Integrand size = 24, antiderivative size = 376 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=-\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}+\frac {48 a A c^{3/2} x \sqrt {a+c x^2}}{5 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {48 a^{5/4} A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{5/4} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 e^3 \sqrt {e x} \sqrt {a+c x^2}} \]
-4/105*(-21*A*c*x+25*B*a)*(c*x^2+a)^(3/2)/e^2/(e*x)^(3/2)-2/35*(-5*B*x+7*A )*(c*x^2+a)^(5/2)/e/(e*x)^(5/2)-8/105*a*c*(-25*B*x+63*A)*(c*x^2+a)^(1/2)/e ^3/(e*x)^(1/2)+48/5*a*A*c^(3/2)*x*(c*x^2+a)^(1/2)/e^3/(a^(1/2)+x*c^(1/2))/ (e*x)^(1/2)-48/5*a^(5/4)*A*c^(5/4)*(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)))*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^3/(e*x)^(1/2)/(c*x^2+a)^(1/2)+8/105*a^(5/ 4)*c^(3/4)*(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))*(25*B*a^(1/2)+63*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^3/(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 = 10.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.23 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=-\frac {2 a^2 x \sqrt {a+c x^2} \left (3 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{4},-\frac {1}{4},-\frac {c x^2}{a}\right )+5 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{4},\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt {1+\frac {c x^2}{a}}} \]
(-2*a^2*x*Sqrt[a + c*x^2]*(3*A*Hypergeometric2F1[-5/2, -5/4, -1/4, -((c*x^ 2)/a)] + 5*B*x*Hypergeometric2F1[-5/2, -3/4, 1/4, -((c*x^2)/a)]))/(15*(e*x )^(7/2)*Sqrt[1 + (c*x^2)/a])
Time = 0.50 (sec) , antiderivative size = 369, 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, 547, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{(e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {4 c \int -\frac {(3 A+5 B x) \left (c x^2+a\right )^{3/2}}{2 (e x)^{3/2}}dx}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \int \frac {(3 A+5 B x) \left (c x^2+a\right )^{3/2}}{(e x)^{3/2}}dx}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 547 |
| \(\displaystyle \frac {2 c \left (-\frac {12 \int -\frac {(5 a B+21 A c x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {6 \int \frac {(5 a B+21 A c x) \sqrt {c x^2+a}}{\sqrt {e x}}dx}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4}{15} a \int \frac {25 a B+63 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {2}{15} a \int \frac {25 a B+63 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {2 a \sqrt {x} \int \frac {25 a B+63 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4 a \sqrt {x} \int \frac {25 a B+63 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-63 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-63 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-63 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-63 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 a B+63 A c x)}{15 e}\right )}{7 e}-\frac {2 \left (a+c x^2\right )^{3/2} (21 A-5 B x)}{7 e \sqrt {e x}}\right )}{3 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (3 A+5 B x)}{15 e (e x)^{5/2}}\) |
(-2*(3*A + 5*B*x)*(a + c*x^2)^(5/2))/(15*e*(e*x)^(5/2)) + (2*c*((-2*(21*A - 5*B*x)*(a + c*x^2)^(3/2))/(7*e*Sqrt[e*x]) + (6*((2*Sqrt[e*x]*(25*a*B + 6 3*A*c*x)*Sqrt[a + c*x^2])/(15*e) + (4*a*Sqrt[x]*(-63*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)*(25*Sqrt[a]*B + 63*A*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*c^(1/4)*Sqrt[ a + c*x^2])))/(15*Sqrt[e*x])))/(7*e)))/(3*e^2)
