\(\int \frac {(c-d x^2)^{3/2}}{(e x)^{3/2} (a-b x^2)^2} \, dx\) [1137]

Optimal result

Integrand size = 30, antiderivative size = 519 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \] Output:

-1/2*(-a*d+5*b*c)*(-d*x^2+c)^(1/2)/a^2/b/e/(e*x)^(1/2)+1/2*(-a*d+b*c)*(-d* 
x^2+c)^(1/2)/a/b/e/(e*x)^(1/2)/(-b*x^2+a)-1/2*c^(3/4)*d^(1/4)*(-a*d+5*b*c) 
*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a^2/b/ 
e^(3/2)/(-d*x^2+c)^(1/2)+1/2*c^(3/4)*d^(1/4)*(-a*d+5*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^2/b/e^(3/2)/(-d*x^2+c 
)^(1/2)-1/4*c^(1/4)*(-a^2*d^2-4*a*b*c*d+5*b^2*c^2)*(1-d*x^2/c)^(1/2)*Ellip 
ticPi(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^(5/2)/b^(3/2)/d^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a^2*d^2 
-4*a*b*c*d+5*b^2*c^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^(5/2)/b^(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 = 11.18 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.38 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (21 a \left (c-d x^2\right ) \left (4 a c-5 b c x^2+a d x^2\right )+7 c (-5 b c+9 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d (-5 b c+a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(x*(21*a*(c - d*x^2)*(4*a*c - 5*b*c*x^2 + a*d*x^2) + 7*c*(-5*b*c + 9*a*d)* 
x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, 
(b*x^2)/a] + 3*d*(-5*b*c + a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*Appell 
F1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*(e*x)^(3/2)*(-a + b* 
x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {368, 27, 968, 27, 1053, 25, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 968

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1053

\(\displaystyle 2 e^3 \left (\frac {-\frac {\int -\frac {b c e x \left (d (5 b c-a d) x^2 e^2+c (5 b c-9 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-a d)}{a \sqrt {e x}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {b c e x \left (d (5 b c-a d) x^2 e^2+c (5 b c-9 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-a d)}{a \sqrt {e x}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {b \int \frac {e x \left (d (5 b c-a d) x^2 e^2+c (5 b c-9 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^2}-\frac {\sqrt {c-d x^2} (5 b c-a d)}{a \sqrt {e x}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*e^3*(((b*c - a*d)*Sqrt[c - d*x^2])/(4*a*b*e^2*Sqrt[e*x]*(a*e^2 - b*e^2*x 
^2)) + (-(((5*b*c - a*d)*Sqrt[c - d*x^2])/(a*Sqrt[e*x])) + (b*(-((c^(3/4)* 
d^(1/4)*(5*b*c - a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4 
)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2])) + (c^(3/4)*d^(1 
/4)*(5*b*c - a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sq 
rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2]) - (c^(1/4)*(5*b^2*c^ 
2 - 4*a*b*c*d - a^2*d^2)*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]*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(5*b^2*c^2 
- 4*a*b*c*d - a^2*d^2)*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]*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2])))/(a*e^2))/(4*a*b*e^4))
 

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 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 968

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

rule 1053

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1344\) vs. \(2(403)=806\).

Time = 1.51 (sec) , antiderivative size = 1345, normalized size of antiderivative = 2.59

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1345\)
default	\(\text {Expression too large to display}\)	\(3867\)

Input:

int((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a)^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*(a*d-b*c)/a^2/e^ 
2*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2*(-d*e*x^2+c*e)/e^2*c/a^2/(x*(-d*e* 
x^2+c*e))^(1/2)-1/2*c*d*(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)/a/e/b*EllipticE(((x+1/ 
d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/4*c*d*(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)/a/e/b*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2 
^(1/2))+5/2*c^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)/e/a^2*EllipticE(((x+1/d*(c*d)^ 
(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-5/4*c^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)/e/a^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+ 
1/8/b^2/e*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/a/b/e*(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))*Ellipt 
icPi(((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-5/8/a^2/e/d*(c*d)^(1/2)*(1+x*d/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {too large to display} \] Input:

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

Output:

(sqrt(e)*( - 10*sqrt(c - d*x**2)*a**2*c*d**2 + 52*sqrt(c - d*x**2)*a*b*c** 
2*d - 10*sqrt(c - d*x**2)*b**2*c**3 + 5*sqrt(x)*int((sqrt(x)*sqrt(c - d*x* 
*2)*x**2)/(a**3*c*d - a**3*d**2*x**2 - 5*a**2*b*c**2 + 3*a**2*b*c*d*x**2 + 
 2*a**2*b*d**2*x**4 + 10*a*b**2*c**2*x**2 - 9*a*b**2*c*d*x**4 - a*b**2*d** 
2*x**6 - 5*b**3*c**2*x**4 + 5*b**3*c*d*x**6),x)*a**5*d**5 - 66*sqrt(x)*int 
((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**3*c*d - a**3*d**2*x**2 - 5*a**2*b*c** 
2 + 3*a**2*b*c*d*x**2 + 2*a**2*b*d**2*x**4 + 10*a*b**2*c**2*x**2 - 9*a*b** 
2*c*d*x**4 - a*b**2*d**2*x**6 - 5*b**3*c**2*x**4 + 5*b**3*c*d*x**6),x)*a** 
4*b*c*d**4 - 5*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**3*c*d - a** 
3*d**2*x**2 - 5*a**2*b*c**2 + 3*a**2*b*c*d*x**2 + 2*a**2*b*d**2*x**4 + 10* 
a*b**2*c**2*x**2 - 9*a*b**2*c*d*x**4 - a*b**2*d**2*x**6 - 5*b**3*c**2*x**4 
 + 5*b**3*c*d*x**6),x)*a**4*b*d**5*x**2 + 288*sqrt(x)*int((sqrt(x)*sqrt(c 
- d*x**2)*x**2)/(a**3*c*d - a**3*d**2*x**2 - 5*a**2*b*c**2 + 3*a**2*b*c*d* 
x**2 + 2*a**2*b*d**2*x**4 + 10*a*b**2*c**2*x**2 - 9*a*b**2*c*d*x**4 - a*b* 
*2*d**2*x**6 - 5*b**3*c**2*x**4 + 5*b**3*c*d*x**6),x)*a**3*b**2*c**2*d**3 
+ 66*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**3*c*d - a**3*d**2*x** 
2 - 5*a**2*b*c**2 + 3*a**2*b*c*d*x**2 + 2*a**2*b*d**2*x**4 + 10*a*b**2*c** 
2*x**2 - 9*a*b**2*c*d*x**4 - a*b**2*d**2*x**6 - 5*b**3*c**2*x**4 + 5*b**3* 
c*d*x**6),x)*a**3*b**2*c*d**4*x**2 - 430*sqrt(x)*int((sqrt(x)*sqrt(c - d*x 
**2)*x**2)/(a**3*c*d - a**3*d**2*x**2 - 5*a**2*b*c**2 + 3*a**2*b*c*d*x*...