\(\int \frac {\sqrt {e x}}{(a-b x^2) \sqrt {c-d x^2}} \, dx\) [1113]

Optimal result

Integrand size = 30, antiderivative size = 203 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

-c^(1/4)*e^(1/2)*(1-d*x^2/c)^(1/2)*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)/a^(1/2)/b^(1/2)/d^(1/4)/(-d*x^ 
2+c)^(1/2)+c^(1/4)*e^(1/2)*(1-d*x^2/c)^(1/2)*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)/a^(1/2)/b^(1/2)/d^(1/ 
4)/(-d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

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

Time = 11.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\frac {2 x \sqrt {e x} \sqrt {\frac {c-d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{3 a \sqrt {c-d x^2}} \] Input:

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

Output:

(2*x*Sqrt[e*x]*Sqrt[(c - d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, ( 
b*x^2)/a])/(3*a*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 27, 993, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

2*e*(-1/2*(c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqr 
t[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(Sqrt[ 
a]*Sqrt[b]*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2]) + (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*Sqrt[a]*Sqrt[b]*d^(1/4)*Sqrt[e]*Sqrt[c - d*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 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 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs. \(2(143)=286\).

Time = 0.89 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.61

method	result	size
default	\(\frac {\left (\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) b c -\sqrt {a b}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}+\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b c +\sqrt {a b}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}\right ) d \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {e x}}{2 \sqrt {-x^{2} d +c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {a b}\, d +\sqrt {c d}\, b \right ) x}\)	\(326\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {e \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\)	\(327\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{b d \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}d x \right ) \] Input:

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

Output:

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