Integrand size = 24, antiderivative size = 339 \[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 \sqrt {e x} (5 A+7 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {7 B e^4 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 \sqrt [4]{a} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (21 \sqrt {a} B+5 A \sqrt {c}\right ) e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} c^{11/4} \sqrt {e x} \sqrt {a+c x^2}} \]
-1/3*e*(e*x)^(5/2)*(B*x+A)/c/(c*x^2+a)^(3/2)-1/6*e^3*(7*B*x+5*A)*(e*x)^(1/ 2)/c^2/(c*x^2+a)^(1/2)+7/2*B*e^4*x*(c*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+x*c^(1 /2))/(e*x)^(1/2)-7/2*a^(1/4)*B*e^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)/c^(11/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/12*e ^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))*(21*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)/a^(1/4)/c^(11/4)/(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 = 139, normalized size of antiderivative = 0.41 \[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {e^3 \sqrt {e x} \left (-5 a A-7 a B x-7 A c x^2-9 B c x^3+5 A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+7 B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{6 c^2 \left (a+c x^2\right )^{3/2}} \]
(e^3*Sqrt[e*x]*(-5*a*A - 7*a*B*x - 7*A*c*x^2 - 9*B*c*x^3 + 5*A*(a + c*x^2) *Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] + 7*B* x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^ 2)/a)]))/(6*c^2*(a + c*x^2)^(3/2))
Time = 0.45 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {549, 27, 549, 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.
| \(\displaystyle \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {e^2 \int \frac {(e x)^{3/2} (5 A+7 B x)}{2 \left (c x^2+a\right )^{3/2}}dx}{3 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(e x)^{3/2} (5 A+7 B x)}{\left (c x^2+a\right )^{3/2}}dx}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \int \frac {5 A+21 B x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{c}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \int \frac {5 A+21 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{2 c}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \int \frac {5 A+21 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{2 c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \int \frac {5 A+21 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \left (\left (\frac {21 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {21 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \left (\left (\frac {21 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {21 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {21 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {21 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {e^2 \left (\frac {e^2 \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {21 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {21 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{c \sqrt {e x}}-\frac {e \sqrt {e x} (5 A+7 B x)}{c \sqrt {a+c x^2}}\right )}{6 c}-\frac {e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
-1/3*(e*(e*x)^(5/2)*(A + B*x))/(c*(a + c*x^2)^(3/2)) + (e^2*(-((e*Sqrt[e*x ]*(5*A + 7*B*x))/(c*Sqrt[a + c*x^2])) + (e^2*Sqrt[x]*((-21*B*(-((Sqrt[x]*S qrt[a + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sq rt[(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] + ((5*A + (21*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])))/(c*Sqrt[e*x])))/(6*c)
3.5.77.3.1 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*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b *x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1]
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 = 0.95 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.18
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (\frac {a \,e^{3} B x}{3 c^{4}}+\frac {a \,e^{3} A}{3 c^{4}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 e^{3} B x}{4 c^{3}}+\frac {7 e^{3} A}{12 c^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {5 A \,e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {7 B \,e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(400\) |
| default | \(\frac {\left (5 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) c \,x^{2}+42 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}-21 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+5 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a +42 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-21 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-18 B \,c^{2} x^{4}-14 A \,c^{2} x^{3}-14 a B c \,x^{2}-10 a A c x \right ) e^{3} \sqrt {e x}}{12 x \,c^{3} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(584\) |
1/e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*((c*x^2+a)*e*x)^(1/2)*((1/3*a*e^3/c^4*B* x+1/3*a*e^3/c^4*A)*(c*e*x^3+a*e*x)^(1/2)/(x^2+a/c)^2-2*x*e*c*(3/4*e^3*B/c^ 3*x+7/12*e^3*A/c^3)/((x^2+a/c)*x*e*c)^(1/2)+5/12*A*e^4/c^3*(-a*c)^(1/2)*(( x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c)^(1/2 )*c)^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*EllipticF(((x+( -a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))+7/4*B*e^4/c^3*(-a*c)^(1/ 2)*((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c) ^(1/2)*c)^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*(-2*(-a*c) ^(1/2)/c*EllipticE(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))+ (-a*c)^(1/2)/c*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^( 1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.56 \[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (A c^{2} e^{3} x^{4} + 2 \, A a c e^{3} x^{2} + A a^{2} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 21 \, {\left (B c^{2} e^{3} x^{4} + 2 \, B a c e^{3} x^{2} + B a^{2} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (9 \, B c^{2} e^{3} x^{3} + 7 \, A c^{2} e^{3} x^{2} + 7 \, B a c e^{3} x + 5 \, A a c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (c^{5} x^{4} + 2 \, a c^{4} x^{2} + a^{2} c^{3}\right )}} \]
1/6*(5*(A*c^2*e^3*x^4 + 2*A*a*c*e^3*x^2 + A*a^2*e^3)*sqrt(c*e)*weierstrass PInverse(-4*a/c, 0, x) - 21*(B*c^2*e^3*x^4 + 2*B*a*c*e^3*x^2 + B*a^2*e^3)* sqrt(c*e)*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (9*B*c^2*e^3*x^3 + 7*A*c^2*e^3*x^2 + 7*B*a*c*e^3*x + 5*A*a*c*e^3)*sqrt(c*x ^2 + a)*sqrt(e*x))/(c^5*x^4 + 2*a*c^4*x^2 + a^2*c^3)
Timed out. \[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]