\(\int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx\) [221]

Optimal result

Integrand size = 27, antiderivative size = 530 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )} \] Output:

-2/15*(2*a*d-b*e)*(a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2)/a/e+2/5*(a+c/x^2+b/x 
)^(1/2)*x*(e*x+d)^(3/2)/e-2/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a^2*d^2+b^2*e^2 
-a*e*(b*d+3*c*e))*(a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(- 
4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*a*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^( 
1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/a^2 
/e^2/(a*(e*x+d)/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(a*x^2+b*x+c)+2/15 
*2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*a*d-b*e)*(a*d^2-e*(b*d-c*e))*(a+c/x^2+b/x)^ 
(1/2)*x*(a*(e*x+d)/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-a*(a*x^2+b*x+ 
c)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*a*x+b)/(-4*a*c+b^2)^(1/2))^(1/2 
)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2) 
)/a^2/e^2/(e*x+d)^(1/2)/(a*x^2+b*x+c)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.39 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.31 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\frac {x \sqrt {a+\frac {c+b x}{x^2}} \left (\frac {4 e^2 \left (-a^2 d^2-b^2 e^2+a e (b d+3 c e)\right )}{\sqrt {d+e x}}+2 a e^2 \sqrt {d+e x} (b e+a (d+3 e x))+\frac {i (d+e x) \sqrt {1-\frac {2 \left (a d^2+e (-b d+c e)\right )}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (a d^2+e (-b d+c e)\right )}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (b^2 e^2 \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a^2 d \left (8 c e^2-d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a e \left (-2 b^2 d e-4 b c e^2+b d \sqrt {\left (b^2-4 a c\right ) e^2}+3 c e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}\right )}{15 a^2 e^3} \] Input:

Integrate[Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x],x]
 

Output:

(x*Sqrt[a + (c + b*x)/x^2]*((4*e^2*(-(a^2*d^2) - b^2*e^2 + a*e*(b*d + 3*c* 
e)))/Sqrt[d + e*x] + 2*a*e^2*Sqrt[d + e*x]*(b*e + a*(d + 3*e*x)) + (I*(d + 
 e*x)*Sqrt[1 - (2*(a*d^2 + e*(-(b*d) + c*e)))/((2*a*d - b*e + Sqrt[(b^2 - 
4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(a*d^2 + e*(-(b*d) + c*e)))/((-2*a*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*a*d - b*e + Sqrt[(b^2 - 4 
*a*c)*e^2])*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*EllipticE[I*ArcSinh[(S 
qrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2 
])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b 
*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) 
 + a^2*d*(8*c*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e*(-2*b^2*d*e - 4*b*c*e 
^2 + b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 3*c*e*Sqrt[(b^2 - 4*a*c)*e^2]))*Ellipti 
cF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b 
^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[(a*d^2 + e*(-(b*d) 
+ c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))))/(15 
*a^2*e^3)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1897, 1162, 1236, 27, 1269, 1172, 321, 327}

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.

\(\displaystyle \int x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \, dx\)

\(\Big \downarrow \) 1897

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \int \sqrt {d+e x} \sqrt {a x^2+b x+c}dx}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 1162

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\int \frac {\sqrt {d+e x} (b d-2 c e+(2 a d-b e) x)}{\sqrt {a x^2+b x+c}}dx}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {2 \int \frac {a d (b d-8 c e)+b e (b d+c e)+2 \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) x}{2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {\int \frac {a d (b d-8 c e)+b e (b d+c e)+2 \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {\frac {2 \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a x^2+b x+c}}dx}{e}-\frac {(2 a d-b e) \left (a d^2-e (b d-c e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{e}}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 e}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}}{3 a}+\frac {2 \sqrt {d+e x} \sqrt {a x^2+b x+c} (2 a d-b e)}{3 a}}{5 e}\right )}{\sqrt {a x^2+b x+c}}\)

Input:

Int[Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x],x]
 

Output:

