Integrand size = 25, antiderivative size = 499 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{a c^2 x}+\frac {\sqrt {b} \left (b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {3 d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2*d^3*(-b*x^2+a)^(1/2)/c^2/(-a*d^2+b*c^2)/(d*x+c)^(1/2)-(d*x+c)^(1/2)*(-b* x^2+a)^(1/2)/a/c^2/x+b^(1/2)*(-3*a*d^2+b*c^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^(1/2)/c^2/(-a*d^2+b*c^2)/(b^(1/2)*(d*x+c)/( b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^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))/a^(1/2 )/c/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+3*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 = 24.80 (sec) , antiderivative size = 1068, normalized size of antiderivative = 2.14 \[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
(Sqrt[a - b*x^2]*((b*c^2*(c + d*x) - a*d^2*(c + 3*d*x))/(a*c^2*(-(b*c^2) + a*d^2)*x) + (b^2*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 4*a*b*c^3*d^2*Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]] + 3*a^2*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 2 *b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 6*a*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + b^2*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]* (c + d*x)^2 - 3*a*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*S qrt[b]*c*(b^(3/2)*c^3 - Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + 3*a^(3/2)*d^ 3)*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*(b^(3/2)*c^3 + 5*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - 3*a ^(3/2)*d^3)*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)*EllipticF[I*ArcSinh[Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - S qrt[a]*d)] + (3*I)*a*b*c^2*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*S qrt[-(((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)] - (3*I )*a^2*d^4*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sq rt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b...
Time = 1.41 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.21, 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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt {a-b x^2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {d^2}{c^2 \sqrt {a-b x^2} (c+d x)^{3/2}}-\frac {d}{c^2 x \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {1}{c x^2 \sqrt {a-b x^2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {a} \sqrt {b} d^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c^2 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {3 d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c^2 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a} c \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 d^3 \sqrt {a-b x^2}}{c^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{a c^2 x}\) |
Input:
Int[1/(x^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
(2*d^3*Sqrt[a - b*x^2])/(c^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - (Sqrt[c + d* x]*Sqrt[a - b*x^2])/(a*c^2*x) + (Sqrt[b]*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a] *EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]* c)/Sqrt[a] + d)])/(Sqrt[a]*c^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[ a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*Sqrt[b]*d^2*Sqrt[c + d*x]*Sqrt[1 - (b *x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/(( Sqrt[b]*c)/Sqrt[a] + d)])/(c^2*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(S qrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (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)])/(Sqrt[a]*c *Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (3*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. \(911\) vs. \(2(414)=828\).
Time = 5.95 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.83
| 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}}{a \,c^{2} x}-\frac {2 \left (-b d \,x^{2}+a d \right ) d^{2}}{\left (a \,d^{2}-b \,c^{2}\right ) c^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 d^{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 \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {d b}{2 a \,c^{2}}-\frac {b \,d^{3}}{\left (a \,d^{2}-b \,c^{2}\right ) c^{2}}\right ) \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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {3 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}}\) | \(912\) |
| risch | \(-\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{a \,c^{2} x}-\frac {d \left (\frac {\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 \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 d \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 )\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{2 c^{2} a \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(952\) |
| default | \(\text {Expression too large to display}\) | \(1839\) |
Input:
int(1/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),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/a/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)*d^2/(a*d^2-b*c^2)/c^2/((x+ c/d)*(-b*d*x^2+a*d))^(1/2)-2*d^2*b/c/(a*d^2-b*c^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)*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*(-1/2/a*d*b/c^2-b *d^3/(a*d^2-b*c^2)/c^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))*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)*Ellipt icF(((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*d^2/c^3*(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)))
Timed out. \[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**2/(d*x+c)**(3/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(1/(x**2*sqrt(a - b*x**2)*(c + d*x)**(3/2)), x)
\[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)*x^2), x)
\[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {1}{x^2\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(1/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)),x)
Output:
int(1/(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {1}{x^2 (c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}+\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b c d x +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b \,d^{2} x^{2}-3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) a c d x -3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{5}-2 b c d \,x^{4}+a \,d^{2} x^{3}-b \,c^{2} x^{3}+2 a c d \,x^{2}+a \,c^{2} x}d x \right ) a \,d^{2} x^{2}}{2 a c x \left (d x +c \right )} \] Input:
int(1/x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a - b*x**2) + int((sqrt(c + d*x)*sqrt(a - b*x**2) *x)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d** 2*x**4),x)*b*c*d*x + int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a*c**2 + 2*a* c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*b*d**2* x**2 - 3*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a *d**2*x**3 - b*c**2*x**3 - 2*b*c*d*x**4 - b*d**2*x**5),x)*a*c*d*x - 3*int( (sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2*x + 2*a*c*d*x**2 + a*d**2*x**3 - b*c**2*x**3 - 2*b*c*d*x**4 - b*d**2*x**5),x)*a*d**2*x**2)/(2*a*c*x*(c + d* x))