Integrand size = 25, antiderivative size = 287 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x}}{a \sqrt {a-b x^2}}-\frac {d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(1/2)/a/(-b*x^2+a)^(1/2)-d*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d)) ^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(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^(1/2)/b^(1/2)/(d*x+c)^ (1/2)/(-b*x^2+a)^(1/2)-2*c*(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 = 22.84 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {c+d x}-i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+2 i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {a-b x^2}} \] Input:
Integrate[Sqrt[c + d*x]/(x*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*Sqrt[c + d*x] - I*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x) *EllipticF[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)] + (2*I)*Sqrt[(d*(Sqrt[a]/Sqrt[b ] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x )*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqr t[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt [a]*d)])/(a*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*Sqrt[a - b*x^2])
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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[Sqrt[c + d*x]/(x*(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]]
Leaf count of result is larger than twice the leaf count of optimal. \(511\) vs. \(2(233)=466\).
Time = 2.66 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {-b d x -b c}{b a \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(512\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b c -\sqrt {a b}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d -2 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b c +2 \sqrt {a b}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d +b d x +b c \right )}{\left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right ) a b}\) | \(646\) |
Input:
int((d*x+c)^(1/2)/x/(-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)/b /a/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+d/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)*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))-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)*d*EllipticPi(((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)))
\[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*x^5 - 2*a*b*x^3 + a^2*x), x)
\[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x}}{x \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(1/2)/x/(-b*x**2+a)**(3/2),x)
Output:
Integral(sqrt(c + d*x)/(x*(a - b*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((-b*x^2 + a)^(3/2)*x), x)
\[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/((-b*x^2 + a)^(3/2)*x), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c+d\,x}}{x\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(1/2)/(x*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(1/2)/(x*(a - b*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x \left (a-b x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}+b^{2} c \,x^{5}-2 a b d \,x^{4}-2 a b c \,x^{3}+a^{2} d \,x^{2}+a^{2} c x}d x \right ) c +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) d \] Input:
int((d*x+c)^(1/2)/x/(-b*x^2+a)^(3/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c*x + a**2*d*x**2 - 2*a*b*c*x** 3 - 2*a*b*d*x**4 + b**2*c*x**5 + b**2*d*x**6),x)*c + int((sqrt(c + d*x)*sq rt(a - b*x**2))/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c* x**4 + b**2*d*x**5),x)*d