Integrand size = 27, antiderivative size = 266 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=-\frac {2 a^{3/4} \sqrt [4]{b} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{d \sqrt {e} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{d^2 \sqrt {e} \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} \left (b c^2-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c d^2 \sqrt {e} \sqrt {a-b x^2}} \] Output:
-2*a^(3/4)*b^(1/4)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4) /e^(1/2),I)/d/e^(1/2)/(-b*x^2+a)^(1/2)+2*a^(1/4)*b^(1/4)*(b^(1/2)*c+a^(1/2 )*d)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/d^ 2/e^(1/2)/(-b*x^2+a)^(1/2)-2*a^(1/4)*(-a*d^2+b*c^2)*(1-b*x^2/a)^(1/2)*Elli pticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2)*d/b^(1/2)/c,I)/b^(1/4) /c/d^2/e^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 22.92 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\frac {2 a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d-2 \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c d x^2+2 i \sqrt {a} \sqrt {b} c d \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )-2 i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+2 i b c^2 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )-2 i a d^2 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d^2 \sqrt {e x} \sqrt {a-b x^2}} \] Input:
Integrate[Sqrt[a - b*x^2]/(Sqrt[e*x]*(c + d*x)),x]
Output:
(2*a*Sqrt[-(Sqrt[a]/Sqrt[b])]*c*d - 2*Sqrt[-(Sqrt[a]/Sqrt[b])]*b*c*d*x^2 + (2*I)*Sqrt[a]*Sqrt[b]*c*d*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh [Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] - (2*I)*Sqrt[a]*d*(Sqrt[b]*c - Sqr t[a]*d)*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqr t[b])]/Sqrt[x]], -1] + (2*I)*b*c^2*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticPi[ -((Sqrt[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], - 1] - (2*I)*a*d^2*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticPi[-((Sqrt[b]*c)/(Sqr t[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1])/(Sqrt[-(Sqrt[a ]/Sqrt[b])]*c*d^2*Sqrt[e*x]*Sqrt[a - b*x^2])
Time = 1.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {616, 27, 1526, 25, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {e \sqrt {a-b x^2}}{c e+d x e}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\sqrt {a-b x^2}}{c e+d x e}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1526 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}-\frac {\int -\frac {b (c e-d e x)}{e^2 \sqrt {a-b x^2}}d\sqrt {e x}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {\int \frac {b (c e-d e x)}{e^2 \sqrt {a-b x^2}}d\sqrt {e x}}{d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {b \int \frac {c e-d e x}{\sqrt {a-b x^2}}d\sqrt {e x}}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {b \left (e \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-\frac {\sqrt {a} d e \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {b \left (e \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {b \left (\frac {e \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}-\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{d^2 e^2}+\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{d^2 e^2}+\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {\sqrt {a} d e \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{d^2 e^2}+\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}+\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a-\frac {b c^2}{d^2}\right ) \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}+\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{d^2 e^2}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{d^2 e^2}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (a-\frac {b c^2}{d^2}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c \sqrt {e} \sqrt {a-b x^2}}\right )\) |
Input:
Int[Sqrt[a - b*x^2]/(Sqrt[e*x]*(c + d*x)),x]
Output:
2*((b*(-((a^(3/4)*d*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)* Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(3/4)*Sqrt[a - b*x^2])) + (a^(1/4)* (c + (Sqrt[a]*d)/Sqrt[b])*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^ (1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*Sqrt[a - b*x^2])))/(d^2 *e^2) + (a^(1/4)*(a - (b*c^2)/d^2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[ a]*d)/(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b ^(1/4)*c*Sqrt[e]*Sqrt[a - b*x^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/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]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[Sqrt[(a_) + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(c*d ^2 + a*e^2)/e^2 Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] - Simp[1/e^2 Int[(c*d - c*e*x^2)/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Time = 0.56 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(\frac {\sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}-2 \sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) c d -2 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) c d +\operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-\operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}\right )}{\sqrt {-b \,x^{2}+a}\, d^{2} \sqrt {e x}\, \left (b c -d \sqrt {a b}\right )}\) | \(337\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} \sqrt {-b e \,x^{3}+a e x}}-\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{d \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(446\) |
Input:
int((-b*x^2+a)^(1/2)/(e*x)^(1/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
(a*b)^(1/2)*2^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/ 2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*(EllipticF(((b*x+(a*b)^(1/ 2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2+EllipticF(((b*x+(a*b)^(1/2))/(a* b)^(1/2))^(1/2),1/2*2^(1/2))*b*c^2-2*(a*b)^(1/2)*EllipticF(((b*x+(a*b)^(1/ 2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*c*d-2*EllipticE(((b*x+(a*b)^(1/2))/(a* b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2+2*(a*b)^(1/2)*EllipticE(((b*x+(a*b)^(1/ 2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*c*d+EllipticPi(((b*x+(a*b)^(1/2))/(a*b )^(1/2))^(1/2),(a*b)^(1/2)*d/(d*(a*b)^(1/2)-b*c),1/2*2^(1/2))*a*d^2-Ellipt icPi(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),(a*b)^(1/2)*d/(d*(a*b)^(1/2)-b* c),1/2*2^(1/2))*b*c^2)/(-b*x^2+a)^(1/2)/d^2/(e*x)^(1/2)/(b*c-d*(a*b)^(1/2) )
Timed out. \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((-b*x^2+a)^(1/2)/(e*x)^(1/2)/(d*x+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\int \frac {\sqrt {a - b x^{2}}}{\sqrt {e x} \left (c + d x\right )}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)/(e*x)**(1/2)/(d*x+c),x)
Output:
Integral(sqrt(a - b*x**2)/(sqrt(e*x)*(c + d*x)), x)
\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{{\left (d x + c\right )} \sqrt {e x}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)/(e*x)^(1/2)/(d*x+c),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)/((d*x + c)*sqrt(e*x)), x)
\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{{\left (d x + c\right )} \sqrt {e x}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)/(e*x)^(1/2)/(d*x+c),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)/((d*x + c)*sqrt(e*x)), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\int \frac {\sqrt {a-b\,x^2}}{\sqrt {e\,x}\,\left (c+d\,x\right )} \,d x \] Input:
int((a - b*x^2)^(1/2)/((e*x)^(1/2)*(c + d*x)),x)
Output:
int((a - b*x^2)^(1/2)/((e*x)^(1/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {e x} (c+d x)} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-b \,x^{2}+a}}{\sqrt {x}\, c +\sqrt {x}\, d x}d x \right )}{e} \] Input:
int((-b*x^2+a)^(1/2)/(e*x)^(1/2)/(d*x+c),x)
Output:
(sqrt(e)*int(sqrt(a - b*x**2)/(sqrt(x)*c + sqrt(x)*d*x),x))/e