3.9.69 \(\int \frac {\sqrt {c-d x^2}}{\sqrt {e x} (a-b x^2)} \, dx\) [869]

3.9.69.1 Optimal result

Integrand size = 30, antiderivative size = 283 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {e x} \left (a-b x^2\right )} \, dx=\frac {2 \sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \]

output

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

3.9.69.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {e x} \left (a-b x^2\right )} \, dx=\frac {2 x \sqrt {c-d x^2} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{a \sqrt {e x} \sqrt {1-\frac {d x^2}{c}}} \]

input

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

output

(2*x*Sqrt[c - d*x^2]*AppellF1[1/4, -1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(a 
*Sqrt[e*x]*Sqrt[1 - (d*x^2)/c])
 

3.9.69.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {368, 27, 922, 765, 762, 925, 27, 1543, 1542}

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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 922

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

input

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

output

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

3.9.69.3.1 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 368

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

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 922

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Simp[b/d 
  Int[1/Sqrt[a + b*x^4], x], x] - Simp[(b*c - a*d)/d   Int[1/(Sqrt[a + b*x^ 
4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 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]
 

3.9.69.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(639\) vs. \(2(213)=426\).

Time = 3.09 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.26

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \left (\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b d \sqrt {c d}-\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a \,d^{2} \sqrt {a b}-\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b^{2} c \sqrt {c d}+\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b c d \sqrt {a b}-\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b d \sqrt {c d}-\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a \,d^{2} \sqrt {a b}+\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b^{2} c \sqrt {c d}+\Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b c d \sqrt {a b}+2 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2} \sqrt {a b}-2 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b c d \sqrt {a b}\right )}{2 \sqrt {-d \,x^{2}+c}\, \sqrt {e x}\, \sqrt {a b}\, \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right )}\)	\(640\)
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-d e \,x^{3}+c e x}}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{2 b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{2 b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(704\)

input

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

output

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

3.9.69.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.9.69.6 Sympy [F]

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

input

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

output

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

3.9.69.7 Maxima [F]

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

input

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

output

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

3.9.69.8 Giac [F]

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

input

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

output

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

3.9.69.9 Mupad [F(-1)]

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

input

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

output

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