(Sqrt[a + c/x^2 + b/x]*x*((2*(d + e*x)^(3/2)*Sqrt[c + b*x + a*x^2])/(5*e) 
- ((2*(2*a*d - b*e)*Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/(3*a) + ((2*Sqrt[ 
2]*Sqrt[b^2 - 4*a*c]*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*Sqrt[d + e*x] 
*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + S 
qrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a* 
c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(a*e*Sqrt[(a*(d + e*x))/(2*a*d 
 - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[c + b*x + a*x^2]) - (2*Sqrt[2]*Sqrt[b^ 
2 - 4*a*c]*(2*a*d - b*e)*(a*d^2 - e*(b*d - c*e))*Sqrt[(a*(d + e*x))/(2*a*d 
 - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c)) 
]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]] 
/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/ 
(a*e*Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2]))/(3*a))/(5*e)))/Sqrt[c + b*x + a 
*x^2]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1897

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) 
+ (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + b/x^ 
n + c/x^(2*n))^FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p])   Int[x^(m 
- 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, 
 d, e, m, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] &&  !IntegerQ[p] && 
  !IntegerQ[q] && PosQ[n]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1710\) vs. \(2(470)=940\).

Time = 1.29 (sec) , antiderivative size = 1711, normalized size of antiderivative = 3.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(1711\)
default	\(\text {Expression too large to display}\)	\(4361\)

Input:

int((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/15*(3*a*e*x+a*d+b*e)*(e*x+d)^(1/2)/a/e*((a*x^2+b*x+c)/x^2)^(1/2)*x-1/15/ 
a/e*(2*(2*a^2*d^2-2*a*b*d*e-6*a*c*e^2+2*b^2*e^2)*(d/e-1/2*(b+(-4*a*c+b^2)^ 
(1/2))/a)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+( 
-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2* 
(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e* 
x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*((-d/e-1/2*(-b+(-4*a*c+b^2)^(1/ 
2))/a)*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-d/e 
+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2) 
)+1/2*(-b+(-4*a*c+b^2)^(1/2))/a*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2 
)^(1/2))/a))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4* 
a*c+b^2)^(1/2))/a))^(1/2)))+2*a*b*d^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*( 
(x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2) 
^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+ 
b^2)^(1/2))/a)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2 
+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2 
)^(1/2))/a))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4* 
a*c+b^2)^(1/2))/a))^(1/2))+2*b*c*e^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*(( 
x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^ 
(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b 
^2)^(1/2))/a)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\frac {2 \, {\left ({\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} \sqrt {a e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right ) + 6 \, {\left (a^{3} d^{2} e - a^{2} b d e^{2} + {\left (a b^{2} - 3 \, a^{2} c\right )} e^{3}\right )} \sqrt {a e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right )\right ) + 3 \, {\left (3 \, a^{3} e^{3} x^{2} + {\left (a^{3} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )}}{45 \, a^{3} e^{3}} \] Input:

integrate((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x, algorithm="fricas")
 

Output:

2/45*((2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9* 
a*b*c)*e^3)*sqrt(a*e)*weierstrassPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 
3*a*c)*e^2)/(a^2*e^2), -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2 
*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(a^3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a 
*e)) + 6*(a^3*d^2*e - a^2*b*d*e^2 + (a*b^2 - 3*a^2*c)*e^3)*sqrt(a*e)*weier 
strassZeta(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^2), -4/27*(2 
*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e 
^3)/(a^3*e^3), weierstrassPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)* 
e^2)/(a^2*e^2), -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e 
^2 + (2*b^3 - 9*a*b*c)*e^3)/(a^3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a*e))) + 
 3*(3*a^3*e^3*x^2 + (a^3*d*e^2 + a^2*b*e^3)*x)*sqrt(e*x + d)*sqrt((a*x^2 + 
 b*x + c)/x^2))/(a^3*e^3)
 

Sympy [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int x \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \] Input:

integrate((a+c/x**2+b/x)**(1/2)*x*(e*x+d)**(1/2),x)
 

Output:

Integral(x*sqrt(d + e*x)*sqrt(a + b/x + c/x**2), x)
 

Maxima [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x \,d x } \] Input:

integrate((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x, x)
 

Giac [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x \,d x } \] Input:

integrate((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int x\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \] Input:

int(x*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2),x)
 

Output:

int(x*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, x \sqrt {e x +d}d x \] Input:

int((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x)
 

Output:

int((a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2),x)