\(\int \frac {\sqrt {a+b x^2}}{\sqrt {e x} (c+d x)} \, dx\) [1342]

Optimal result

Integrand size = 26, antiderivative size = 537 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {e x} (c+d x)} \, dx=\frac {2 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2}}{d e \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {\sqrt {b c^2+a d^2} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^{3/2} \sqrt {e}}-\frac {2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{d \sqrt {e} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{a} b^{3/4} c \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{d \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \sqrt {a+b x^2}}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2}{4 \sqrt {a} \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c d^2 \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \sqrt {a+b x^2}} \] Output:

2*b^(1/2)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/d/e/(a^(1/2)+b^(1/2)*x)+(a*d^2+b*c^2 
)^(1/2)*arctan((a*d^2+b*c^2)^(1/2)*(e*x)^(1/2)/c^(1/2)/d^(1/2)/e^(1/2)/(b* 
x^2+a)^(1/2))/c^(1/2)/d^(3/2)/e^(1/2)-2*a^(1/4)*b^(1/4)*(a^(1/2)+b^(1/2)*x 
)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*( 
e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))/d/e^(1/2)/(b*x^2+a)^(1/2)+2*a^(1 
/4)*b^(3/4)*c*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)* 
InverseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2)) 
/d/(b^(1/2)*c-a^(1/2)*d)/e^(1/2)/(b*x^2+a)^(1/2)-1/2*(b^(1/2)*c+a^(1/2)*d) 
*(a*d^2+b*c^2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2) 
*EllipticPi(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),-1/4*(b^(1/ 
2)*c-a^(1/2)*d)^2/a^(1/2)/b^(1/2)/c/d,1/2*2^(1/2))/a^(1/4)/b^(1/4)/c/d^2/( 
b^(1/2)*c-a^(1/2)*d)/e^(1/2)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.03 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {e x} (c+d x)} \, dx=\frac {2 a \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} c d+2 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} b c d x^2-2 \sqrt {a} \sqrt {b} c d \sqrt {1+\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+2 \sqrt {a} d \left (\sqrt {b} c+i \sqrt {a} d\right ) \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )-2 i b c^2 \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {i \sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )-2 i a d^2 \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {i \sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} c d^2 \sqrt {e x} \sqrt {a+b x^2}} \] Input:

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

Output:

(2*a*Sqrt[(I*Sqrt[a])/Sqrt[b]]*c*d + 2*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*c*d*x^2 
 - 2*Sqrt[a]*Sqrt[b]*c*d*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[S 
qrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + 2*Sqrt[a]*d*(Sqrt[b]*c + I*Sqrt[a 
]*d)*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt 
[b]]/Sqrt[x]], -1] - (2*I)*b*c^2*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticPi[(( 
-I)*Sqrt[b]*c)/(Sqrt[a]*d), I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], 
-1] - (2*I)*a*d^2*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticPi[((-I)*Sqrt[b]*c)/ 
(Sqrt[a]*d), I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/(Sqrt[(I*S 
qrt[a])/Sqrt[b]]*c*d^2*Sqrt[e*x]*Sqrt[a + b*x^2])
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {616, 27, 1524, 27, 1512, 27, 761, 1510, 2221}

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.

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

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1524

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1512

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

\(\displaystyle 2 \left (\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} c \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b x^2}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a}}\right )}{d e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}-\frac {\left (a d^2+b c^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a} d e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}\right )\)

\(\Big \downarrow \) 1510

\(\displaystyle 2 \left (\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} c \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b x^2}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {a}}\right )}{d e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}-\frac {\left (a d^2+b c^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a} d e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}\right )\)

\(\Big \downarrow \) 2221

\(\displaystyle 2 \left (\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} c \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b x^2}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {a}}\right )}{d e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}-\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} d e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}\right )\)

Input:

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

Output:

2*((Sqrt[b]*(-(((Sqrt[b]*c - Sqrt[a]*d)*(-((e^2*Sqrt[e*x]*Sqrt[a + b*x^2]) 
/(Sqrt[a]*e + Sqrt[b]*e*x)) + (a^(1/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*S 
qrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticE[2*ArcTan[(b 
^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])))/Sq 
rt[a]) + (b^(1/4)*c*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2* 
x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/( 
a^(1/4)*Sqrt[e])], 1/2])/(a^(1/4)*Sqrt[a + b*x^2])))/(((Sqrt[b]*c)/Sqrt[a] 
 - d)*d*e^2) - ((b*c^2 + a*d^2)*(-1/2*((Sqrt[b]*c - Sqrt[a]*d)*Sqrt[e]*Arc 
Tan[(Sqrt[b*c^2 + a*d^2]*Sqrt[e*x])/(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^ 
2])])/(Sqrt[c]*Sqrt[d]*Sqrt[b*c^2 + a*d^2]) + ((Sqrt[b]*c + Sqrt[a]*d)*(Sq 
rt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^ 
2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)^2)/(Sqrt[b]*c*d), 2* 
ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(1/4)*b^(1/4)*c* 
d*Sqrt[e]*Sqrt[a + b*x^2])))/(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)*d*e))
 

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 616

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) - 
 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; 
FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1512

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q 
 Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c 
, d, e}, x] && PosQ[c/a]
 

rule 1524

Int[Sqrt[(a_) + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[(c*d^2 + a*e^2)/(e*(e - d*q))   Int[(1 + q*x^2)/((d + e* 
x^2)*Sqrt[a + c*x^4]), x], x] - Simp[1/(e*(e - d*q))   Int[(c*d + a*e*q - ( 
c*e - a*d*q^3)*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && N 
eQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (2 \sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c d -2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c d +\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}-2 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+\operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {-a b}\, d}{\sqrt {-a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+\operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {-a b}\, d}{\sqrt {-a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}\right )}{\sqrt {b \,x^{2}+a}\, d^{2} \sqrt {e x}\, \left (b c -\sqrt {-a b}\, d \right )}\)	\(363\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {c \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} \sqrt {b e \,x^{3}+a e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{d \sqrt {b e \,x^{3}+a e x}}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{b \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(471\)

Input:

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

Output:

2^(1/2)*(-a*b)^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b) 
^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*(2*(-a*b)^(1/2)*El 
lipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*c*d-2*(-a*b)^ 
(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*c*d+E 
llipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2-Ellipt 
icF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*b*c^2-2*EllipticE 
(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2+EllipticPi(((b 
*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),(-a*b)^(1/2)*d/((-a*b)^(1/2)*d-b*c),1 
/2*2^(1/2))*a*d^2+EllipticPi(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),(-a*b 
)^(1/2)*d/((-a*b)^(1/2)*d-b*c),1/2*2^(1/2))*b*c^2)/(b*x^2+a)^(1/2)/d^2/(e* 
x)^(1/2)/(b*c-(-a*b)^(1/2)*d)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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