Integrand size = 25, antiderivative size = 451 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x}}{a x \sqrt {a-b x^2}}-\frac {2 \sqrt {c+d x} \sqrt {a-b x^2}}{a^2 x}+\frac {2 \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {b} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(1/2)/a/x/(-b*x^2+a)^(1/2)-2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/x+ 2*b^(1/2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/ 2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2) /(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-2*b^(1/2)* c*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*Elliptic F(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^ (1/2)*d))^(1/2))/a^(3/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-d*(b^(1/2)*(d*x+c) /(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/ 2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1 /2))/a/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.92 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {(c+d x) \left (a-2 b x^2\right )}{a^2 x \left (a-b x^2\right )}-\frac {-2 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+2 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+4 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-2 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+2 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (2 \sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i a d^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2 c d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{\sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]/(x^2*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(-(((c + d*x)*(a - 2*b*x^2))/(a^2*x*(a - b*x^2))) - (-2*b *c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 2*a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt [b]] + 4*b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 2*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + (2*I)*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d) *Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d *x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*d*(2*Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Ell ipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*a*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^ (3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(a^2*c*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/Sqrt[ c + d*x]
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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[Sqrt[c + d*x]/(x^2*(a - b*x^2)^(3/2)),x]
Output:
$Aborted
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.96 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.43
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {\left (-b d x -b c \right ) x}{a^{2} \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{a^{2} x}-\frac {2 d b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{a^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, c}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(644\) |
| risch | \(-\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{a^{2} x}+\frac {\left (-\frac {d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {a d \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-2 a b \left (\frac {\left (-b d x -b c \right ) x}{b a \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{2 a b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{2 a^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(758\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (2 a b \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) c \,d^{2} x -2 \sqrt {a b}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2} d x -\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2} x +\sqrt {a b}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a \,d^{3} x -2 a b \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) c \,d^{2} x +2 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3} x +2 b^{2} c \,x^{3} d^{2}+2 b^{2} c^{2} d \,x^{2}-a b c \,d^{2} x -a b \,c^{2} d \right )}{c x \,a^{2} b d \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\) | \(990\) |
Input:
int((d*x+c)^(1/2)/x^2/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-(-b*d*x-b*c)/a ^2*x/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-1/a^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/ 2)/x-2/a^2*d*b*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2) *((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/( -c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b *(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*( a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c /d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^( 1/2)))^(1/2)))-d^2/a*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2))) ^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1 /2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/c*El lipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d ,((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*x^6 - 2*a*b*x^4 + a^2*x^2), x )
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x}}{x^{2} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(1/2)/x**2/(-b*x**2+a)**(3/2),x)
Output:
Integral(sqrt(c + d*x)/(x**2*(a - b*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((-b*x^2 + a)^(3/2)*x^2), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/((-b*x^2 + a)^(3/2)*x^2), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c+d\,x}}{x^2\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(1/2)/(x^2*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(1/2)/(x^2*(a - b*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} x^{6}-2 a b \,x^{4}+a^{2} x^{2}}d x \] Input:
int((d*x+c)^(1/2)/x^2/(-b*x^2+a)^(3/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*x**2 - 2*a*b*x**4 + b**2*x**6), x)