Integrand size = 25, antiderivative size = 456 \[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=-\frac {3 d \sqrt {a-b x^2}}{c^2 \sqrt {c+d x}}-\frac {\sqrt {a-b x^2}}{c x \sqrt {c+d x}}+\frac {3 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {3 a d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-3*d*(-b*x^2+a)^(1/2)/c^2/(d*x+c)^(1/2)-(-b*x^2+a)^(1/2)/c/x/(d*x+c)^(1/2) +3*a^(1/2)*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) )/c^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-a^(1/ 2)*b^(1/2)*(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))/c/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+3*a*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))/c^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.95 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {c+3 d x}{c^2 x}+\frac {3 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-3 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}-6 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)+3 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2-3 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \left (2 b c^2-3 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+3 i a d^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{c^3 d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{\sqrt {c+d x}} \] Input:
Integrate[Sqrt[a - b*x^2]/(x^2*(c + d*x)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(-((c + 3*d*x)/(c^2*x)) + (3*b*c^3*Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]] - 3*a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 6*b*c^2*Sqrt[-c + (S qrt[a]*d)/Sqrt[b]]*(c + d*x) + 3*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d *x)^2 - (3*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]], (S qrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*(2*b*c^2 - 3*Sqrt[a]*Sq rt[b]*c*d - 3*a*d^2)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sq rt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt [b]*c - Sqrt[a]*d)] + (3*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)*EllipticP i[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(c ^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/Sqrt[c + d*x]
Time = 1.52 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {637, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {d^2 \sqrt {a-b x^2}}{c^2 (c+d x)^{3/2}}-\frac {d \sqrt {a-b x^2}}{c^2 x \sqrt {c+d x}}+\frac {\sqrt {a-b x^2}}{c x^2 \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 a d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 d \sqrt {a-b x^2}}{c^2 \sqrt {c+d x}}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}\) |
Input:
Int[Sqrt[a - b*x^2]/(x^2*(c + d*x)^(3/2)),x]
Output:
(-2*d*Sqrt[a - b*x^2])/(c^2*Sqrt[c + d*x]) - (Sqrt[c + d*x]*Sqrt[a - b*x^2 ])/(c^2*x) + (3*Sqrt[a]*Sqrt[b]*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Elliptic E[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a ] + d)])/(c^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b *x^2]) - (Sqrt[a]*Sqrt[b]*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d) ]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[ 2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(c*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (3*a*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2) /a]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a ]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(c^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(846\) vs. \(2(371)=742\).
Time = 3.67 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.86
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{c^{2} x}-\frac {2 \left (-b d \,x^{2}+a d \right )}{c^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 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}}}\, \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 )}{c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 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 )}{c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {3 a \,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 )}{c^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(847\) |
| risch | \(-\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{c^{2} x}-\frac {\left (\left (-2 a \,d^{2}+2 b \,c^{2}\right ) \left (-\frac {2 \left (-b d \,x^{2}+a d \right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {c \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}}}\, \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 )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\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 )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a 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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 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}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{2 c^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(957\) |
| default | \(\text {Expression too large to display}\) | \(1128\) |
Input:
int((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(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)*(-1/c^2/x*(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2*(-b*d*x^2+a*d)/c^2/((x+c/d)*(-b*d*x^2+a*d)) ^(1/2)-2*b/c*(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))-3/c^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)*Ell ipticF(((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)))+3*a/c^3*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)*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)))
\[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*sqrt(d*x + c)/(d^2*x^4 + 2*c*d*x^3 + c^2*x^2), x )
\[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a - b x^{2}}}{x^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)/x**2/(d*x+c)**(3/2),x)
Output:
Integral(sqrt(a - b*x**2)/(x**2*(c + d*x)**(3/2)), x)
\[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)/((d*x + c)^(3/2)*x^2), x)
\[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)/((d*x + c)^(3/2)*x^2), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a-b\,x^2}}{x^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((a - b*x^2)^(1/2)/(x^2*(c + d*x)^(3/2)),x)
Output:
int((a - b*x^2)^(1/2)/(x^2*(c + d*x)^(3/2)), x)
\[ \int \frac {\sqrt {a-b x^2}}{x^2 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}}{x^{2} \left (d x +c \right )^{\frac {3}{2}}}d x \] Input:
int((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(3/2),x)
Output:
int((-b*x^2+a)^(1/2)/x^2/(d*x+c)^(3/2),x)