3.5.55.3.1 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[((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[((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 = 0.72 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\frac {48 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}}{5}-\frac {24 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}}{5}+\frac {40 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{21}+\frac {2 B \,c^{3} x^{7}}{7}+\frac {2 A \,c^{3} x^{6}}{5}+\frac {38 a B \,c^{2} x^{5}}{21}-\frac {22 a A \,c^{2} x^{4}}{5}+\frac {6 a^{2} B c \,x^{3}}{7}-\frac {26 a^{2} A c \,x^{2}}{5}-\frac {2 a^{3} B x}{3}-\frac {2 A \,a^{3}}{5}}{x^{2} \sqrt {c \,x^{2}+a}\, e^{3} \sqrt {e x}}\) | \(367\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-15 B \,c^{2} x^{5}-21 A \,c^{2} x^{4}-80 a B c \,x^{3}+252 a A c \,x^{2}+35 a^{2} B x +21 A \,a^{2}\right )}{105 x^{2} e^{3} \sqrt {e x}}+\frac {8 a c \left (\frac {25 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {63 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(376\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 a^{2} A \sqrt {c e \,x^{3}+a e x}}{5 e^{4} x^{3}}-\frac {2 a^{2} B \sqrt {c e \,x^{3}+a e x}}{3 e^{4} x^{2}}-\frac {24 \left (c e \,x^{2}+a e \right ) A a c}{5 e^{4} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {2 B \,c^{2} x^{2} \sqrt {c e \,x^{3}+a e x}}{7 e^{4}}+\frac {2 A \,c^{2} x \sqrt {c e \,x^{3}+a e x}}{5 e^{4}}+\frac {32 B a c \sqrt {c e \,x^{3}+a e x}}{21 e^{4}}+\frac {40 B \,a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 e^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {24 A a c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 e^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(457\) |
2/105/x^2*(504*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^2*c*x^2-252*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^2*c*x^2+100*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^2*x^2+15*B*c^3*x^7+21*A*c^3*x^6+95*a*B*c^2*x^5-231*a*A*c^2*x^4+45*a^ 2*B*c*x^3-273*a^2*A*c*x^2-35*a^3*B*x-21*A*a^3)/(c*x^2+a)^(1/2)/e^3/(e*x)^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\frac {2 \, {\left (200 \, \sqrt {c e} B a^{2} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 504 \, \sqrt {c e} A a c x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (15 \, B c^{2} x^{5} + 21 \, A c^{2} x^{4} + 80 \, B a c x^{3} - 252 \, A a c x^{2} - 35 \, B a^{2} x - 21 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{105 \, e^{4} x^{3}} \]
2/105*(200*sqrt(c*e)*B*a^2*x^3*weierstrassPInverse(-4*a/c, 0, x) - 504*sqr t(c*e)*A*a*c*x^3*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (15*B*c^2*x^5 + 21*A*c^2*x^4 + 80*B*a*c*x^3 - 252*A*a*c*x^2 - 35*B* a^2*x - 21*A*a^2)*sqrt(c*x^2 + a)*sqrt(e*x))/(e^4*x^3)
Result contains complex when optimal does not.
Time = 42.81 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\frac {A a^{\frac {5}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {A a^{\frac {3}{2}} c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {A \sqrt {a} c^{2} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {5}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B a^{\frac {3}{2}} c \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {7}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} c^{2} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {9}{4}\right )} \]
A*a**(5/2)*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), c*x**2*exp_polar(I*pi) /a)/(2*e**(7/2)*x**(5/2)*gamma(-1/4)) + A*a**(3/2)*c*gamma(-1/4)*hyper((-1 /2, -1/4), (3/4,), c*x**2*exp_polar(I*pi)/a)/(e**(7/2)*sqrt(x)*gamma(3/4)) + A*sqrt(a)*c**2*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*ex p_polar(I*pi)/a)/(2*e**(7/2)*gamma(7/4)) + B*a**(5/2)*gamma(-3/4)*hyper((- 3/4, -1/2), (1/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(7/2)*x**(3/2)*gamma(1 /4)) + B*a**(3/2)*c*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**2*e xp_polar(I*pi)/a)/(e**(7/2)*gamma(5/4)) + B*sqrt(a)*c**2*x**(5/2)*gamma(5/ 4)*hyper((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(7/2)*gamma( 9/4))
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{7/2}} \,d x \]