(-a*c)^(1/2))/( -a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2-45*A*c^3*x^5+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^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^3-49*a*B*c^2*x^4-75*a*A *c^2*x^3-14*a^2*B*c*x^2-30*a^2*A*c*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.29 \[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=-\frac {2 \, {\left (30 \, \sqrt {c e} A a^{2} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 42 \, \sqrt {c e} B a^{2} e {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (35 \, B c^{2} e x^{3} + 45 \, A c^{2} e x^{2} + 14 \, B a c e x + 30 \, A a c e\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{315 \, c^{2}} \]
-2/315*(30*sqrt(c*e)*A*a^2*e*weierstrassPInverse(-4*a/c, 0, x) - 42*sqrt(c *e)*B*a^2*e*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (35*B*c^2*e*x^3 + 45*A*c^2*e*x^2 + 14*B*a*c*e*x + 30*A*a*c*e)*sqrt(c*x^2 + a)*sqrt(e*x))/c^2
Result contains complex when optimal does not.
Time = 4.81 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.27 \[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {A \sqrt {a} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
A*sqrt(a)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2*e xp_polar(I*pi)/a)/(2*gamma(9/4)) + B*sqrt(a)*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))
\[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
\[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx=\int {\left (e\,x\right )}^{3/2}\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \]