Integrand size = 24, antiderivative size = 377 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\frac {48 a B c^{3/2} x \sqrt {a+c x^2}}{5 e^4 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {8 c (63 a B-25 A c x) \sqrt {a+c x^2}}{105 e^4 \sqrt {e x}}-\frac {4 (21 a B+25 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{5/2}}-\frac {2 (5 A-7 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{7/2}}-\frac {48 a^{5/4} B c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^4 \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{3/4} \left (63 \sqrt {a} B+25 A \sqrt {c}\right ) c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 e^4 \sqrt {e x} \sqrt {a+c x^2}} \]
-4/105*(25*A*c*x+21*B*a)*(c*x^2+a)^(3/2)/e^2/(e*x)^(5/2)-2/35*(-7*B*x+5*A) *(c*x^2+a)^(5/2)/e/(e*x)^(7/2)-8/105*c*(-25*A*c*x+63*B*a)*(c*x^2+a)^(1/2)/ e^4/(e*x)^(1/2)+48/5*a*B*c^(3/2)*x*(c*x^2+a)^(1/2)/e^4/(a^(1/2)+x*c^(1/2)) /(e*x)^(1/2)-48/5*a^(5/4)*B*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^4/(e*x)^(1/2)/(c*x^2+a)^(1/2)+8/105*a^(3 /4)*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)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))) ,1/2*2^(1/2))*(63*B*a^(1/2)+25*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^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 = 10.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=-\frac {2 a^2 \sqrt {e x} \sqrt {a+c x^2} \left (5 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {7}{4},-\frac {3}{4},-\frac {c x^2}{a}\right )+7 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{4},-\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{35 e^5 x^4 \sqrt {1+\frac {c x^2}{a}}} \]
(-2*a^2*Sqrt[e*x]*Sqrt[a + c*x^2]*(5*A*Hypergeometric2F1[-5/2, -7/4, -3/4, -((c*x^2)/a)] + 7*B*x*Hypergeometric2F1[-5/2, -5/4, -1/4, -((c*x^2)/a)])) /(35*e^5*x^4*Sqrt[1 + (c*x^2)/a])
Time = 0.49 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.97, 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, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{(e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {4 c \int -\frac {(5 A+7 B x) \left (c x^2+a\right )^{3/2}}{2 (e x)^{5/2}}dx}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \int \frac {(5 A+7 B x) \left (c x^2+a\right )^{3/2}}{(e x)^{5/2}}dx}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle \frac {2 c \left (-\frac {4 c \int -\frac {(5 A+21 B x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \int \frac {(5 A+21 B x) \sqrt {c x^2+a}}{\sqrt {e x}}dx}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4}{15} a \int \frac {25 A+63 B x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {2}{15} a \int \frac {25 A+63 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {2 a \sqrt {x} \int \frac {25 A+63 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4 a \sqrt {x} \int \frac {25 A+63 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4 a \sqrt {x} \left (\left (\frac {63 \sqrt {a} B}{\sqrt {c}}+25 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {63 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4 a \sqrt {x} \left (\left (\frac {63 \sqrt {a} B}{\sqrt {c}}+25 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {63 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {63 \sqrt {a} B}{\sqrt {c}}+25 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {63 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 c \left (\frac {2 c \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {63 \sqrt {a} B}{\sqrt {c}}+25 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {63 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (25 A+63 B x)}{15 e}\right )}{e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+21 B x)}{3 e (e x)^{3/2}}\right )}{7 e^2}-\frac {2 \left (a+c x^2\right )^{5/2} (5 A+7 B x)}{35 e (e x)^{7/2}}\) |
(-2*(5*A + 7*B*x)*(a + c*x^2)^(5/2))/(35*e*(e*x)^(7/2)) + (2*c*((-2*(5*A + 21*B*x)*(a + c*x^2)^(3/2))/(3*e*(e*x)^(3/2)) + (2*c*((2*Sqrt[e*x]*(25*A + 63*B*x)*Sqrt[a + c*x^2])/(15*e) + (4*a*Sqrt[x]*((-63*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])))/Sqrt[c] + ((25*A + (63*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)*S qrt[a + c*x^2])))/(15*Sqrt[e*x])))/e^2))/(7*e^2)
