Integrand size = 24, antiderivative size = 432 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac {11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {77 A \sqrt {a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac {77 A c \sqrt {a+c x^2}}{10 a^4 e^3 \sqrt {e x}}-\frac {77 A c^{3/2} x \sqrt {a+c x^2}}{10 a^4 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {77 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{15/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (25 \sqrt {a} B+77 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{20 a^{15/4} e^3 \sqrt {e x} \sqrt {a+c x^2}} \]
1/3*(B*x+A)/a/e/(e*x)^(5/2)/(c*x^2+a)^(3/2)+1/6*(9*B*x+11*A)/a^2/e/(e*x)^( 5/2)/(c*x^2+a)^(1/2)-77/30*A*(c*x^2+a)^(1/2)/a^3/e/(e*x)^(5/2)-5/2*B*(c*x^ 2+a)^(1/2)/a^3/e^2/(e*x)^(3/2)+77/10*A*c*(c*x^2+a)^(1/2)/a^4/e^3/(e*x)^(1/ 2)-77/10*A*c^(3/2)*x*(c*x^2+a)^(1/2)/a^4/e^3/(a^(1/2)+x*c^(1/2))/(e*x)^(1/ 2)+77/10*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)/a^(15/4)/e^3/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/20*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)+77*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^(15/4)/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 = 10.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.32 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {x \left (65 a A+55 a B x+55 A c x^2+45 B c x^3-77 A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},-\frac {c x^2}{a}\right )-75 B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{30 a^2 (e x)^{7/2} \left (a+c x^2\right )^{3/2}} \]
(x*(65*a*A + 55*a*B*x + 55*A*c*x^2 + 45*B*c*x^3 - 77*A*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[-5/4, 1/2, -1/4, -((c*x^2)/a)] - 75*B*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((c*x^2)/a )]))/(30*a^2*(e*x)^(7/2)*(a + c*x^2)^(3/2))
Time = 0.60 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {551, 27, 551, 27, 553, 27, 553, 27, 553, 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.
| \(\displaystyle \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}-\frac {\int -\frac {11 A+9 B x}{2 (e x)^{7/2} \left (c x^2+a\right )^{3/2}}dx}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 A+9 B x}{(e x)^{7/2} \left (c x^2+a\right )^{3/2}}dx}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {\int -\frac {77 A+45 B x}{2 (e x)^{7/2} \sqrt {c x^2+a}}dx}{a}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {77 A+45 B x}{(e x)^{7/2} \sqrt {c x^2+a}}dx}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {3 (75 a B-77 A c x)}{2 (e x)^{5/2} \sqrt {c x^2+a}}dx}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {75 a B-77 A c x}{(e x)^{5/2} \sqrt {c x^2+a}}dx}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {2 \int \frac {3 a c (77 A+25 B x)}{2 (e x)^{3/2} \sqrt {c x^2+a}}dx}{3 a e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \int \frac {77 A+25 B x}{(e x)^{3/2} \sqrt {c x^2+a}}dx}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (-\frac {2 \int -\frac {25 a B+77 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {\int \frac {25 a B+77 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{a e}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {\sqrt {x} \int \frac {25 a B+77 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {2 \sqrt {x} \int \frac {25 a B+77 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B+77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-77 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (25 \sqrt {a} B+77 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-77 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {c \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+77 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-77 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{a e \sqrt {e x}}-\frac {154 A \sqrt {a+c x^2}}{a e \sqrt {e x}}\right )}{e}-\frac {50 B \sqrt {a+c x^2}}{e (e x)^{3/2}}\right )}{5 a e}-\frac {154 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}}{2 a}+\frac {11 A+9 B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}}{6 a}+\frac {A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}\) |
