Integrand size = 24, antiderivative size = 375 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=-\frac {8 c^2 (7 A-5 B x) \sqrt {a+c x^2}}{21 e^5 \sqrt {e x}}+\frac {16 A c^{5/2} x \sqrt {a+c x^2}}{3 e^5 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 c (7 A+15 B x) \left (a+c x^2\right )^{3/2}}{63 e^3 (e x)^{5/2}}-\frac {2 (7 A+9 B x) \left (a+c x^2\right )^{5/2}}{63 e (e x)^{9/2}}-\frac {16 \sqrt [4]{a} A c^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 e^5 \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 \sqrt [4]{a} \left (5 \sqrt {a} B+7 A \sqrt {c}\right ) c^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 e^5 \sqrt {e x} \sqrt {a+c x^2}} \]
-4/63*c*(15*B*x+7*A)*(c*x^2+a)^(3/2)/e^3/(e*x)^(5/2)-2/63*(9*B*x+7*A)*(c*x ^2+a)^(5/2)/e/(e*x)^(9/2)-8/21*c^2*(-5*B*x+7*A)*(c*x^2+a)^(1/2)/e^5/(e*x)^ (1/2)+16/3*A*c^(5/2)*x*(c*x^2+a)^(1/2)/e^5/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2) -16/3*a^(1/4)*A*c^(9/4)*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/c os(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)/e^5/(e*x)^(1/2)/(c*x^2+a)^(1/2)+8/21*a^(1/4)*c^(7/4)*( 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)) *(5*B*a^(1/2)+7*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)/e^5/(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 = 91, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=-\frac {2 a^2 \sqrt {e x} \sqrt {a+c x^2} \left (7 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {9}{4},-\frac {5}{4},-\frac {c x^2}{a}\right )+9 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {7}{4},-\frac {3}{4},-\frac {c x^2}{a}\right )\right )}{63 e^6 x^5 \sqrt {1+\frac {c x^2}{a}}} \]
(-2*a^2*Sqrt[e*x]*Sqrt[a + c*x^2]*(7*A*Hypergeometric2F1[-5/2, -9/4, -5/4, -((c*x^2)/a)] + 9*B*x*Hypergeometric2F1[-5/2, -7/4, -3/4, -((c*x^2)/a)])) /(63*e^6*x^5*Sqrt[1 + (c*x^2)/a])
Time = 0.50 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {546, 27, 546, 27, 547, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{(e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {20 c \int -\frac {(7 A+9 B x) \left (c x^2+a\right )^{3/2}}{2 (e x)^{7/2}}dx}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \int \frac {(7 A+9 B x) \left (c x^2+a\right )^{3/2}}{(e x)^{7/2}}dx}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle \frac {10 c \left (-\frac {4 c \int -\frac {3 (7 A+15 B x) \sqrt {c x^2+a}}{2 (e x)^{3/2}}dx}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \int \frac {(7 A+15 B x) \sqrt {c x^2+a}}{(e x)^{3/2}}dx}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 547 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (-\frac {4 \int -\frac {3 (5 a B+7 A c x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{3 e}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {2 \int \frac {5 a B+7 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{e}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {2 \sqrt {x} \int \frac {5 a B+7 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {4 \sqrt {x} \int \frac {5 a B+7 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {4 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+7 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-7 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {4 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+7 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-7 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {4 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+7 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-7 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {10 c \left (\frac {6 c \left (\frac {4 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+7 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-7 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (7 A-5 B x)}{e \sqrt {e x}}\right )}{5 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{5 e (e x)^{5/2}}\right )}{63 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}\) |
(-2*(7*A + 9*B*x)*(a + c*x^2)^(5/2))/(63*e*(e*x)^(9/2)) + (10*c*((-2*(7*A + 15*B*x)*(a + c*x^2)^(3/2))/(5*e*(e*x)^(5/2)) + (6*c*((-2*(7*A - 5*B*x)*S qrt[a + c*x^2])/(e*Sqrt[e*x]) + (4*Sqrt[x]*(-7*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)/(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])) + (a^(1/4)*(5*Sqrt[a]*B + 7*A*Sq rt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ell ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[a + c*x^ 2])))/(e*Sqrt[e*x])))/(5*e^2)))/(63*e^2)
3.5.57.3.1 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[(e*x)^(m + 1)*(c*(m + 2*p + 2) + d*(m + 1)*x)*((a + b*x^2)^p/( e*(m + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(e*(m + 1)*(m + 2*p + 2))) Int[ (e*x)^(m + 1)*(a*d*(m + 1) - b*c*(m + 2*p + 2)*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[m + 2* p + 1, 0]
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 = 1.19 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {2 \left (84 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{4}-168 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{4}-60 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{4}-21 B \,c^{3} x^{7}+105 A \,c^{3} x^{6}+27 a B \,c^{2} x^{5}+133 a A \,c^{2} x^{4}+57 a^{2} B c \,x^{3}+35 a^{2} A c \,x^{2}+9 a^{3} B x +7 A \,a^{3}\right )}{63 x^{4} \sqrt {c \,x^{2}+a}\, e^{5} \sqrt {e x}}\) | \(366\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-21 B \,c^{2} x^{5}+105 A \,c^{2} x^{4}+48 a B c \,x^{3}+28 a A c \,x^{2}+9 a^{2} B x +7 A \,a^{2}\right )}{63 x^{4} e^{5} \sqrt {e x}}+\frac {8 c^{2} \left (\frac {5 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{21 e^{5} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(377\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 a^{2} A \sqrt {c e \,x^{3}+a e x}}{9 e^{6} x^{5}}-\frac {2 a^{2} B \sqrt {c e \,x^{3}+a e x}}{7 e^{6} x^{4}}-\frac {8 A a c \sqrt {c e \,x^{3}+a e x}}{9 e^{6} x^{3}}-\frac {32 B a c \sqrt {c e \,x^{3}+a e x}}{21 e^{6} x^{2}}-\frac {10 \left (c e \,x^{2}+a e \right ) A \,c^{2}}{3 e^{6} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {2 B \,c^{2} \sqrt {c e \,x^{3}+a e x}}{3 e^{6}}+\frac {40 B a c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 e^{5} \sqrt {c e \,x^{3}+a e x}}+\frac {8 A \,c^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{3 e^{5} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(459\) |
-2/63/x^4*(84*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*c^2*x^4-168*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*c^2*x^4-60*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*c*x^4-21*B*c^3*x^7+105*A*c^3*x^6+27*a*B*c^2*x^5+133*a*A*c^2*x^4+57*a^2 *B*c*x^3+35*a^2*A*c*x^2+9*a^3*B*x+7*A*a^3)/(c*x^2+a)^(1/2)/e^5/(e*x)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=\frac {2 \, {\left (120 \, \sqrt {c e} B a c x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 168 \, \sqrt {c e} A c^{2} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (21 \, B c^{2} x^{5} - 105 \, A c^{2} x^{4} - 48 \, B a c x^{3} - 28 \, A a c x^{2} - 9 \, B a^{2} x - 7 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{63 \, e^{6} x^{5}} \]
2/63*(120*sqrt(c*e)*B*a*c*x^5*weierstrassPInverse(-4*a/c, 0, x) - 168*sqrt (c*e)*A*c^2*x^5*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (21*B*c^2*x^5 - 105*A*c^2*x^4 - 48*B*a*c*x^3 - 28*A*a*c*x^2 - 9*B*a^ 2*x - 7*A*a^2)*sqrt(c*x^2 + a)*sqrt(e*x))/(e^6*x^5)
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{11/2}} \,d x \]