3.5.56.3.1 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 + 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 = 0.80 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {\frac {40 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{3}}{21}+\frac {48 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{3}}{5}-\frac {24 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{3}}{5}+\frac {2 B \,c^{3} x^{7}}{5}+\frac {2 A \,c^{3} x^{6}}{3}-\frac {22 a B \,c^{2} x^{5}}{5}-\frac {6 a A \,c^{2} x^{4}}{7}-\frac {26 a^{2} B c \,x^{3}}{5}-\frac {38 a^{2} A c \,x^{2}}{21}-\frac {2 a^{3} B x}{5}-\frac {2 A \,a^{3}}{7}}{x^{3} \sqrt {c \,x^{2}+a}\, e^{4} \sqrt {e x}}\) | \(366\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-21 B \,c^{2} x^{5}-35 A \,c^{2} x^{4}+252 a B c \,x^{3}+80 a A c \,x^{2}+21 a^{2} B x +15 A \,a^{2}\right )}{105 x^{3} e^{4} \sqrt {e x}}+\frac {8 a \,c^{2} \left (\frac {25 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {63 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 e^{4} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(380\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 a^{2} A \sqrt {c e \,x^{3}+a e x}}{7 e^{5} x^{4}}-\frac {2 a^{2} B \sqrt {c e \,x^{3}+a e x}}{5 e^{5} x^{3}}-\frac {32 A a c \sqrt {c e \,x^{3}+a e x}}{21 e^{5} x^{2}}-\frac {24 \left (c e \,x^{2}+a e \right ) B a c}{5 e^{5} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {2 B \,c^{2} x \sqrt {c e \,x^{3}+a e x}}{5 e^{5}}+\frac {2 A \,c^{2} \sqrt {c e \,x^{3}+a e x}}{3 e^{5}}+\frac {40 A a c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 e^{4} \sqrt {c e \,x^{3}+a e x}}+\frac {24 B a c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 e^{4} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(456\) |
2/105/x^3*(100*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)*Ell ipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c*x^3+504*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^2*c*x^3-252*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^2*c*x^3+21*B*c^3*x^7+35*A*c^3*x^6-231*a*B*c^2*x^5-45*a*A*c^2*x^4-273*a ^2*B*c*x^3-95*a^2*A*c*x^2-21*a^3*B*x-15*A*a^3)/(c*x^2+a)^(1/2)/e^4/(e*x)^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\frac {2 \, {\left (200 \, \sqrt {c e} A a c x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 504 \, \sqrt {c e} B a c x^{4} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (21 \, B c^{2} x^{5} + 35 \, A c^{2} x^{4} - 252 \, B a c x^{3} - 80 \, A a c x^{2} - 21 \, B a^{2} x - 15 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{105 \, e^{5} x^{4}} \]
2/105*(200*sqrt(c*e)*A*a*c*x^4*weierstrassPInverse(-4*a/c, 0, x) - 504*sqr t(c*e)*B*a*c*x^4*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (21*B*c^2*x^5 + 35*A*c^2*x^4 - 252*B*a*c*x^3 - 80*A*a*c*x^2 - 21*B* a^2*x - 15*A*a^2)*sqrt(c*x^2 + a)*sqrt(e*x))/(e^5*x^4)
Result contains complex when optimal does not.
Time = 100.71 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\frac {A a^{\frac {5}{2}} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {A a^{\frac {3}{2}} c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {A \sqrt {a} c^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B a^{\frac {5}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B a^{\frac {3}{2}} c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {9}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} c^{2} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} \Gamma \left (\frac {7}{4}\right )} \]
A*a**(5/2)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), c*x**2*exp_polar(I*pi) /a)/(2*e**(9/2)*x**(7/2)*gamma(-3/4)) + A*a**(3/2)*c*gamma(-3/4)*hyper((-3 /4, -1/2), (1/4,), c*x**2*exp_polar(I*pi)/a)/(e**(9/2)*x**(3/2)*gamma(1/4) ) + A*sqrt(a)*c**2*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**2*ex p_polar(I*pi)/a)/(2*e**(9/2)*gamma(5/4)) + B*a**(5/2)*gamma(-5/4)*hyper((- 5/4, -1/2), (-1/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*x**(5/2)*gamma( -1/4)) + B*a**(3/2)*c*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), c*x**2*exp_p olar(I*pi)/a)/(e**(9/2)*sqrt(x)*gamma(3/4)) + B*sqrt(a)*c**2*x**(3/2)*gamm a(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*ga mma(7/4))
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{9/2}} \,d x \]