(A + B*x)/(3*a*e*(e*x)^(5/2)*(a + c*x^2)^(3/2)) + ((11*A + 9*B*x)/(a*e*(e* x)^(5/2)*Sqrt[a + c*x^2]) + ((-154*A*Sqrt[a + c*x^2])/(5*a*e*(e*x)^(5/2)) + (3*((-50*B*Sqrt[a + c*x^2])/(e*(e*x)^(3/2)) - (c*((-154*A*Sqrt[a + c*x^2 ])/(a*e*Sqrt[e*x]) + (2*Sqrt[x]*(-77*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 + 77*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])))/(a *e*Sqrt[e*x])))/e))/(5*a*e))/(2*a))/(6*a)
3.5.84.3.1 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 + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[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 = 2.68 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.11
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (\frac {A x}{3 a^{3} e^{4}}-\frac {B}{3 a^{2} e^{4} c}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (-\frac {5 A c x}{4 a^{4} e^{4}}+\frac {11 B}{12 a^{3} e^{4}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {2 A \sqrt {c e \,x^{3}+a e x}}{5 a^{3} e^{4} x^{3}}-\frac {2 B \sqrt {c e \,x^{3}+a e x}}{3 a^{3} e^{4} x^{2}}+\frac {26 \left (c e \,x^{2}+a e \right ) A c}{5 a^{4} e^{4} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{4 a^{3} e^{3} \sqrt {c e \,x^{3}+a e x}}-\frac {77 c A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{20 a^{4} e^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(481\) |
| default | \(\frac {231 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{4}-462 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{4}-75 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{4}+231 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}-462 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}-75 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}+462 A \,c^{3} x^{6}-150 a B \,c^{2} x^{5}+770 a A \,c^{2} x^{4}-210 a^{2} B c \,x^{3}+264 a^{2} A c \,x^{2}-40 a^{3} B x -24 A \,a^{3}}{60 x^{2} a^{4} e^{3} \sqrt {e x}\, \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(632\) |
| risch | \(\text {Expression too large to display}\) | \(1038\) |
((c*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(c*x^2+a)^(1/2)*((1/3/a^3/e^4*A*x-1/3/a^ 2/e^4/c*B)*(c*e*x^3+a*e*x)^(1/2)/(x^2+a/c)^2-2*x*e*c*(-5/4/a^4/e^4*A*c*x+1 1/12/a^3/e^4*B)/((x^2+a/c)*x*e*c)^(1/2)-2/5/a^3/e^4*A*(c*e*x^3+a*e*x)^(1/2 )/x^3-2/3/a^3/e^4*B*(c*e*x^3+a*e*x)^(1/2)/x^2+26/5*(c*e*x^2+a*e)/a^4/e^4*A *c/(x*(c*e*x^2+a*e))^(1/2)-5/4/a^3*B/e^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))-77/20*c/a^4*A/e^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*Ellip ticE(((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 = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.50 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=-\frac {75 \, {\left (B a c^{2} x^{7} + 2 \, B a^{2} c x^{5} + B a^{3} x^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 231 \, {\left (A c^{3} x^{7} + 2 \, A a c^{2} x^{5} + A a^{2} c x^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (231 \, A c^{3} x^{6} - 75 \, B a c^{2} x^{5} + 385 \, A a c^{2} x^{4} - 105 \, B a^{2} c x^{3} + 132 \, A a^{2} c x^{2} - 20 \, B a^{3} x - 12 \, A a^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{30 \, {\left (a^{4} c^{2} e^{4} x^{7} + 2 \, a^{5} c e^{4} x^{5} + a^{6} e^{4} x^{3}\right )}} \]
-1/30*(75*(B*a*c^2*x^7 + 2*B*a^2*c*x^5 + B*a^3*x^3)*sqrt(c*e)*weierstrassP Inverse(-4*a/c, 0, x) - 231*(A*c^3*x^7 + 2*A*a*c^2*x^5 + A*a^2*c*x^3)*sqrt (c*e)*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (231 *A*c^3*x^6 - 75*B*a*c^2*x^5 + 385*A*a*c^2*x^4 - 105*B*a^2*c*x^3 + 132*A*a^ 2*c*x^2 - 20*B*a^3*x - 12*A*a^3)*sqrt(c*x^2 + a)*sqrt(e*x))/(a^4*c^2*e^4*x ^7 + 2*a^5*c*e^4*x^5 + a^6*e^4*x^3)
Timed out. \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{7/2}\,{\left (c\,x^2+a\right )}^{5/2}} \,d x \]