\(\int \frac {1}{(e x)^{3/2} (a-b x^2) \sqrt {c-d x^2}} \, dx\) [1114]

Optimal result

Integrand size = 30, antiderivative size = 379 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}-\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \] Output:

-2*(-d*x^2+c)^(1/2)/a/c/e/(e*x)^(1/2)-2*d^(1/4)*(1-d*x^2/c)^(1/2)*Elliptic 
E(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2 
)+2*d^(1/4)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2 
),I)/a/c^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2)-b^(1/2)*c^(1/4)*(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^(3/2)/d^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2)+b^(1/2)*c^(1/4)*(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^(3/2)/d^(1/4)/e^(3/2)/(-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 = 11.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\frac {x \left (-42 a \left (c-d x^2\right )+14 (b c-a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+6 b d x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 c (e x)^{3/2} \sqrt {c-d x^2}} \] Input:

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

Output:

(x*(-42*a*(c - d*x^2) + 14*(b*c - a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/ 
4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*b*d*x^4*Sqrt[1 - (d*x^2)/c]*Appe 
llF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*c*(e*x)^(3/2)*Sqrt 
[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {368, 27, 980, 27, 1054, 2009}

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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 e \left (\frac {-\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}}{a c e^4}-\frac {\sqrt {c-d x^2}}{a c e^2 \sqrt {e x}}\right )\)

Input:

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

Output:

2*e*(-(Sqrt[c - d*x^2]/(a*c*e^2*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*e^(3/2)* 
Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e]) 
], -1])/Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*El 
lipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2 
] - (Sqrt[b]*c^(5/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqr 
t[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], - 
1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[b]*c^(5/4)*e^(3/2)*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]*d^(1/4)*Sqrt[c - d*x^2]) 
)/(a*c*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 980

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

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.40

method	result	size
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} c a \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {2 \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\frac {\sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e a d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e a d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-x^{2} d +c}}\)	\(530\)
default	\(\frac {\left (\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\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^{2}+\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \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}\, c -4 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a c d +4 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}+2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a c d -2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}+\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \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^{2}-\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \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}\, c -4 a \,d^{2} x^{2}+4 b c d \,x^{2}+4 a c d -4 b \,c^{2}\right ) d b}{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 ) c a e \sqrt {e x}}\)	\(824\)

Input:

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

Output:

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2*(-d*e*x^2+c*e)/e^2 
/c/a/(x*(-d*e*x^2+c*e))^(1/2)+2/e/a*(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)*EllipticE( 
((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/e/a*(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))-1/2/e/a/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))-1/2/e/a/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))*El 
lipticPi(((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)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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