Integrand size = 25, antiderivative size = 422 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{3/2}}{a \sqrt {a-b x^2}}+\frac {d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(3/2)/a/(-b*x^2+a)^(1/2)+d*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*Ellipti cE(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)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^ (1/2)/(-b*x^2+a)^(1/2)-2*c*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^2*(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 = 23.98 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {i d \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {a}+\sqrt {b} x\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}}}-i \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+\sqrt {b} \left ((c+d x) \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}+2 i c \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (1+\frac {\sqrt {a} d}{\sqrt {b} c},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )\right )}{a \sqrt {b} \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}} \] Input:
Integrate[(c + d*x)^(3/2)/(x*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[c + d*x]*((I*d*Sqrt[(d*(Sqrt[a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[a]*d )]*(Sqrt[a] + Sqrt[b]*x)*EllipticE[I*ArcSinh[Sqrt[-((Sqrt[b]*(c + d*x))/(S qrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] )/Sqrt[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqrt[a]*d)] - I*(Sqrt[b]* c + Sqrt[a]*d)*Sqrt[(d*(Sqrt[a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqr t[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]*EllipticF[I*ArcSin h[Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c + Sqrt [a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + Sqrt[b]*((c + d*x)*Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))] + (2*I)*c*Sqrt[(d*(Sqrt[a] - Sqrt[b]*x))/( Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqr t[a]*d)]*EllipticPi[1 + (Sqrt[a]*d)/(Sqrt[b]*c), I*ArcSinh[Sqrt[-((Sqrt[b] *(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])))/(a*Sqrt[b]*Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[ a]*d))]*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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[(c + d*x)^(3/2)/(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. \(831\) vs. \(2(341)=682\).
Time = 3.11 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.97
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {d x}{2 a b}+\frac {c}{2 a b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {c 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 {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}}}\, \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 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 c \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}}\) | \(832\) |
| default | \(\text {Expression too large to display}\) | \(1076\) |
Input:
int((d*x+c)^(3/2)/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)*(-2*(-b*d*x-b*c) *(1/2*d/a/b*x+1/2*c/a/b)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+c*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))-1/a*d^2* (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))*E llipticE(((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)))- 2*c/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)))
\[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)/(b^2*x^5 - 2*a*b*x^3 + a^2*x), x )
\[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(3/2)/x/(-b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x)**(3/2)/(x*(a - b*x**2)**(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x), x)
\[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(3/2)/(x*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(3/2)/(x*(a - b*x^2)^(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x \left (a-b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, d x +4 \left (\int \frac {\sqrt {d x +c}}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a^{2} c^{2} d -4 \left (\int \frac {\sqrt {d x +c}}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a b \,c^{2} d \,x^{2}+\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{3}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) a b \,d^{2}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{3}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} d^{2} x^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}+b^{2} c \,x^{5}-2 a b d \,x^{4}-2 a b c \,x^{3}+a^{2} d \,x^{2}+a^{2} c x}d x \right ) a^{2} c^{2}-3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}+b^{2} c \,x^{5}-2 a b d \,x^{4}-2 a b c \,x^{3}+a^{2} d \,x^{2}+a^{2} c x}d x \right ) a b \,c^{2} x^{2}-4 \left (\int \frac {\sqrt {d x +c}\, x}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a^{2} c \,d^{2}+4 \left (\int \frac {\sqrt {d x +c}\, x}{\sqrt {-b \,x^{2}+a}\, a \,c^{2}-\sqrt {-b \,x^{2}+a}\, a \,d^{2} x^{2}-\sqrt {-b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{4}}d x \right ) a b c \,d^{2} x^{2}}{3 a \left (-b \,x^{2}+a \right )} \] Input:
int((d*x+c)^(3/2)/x/(-b*x^2+a)^(3/2),x)
Output:
(2*sqrt(c + d*x)*sqrt(a - b*x**2)*d*x + 4*int(sqrt(c + d*x)/(sqrt(a - b*x* *2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2*c**2*d - 4*int(sqrt(c + d*x)/(sqrt( a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c** 2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a*b*c**2*d*x**2 + int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(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)*a*b*d**2 - int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(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)*b**2*d**2*x**2 + 3*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)*a**2*c**2 - 3*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)*a*b*c**2*x**2 - 4*int((sqrt(c + d*x)*x)/(sqrt(a - b*x**2)*a*c**2 - sqr t(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2 )*b*d**2*x**4),x)*a**2*c*d**2 + 4*int((sqrt(c + d*x)*x)/(sqrt(a - b*x**2)* a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqr t(a - b*x**2)*b*d**2*x**4),x)*a*b*c*d**2*x**2)/(3*a*(a - b*x**2))