\(\int \frac {1}{\sqrt {e x} (a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [1150]

Optimal result

Integrand size = 30, antiderivative size = 426 \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d) \sqrt {e x}}{2 a c (b c-a d)^2 e \sqrt {c-d x^2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {d^{3/4} (b c+2 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a c^{3/4} (b c-a d)^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {3 b \sqrt [4]{c} (b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {3 b \sqrt [4]{c} (b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^2 \sqrt {e} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 11.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {5 a x \left (-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (-c+d x^2\right )\right )+5 \left (3 b^2 c^2-8 a b c d+2 a^2 d^2\right ) x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (b c+2 a d) x^3 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{10 a^2 c (b c-a d)^2 \sqrt {e x} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(5*a*x*(-2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*x^2)) + 5*(3*b^2*c^2 - 
8*a*b*c*d + 2*a^2*d^2)*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/ 
2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(b*c + 2*a*d)*x^3*(a - b*x^2)*Sqrt[ 
1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(10*a^2*c 
*(b*c - a*d)^2*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {368, 27, 931, 27, 1024, 27, 1021, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1024

\(\displaystyle 2 e^3 \left (\frac {\frac {d \sqrt {e x} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (3 b^2 c^2-8 a b d c+2 a^2 d^2\right ) e^2-b d (b c+2 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (3 b^2 c^2-8 a b d c+2 a^2 d^2\right ) e^2-b d (b c+2 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

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

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 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 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

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. \(938\) vs. \(2(338)=676\).

Time = 2.54 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.20

method	result	size
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {b^{2} \sqrt {-d e \,x^{3}+c e x}}{2 a \left (a d -b c \right )^{2} e \left (-b \,x^{2}+a \right )}+\frac {d^{2} x}{c \left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {\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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b}{4 \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )^{2} a}+\frac {d \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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )^{2} c}+\frac {9 b \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 )}{8 \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 b^{2} \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 ) c}{8 a \left (a d -b c \right )^{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 {9 b \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 )}{8 \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 b^{2} \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 ) c}{8 a \left (a d -b c \right )^{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 {-x^{2} d +c}}\)	\(939\)
default	\(\text {Expression too large to display}\)	\(2542\)

Input:

int(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(1/2*b^2/a/(a*d-b*c)^2 
/e*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+d^2*x/c/(a*d-b*c)^2/(-(x^2-c/d)*d*e*x 
)^(1/2)+1/4*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1 
/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d 
)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b/(a*d-b*c)^2/a+1/2*d*(c*d)^(1/ 
2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2) 
)^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2 
))^(1/2),1/2*2^(1/2))/(a*d-b*c)^2/c+9/8/(a*d-b*c)^2*b/(a*b)^(1/2)*(c*d)^(1 
/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2 
))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*Ellipti 
cPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d) 
^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))-3/8/a/(a*d-b*c)^2*b^2/(a*b)^(1/2)/d*( 
c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c* 
d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))* 
EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/ 
d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))*c-9/8/(a*d-b*c)^2*b/(a*b)^(1/2 
)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/ 
(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2 
))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/( 
-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))+3/8/a/(a*d-b*c)^2*b^2/(a...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(1/2)/(-b*x**2+a)**2/(-d*x**2+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}} \,d x } \] Input:

integrate(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*sqrt(e*x)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}} \,d x } \] Input:

integrate(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*sqrt(e*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {e\,x}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:

int(1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x)
 

Output:

int(1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c^{2}-2 \sqrt {x}\, a^{2} c d \,x^{2}+\sqrt {x}\, a^{2} d^{2} x^{4}-2 \sqrt {x}\, a b \,c^{2} x^{2}+4 \sqrt {x}\, a b c d \,x^{4}-2 \sqrt {x}\, a b \,d^{2} x^{6}+\sqrt {x}\, b^{2} c^{2} x^{4}-2 \sqrt {x}\, b^{2} c d \,x^{6}+\sqrt {x}\, b^{2} d^{2} x^{8}}d x \right )}{e} \] Input:

int(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x)
 

Output:

(sqrt(e)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 
 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x* 
*4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d 
*x**6 + sqrt(x)*b**2*d**2*x**8),x))/e