Integrand size = 24, antiderivative size = 300 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=-\frac {2 (3 A-B x) \sqrt {a+c x^2}}{3 e \sqrt {e x}}+\frac {4 A \sqrt {c} x \sqrt {a+c x^2}}{e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 \sqrt [4]{a} A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 \sqrt [4]{a} \left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]
-2/3*(-B*x+3*A)*(c*x^2+a)^(1/2)/e/(e*x)^(1/2)+4*A*x*c^(1/2)*(c*x^2+a)^(1/2 )/e/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-4*a^(1/4)*A*c^(1/4)*(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)))*Elli pticE(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/(e*x)^(1/2)/(c*x^2+a )^(1/2)+2/3*a^(1/4)*(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))*(B*a^(1/2)+3*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)/c^(1/4)/e/(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 = 9.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.27 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\frac {2 x \sqrt {a+c x^2} \left (-A \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\frac {c x^2}{a}\right )+B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )\right )}{(e x)^{3/2} \sqrt {1+\frac {c x^2}{a}}} \]
(2*x*Sqrt[a + c*x^2]*(-(A*Hypergeometric2F1[-1/2, -1/4, 3/4, -((c*x^2)/a)] ) + B*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/a)]))/((e*x)^(3/2)*Sqr t[1 + (c*x^2)/a])
Time = 0.39 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (A+B x)}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 547 |
| \(\displaystyle -\frac {4 \int -\frac {a B+3 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{3 e}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {a B+3 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{3 e}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 \sqrt {x} \int \frac {a B+3 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {4 \sqrt {x} \int \frac {a B+3 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {4 \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B+3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-3 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B+3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {4 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {4 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}-3 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{3 e \sqrt {e x}}-\frac {2 \sqrt {a+c x^2} (3 A-B x)}{3 e \sqrt {e x}}\) |
(-2*(3*A - B*x)*Sqrt[a + c*x^2])/(3*e*Sqrt[e*x]) + (4*Sqrt[x]*(-3*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)*(S qrt[a]*B + 3*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])))/(3*e*Sqrt[e*x])
3.5.38.3.1 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*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 = 0.54 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(-\frac {2 \left (3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c -6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c -B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a -B \,c^{2} x^{3}+3 A \,c^{2} x^{2}-a B c x +3 A a c \right )}{3 \sqrt {c \,x^{2}+a}\, e \sqrt {e x}\, c}\) | \(310\) |
| risch | \(-\frac {2 \left (-B x +3 A \right ) \sqrt {c \,x^{2}+a}}{3 e \sqrt {e x}}+\frac {\left (\frac {2 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 c \sqrt {c e \,x^{3}+a e x}}+\frac {2 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(330\) |
| elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right ) A}{e^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {2 B \sqrt {c e \,x^{3}+a e x}}{3 e^{2}}+\frac {2 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 e c \sqrt {c e \,x^{3}+a e x}}+\frac {2 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(354\) |
-2/3*(3*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-6*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-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*c)^(1/2)*a-B*c^2*x^3+3*A*c^2*x ^2-a*B*c*x+3*A*a*c)/(c*x^2+a)^(1/2)/e/(e*x)^(1/2)/c
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.27 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {c e} B a x {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 6 \, \sqrt {c e} A c x {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (B c x - 3 \, A c\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{3 \, c e^{2} x} \]
2/3*(2*sqrt(c*e)*B*a*x*weierstrassPInverse(-4*a/c, 0, x) - 6*sqrt(c*e)*A*c *x*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (B*c*x - 3*A*c)*sqrt(c*x^2 + a)*sqrt(e*x))/(c*e^2*x)
Result contains complex when optimal does not.
Time = 3.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
A*sqrt(a)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), c*x**2*exp_polar(I*pi)/a )/(2*e**(3/2)*sqrt(x)*gamma(3/4)) + B*sqrt(a)*sqrt(x)*gamma(1/4)*hyper((-1 /2, 1/4), (5/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(3/2)*gamma(5/4))
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{3/2}} \,d x \]