\(\int \frac {\sqrt {e x} \sqrt {a-b x^2}}{c+d x} \, dx\) [1348]

Optimal result

Integrand size = 27, antiderivative size = 304 \[ \int \frac {\sqrt {e x} \sqrt {a-b x^2}}{c+d x} \, dx=\frac {2 \sqrt {e x} \sqrt {a-b x^2}}{3 d}+\frac {2 a^{3/4} \sqrt [4]{b} c \sqrt {e} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{d^2 \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} \left (3 b c^2+3 \sqrt {a} \sqrt {b} c d-2 a d^2\right ) \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{3 \sqrt [4]{b} d^3 \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} \left (b c^2-a d^2\right ) \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.73 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {e x} \sqrt {a-b x^2}}{c+d x} \, dx=\frac {2 \sqrt {e x} \sqrt {a-b x^2} \left (d^2+\frac {-3 a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d+3 \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c d x^2-3 i \sqrt {a} \sqrt {b} c d \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+i \sqrt {a} d \left (3 \sqrt {b} c-\sqrt {a} d\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )-3 i b c^2 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+3 i a d^2 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} x \left (a-b x^2\right )}\right )}{3 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {616, 27, 1633, 25, 1543, 1542, 2427, 25, 27, 1513, 27, 765, 762, 1390, 1389, 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.

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

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1633

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

\(\Big \downarrow \) 2427

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2}}-\frac {-\frac {e^2 \int -\frac {b \left (\left (3 b c^2-2 a d^2\right ) e-3 b c d e x\right )}{d e^3 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {1}{3} d \sqrt {e x} \sqrt {a-b x^2}}{d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2}}-\frac {\frac {e^2 \int \frac {b \left (\left (3 b c^2-2 a d^2\right ) e-3 b c d e x\right )}{d e^3 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {1}{3} d \sqrt {e x} \sqrt {a-b x^2}}{d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2}}-\frac {\frac {\int \frac {\left (3 b c^2-2 a d^2\right ) e-3 b c d e x}{\sqrt {a-b x^2}}d\sqrt {e x}}{3 d e}-\frac {1}{3} d \sqrt {e x} \sqrt {a-b x^2}}{d^2}\right )\)

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (3 \sqrt {a} \sqrt {b} c d-2 a d^2+3 b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 a^{3/4} \sqrt [4]{b} c d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a-b x^2}}}{3 d e}-\frac {1}{3} d \sqrt {e x} \sqrt {a-b x^2}}{d^2}\right )\)

Input:

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

Output:

2*(-((-1/3*(d*Sqrt[e*x]*Sqrt[a - b*x^2]) + ((-3*a^(3/4)*b^(1/4)*c*d*e^(3/2 
)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e 
])], -1])/Sqrt[a - b*x^2] + (a^(1/4)*(3*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d - 2* 
a*d^2)*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[e*x])/(a 
^(1/4)*Sqrt[e])], -1])/(b^(1/4)*Sqrt[a - b*x^2]))/(3*d*e))/d^2) + (a^(1/4) 
*(b*c^2 - a*d^2)*Sqrt[e]*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d)/(Sqr 
t[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*d^3 
*Sqrt[a - b*x^2]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

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

rule 1542

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

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

rule 1633

Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] 
 :> Simp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1))   Int[1/((d + 
 e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[1/e^(2*p + 1)   Int[(1/Sqrt[a + c*x 
^4])*ExpandToSum[(e^(2*p + 1)*x^m*(a + c*x^4)^(p + 1/2) - (-d/e)^(m/2)*(c*d 
^2 + a*e^2)^(p + 1/2))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && 
 IGtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
 

rule 2427

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x 
]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 
1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1))   Int[ExpandToSum[b*(q 
 + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, 
 x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ 
[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(234)=468\).

Time = 1.10 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.64

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 \sqrt {-b e \,x^{3}+a e x}}{3 d}+\frac {\left (\frac {\left (a \,d^{2}-b \,c^{2}\right ) e}{d^{3}}-\frac {a e}{3 d}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {c e \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{d^{2} \sqrt {-b e \,x^{3}+a e x}}-\frac {c \left (a \,d^{2}-b \,c^{2}\right ) e \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{4} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\)	\(499\)
risch	\(\frac {2 \sqrt {-b \,x^{2}+a}\, x e}{3 d \sqrt {e x}}+\frac {\left (\frac {\frac {2 a \,d^{2} \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 )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {3 c^{2} \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 )}{\sqrt {-b e \,x^{3}+a e x}}+\frac {3 d 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}}}\, \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 )}{\sqrt {-b e \,x^{3}+a e x}}}{d^{2}}-\frac {3 c \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 {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right ) e \sqrt {\left (-b \,x^{2}+a \right ) e x}}{3 d \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(587\)
default	\(-\frac {\sqrt {e x}\, \left (2 \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{3} \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}}}-6 \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2} d \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}}}+\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,d^{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}}}+3 \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b \,c^{3} \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}}}+6 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2} d \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}}}-6 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,d^{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}}}+3 \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,d^{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}}}-3 \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b \,c^{3} \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}}}+2 b^{2} c \,x^{3} d^{2}-2 b \,d^{3} x^{3} \sqrt {a b}-2 a b c \,d^{2} x +2 a \,d^{3} x \sqrt {a b}\right )}{3 \sqrt {-b \,x^{2}+a}\, d^{3} x \left (b c -d \sqrt {a b}\right )}\)	\(785\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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