\(\int \frac {(c-d x^2)^{3/2}}{(e x)^{5/2} (a-b x^2)^2} \, dx\) [1134]

Optimal result

Integrand size = 30, antiderivative size = 412 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=-\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (7 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 b e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}} \] Output:

-1/6*(-3*a*d+7*b*c)*(-d*x^2+c)^(1/2)/a^2/b/e/(e*x)^(3/2)+1/2*(-a*d+b*c)*(- 
d*x^2+c)^(1/2)/a/b/e/(e*x)^(3/2)/(-b*x^2+a)+1/6*c^(1/4)*d^(3/4)*(-3*a*d+7* 
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^(5/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a*d+b*c)*(-a*d+7*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^3/b/d^(1/4)/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a*d 
+b*c)*(-a*d+7*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^3/b/d^(1/4)/e^(5/2)/(-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 = 199, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (5 a \left (c-d x^2\right ) \left (4 a c-7 b c x^2+3 a d x^2\right )+5 c (-21 b c+17 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-d (-7 b c+3 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

(x*(5*a*(c - d*x^2)*(4*a*c - 7*b*c*x^2 + 3*a*d*x^2) + 5*c*(-21*b*c + 17*a* 
d)*x^2*(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] - d*(-7*b*c + 3*a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*App 
ellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*(e*x)^(5/2)*(-a + 
b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {368, 27, 968, 27, 1053, 25, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 968

\(\displaystyle 2 e^3 \left (\frac {\int \frac {c (7 b c-3 a d) e^2-d (5 b c-a d) e^2 x^2}{e^4 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a b e^2}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\int \frac {c (7 b c-3 a d) e^2-d (5 b c-a d) e^2 x^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 e^3 \left (\frac {-\frac {\int -\frac {b c \left (c (21 b c-17 a d) e^2-d (7 b c-3 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {b c \left (c (21 b c-17 a d) e^2-d (7 b c-3 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {b \int \frac {c (21 b c-17 a d) e^2-d (7 b c-3 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d (7 b c-3 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {3 e^2 (b c-a d) (7 b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {\frac {b \left (\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}+\frac {3 e^2 (b c-a d) (7 b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}\right )}{3 a e^2}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{3 a (e x)^{3/2}}}{4 a b e^4}+\frac {\sqrt {c-d x^2} (b c-a d)}{4 a b e^2 (e x)^{3/2} \left (a e^2-b e^2 x^2\right )}\right )\)

Input:

Int[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

2*e^3*(((b*c - a*d)*Sqrt[c - d*x^2])/(4*a*b*e^2*(e*x)^(3/2)*(a*e^2 - b*e^2 
*x^2)) + (-1/3*((7*b*c - 3*a*d)*Sqrt[c - d*x^2])/(a*(e*x)^(3/2)) + (b*((c^ 
(1/4)*d^(3/4)*(7*b*c - 3*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin 
[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2]) + (3*(b* 
c - a*d)*(7*b*c - a*d)*e^2*((c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqr 
t[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]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqr 
t[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/b 
))/(3*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 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 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 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 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 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. \(1190\) vs. \(2(324)=648\).

Time = 1.78 (sec) , antiderivative size = 1191, normalized size of antiderivative = 2.89

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1191\)
default	\(\text {Expression too large to display}\)	\(3472\)

Input:

int((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-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/e^3*(a*d-b*c)/a^ 
2*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2/3/e^3*c/a^2*(-d*e*x^3+c*e*x)^(1/2)/x 
^2-1/4*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/e^2/b+7/12*(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))*c/e^2/a^2-1/8/e^2/b/(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))+1/a/e^2/(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))*c-7/8/a^2/e^2*b/(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^2+1/8/e^2/b/(a*b)^(1/2)*d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \] Input:

integrate((-d*x**2+c)**(3/2)/(e*x)**(5/2)/(-b*x**2+a)**2,x)
 

Output:

Integral((c - d*x**2)**(3/2)/((e*x)**(5/2)*(-a + b*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*(e*x)^(5/2)), x)
 

Giac [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a)^2,x, algorithm="giac")
 

Output:

integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*(e*x)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2} \,d x \] Input:

int((c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2),x)
 

Output:

int((c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {too large to display} \] Input:

int((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a)^2,x)
 

Output:

(sqrt(e)*( - 2*sqrt(c - d*x**2)*a**2*c*d**2 - 12*sqrt(c - d*x**2)*a*b*c**2 
*d + 14*sqrt(c - d*x**2)*b**2*c**3 - 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt( 
x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 + 7*sqrt(x)*a**2*b*c**2 - 9*sqrt(x)*a 
**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 - 14*sqrt(x)*a*b**2*c**2*x**2 
+ 15*sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 + 7*sqrt(x)*b**3*c 
**2*x**4 - 7*sqrt(x)*b**3*c*d*x**6),x)*a**5*c*d**4*x - 58*sqrt(x)*int(sqrt 
(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 + 7*sqrt(x)*a**2*b 
*c**2 - 9*sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 - 14*sqrt(x 
)*a*b**2*c**2*x**2 + 15*sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 
 + 7*sqrt(x)*b**3*c**2*x**4 - 7*sqrt(x)*b**3*c*d*x**6),x)*a**4*b*c**2*d**3 
*x + 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2* 
x**2 + 7*sqrt(x)*a**2*b*c**2 - 9*sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2* 
b*d**2*x**4 - 14*sqrt(x)*a*b**2*c**2*x**2 + 15*sqrt(x)*a*b**2*c*d*x**4 - s 
qrt(x)*a*b**2*d**2*x**6 + 7*sqrt(x)*b**3*c**2*x**4 - 7*sqrt(x)*b**3*c*d*x* 
*6),x)*a**4*b*c*d**4*x**3 - 84*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**3* 
c*d - sqrt(x)*a**3*d**2*x**2 + 7*sqrt(x)*a**2*b*c**2 - 9*sqrt(x)*a**2*b*c* 
d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 - 14*sqrt(x)*a*b**2*c**2*x**2 + 15*sqr 
t(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 + 7*sqrt(x)*b**3*c**2*x**4 
 - 7*sqrt(x)*b**3*c*d*x**6),x)*a**3*b**2*c**3*d**2*x + 58*sqrt(x)*int(sqrt 
(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 + 7*sqrt(x)*a**...