\(\int \frac {(e x)^{11/2}}{(a-b x^2)^2 \sqrt {c-d x^2}} \, dx\) [1138]

Optimal result

Integrand size = 30, antiderivative size = 443 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {(4 b c-7 a d) e^5 \sqrt {e x} \sqrt {c-d x^2}}{6 b^2 d (b c-a d)}+\frac {a e^3 (e x)^{5/2} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} \left (4 b^2 c^2+20 a b c d-21 a^2 d^2\right ) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b^3 d^{5/4} (b c-a d) \sqrt {c-d x^2}}-\frac {a \sqrt [4]{c} (9 b c-7 a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^3 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {a \sqrt [4]{c} (9 b c-7 a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^3 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \] Output:

-1/6*(-7*a*d+4*b*c)*e^5*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2/d/(-a*d+b*c)+1/2* 
a*e^3*(e*x)^(5/2)*(-d*x^2+c)^(1/2)/b/(-a*d+b*c)/(-b*x^2+a)+1/6*c^(1/4)*(-2 
1*a^2*d^2+20*a*b*c*d+4*b^2*c^2)*e^(11/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/ 
4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/b^3/d^(5/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)- 
1/4*a*c^(1/4)*(-7*a*d+9*b*c)*e^(11/2)*(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)/b^3/d^(1/ 
4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/4*a*c^(1/4)*(-7*a*d+9*b*c)*e^(11/2)*(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)/b^3/d^(1/4)/(-a*d+b*c)/(-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 = 11.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {e^5 \sqrt {e x} \left (-5 a \left (c-d x^2\right ) \left (7 a^2 d+4 b^2 c x^2-4 a b \left (c+d x^2\right )\right )+5 a c (-4 b c+7 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-\left (4 b^2 c^2+20 a b c d-21 a^2 d^2\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a b^2 d (-b c+a d) \left (a-b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[(e*x)^(11/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]
 

Output:

(e^5*Sqrt[e*x]*(-5*a*(c - d*x^2)*(7*a^2*d + 4*b^2*c*x^2 - 4*a*b*(c + d*x^2 
)) + 5*a*c*(-4*b*c + 7*a*d)*(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] - (4*b^2*c^2 + 20*a*b*c*d - 21*a^2*d^2) 
*x^2*(-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]))/(30*a*b^2*d*(-(b*c) + a*d)*(a - b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {368, 27, 970, 1052, 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.

\(\displaystyle \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^{10} x^6}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {e^6 x^6}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\)

\(\Big \downarrow \) 970

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

\(\Big \downarrow \) 1052

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \int \frac {a c (4 b c-7 a d) e^2-\left (4 b^2 c^2+20 a b d c-21 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}-\frac {3 a^2 d e^2 (9 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2} \sqrt {c-d x^2}}{4 b (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (4 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+20 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 a^2 d e^2 (9 b c-7 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}\right )}{3 b d}}{4 b (b c-a d)}\right )\)

Input:

Int[(e*x)^(11/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]
 

Output:

2*e^3*((a*e^2*(e*x)^(5/2)*Sqrt[c - d*x^2])/(4*b*(b*c - a*d)*(a*e^2 - b*e^2 
*x^2)) - (((4*b*c - 7*a*d)*e^2*Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b*d) - (e^2*( 
(c^(1/4)*(4*b^2*c^2 + 20*a*b*c*d - 21*a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c] 
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*d^(1/4)* 
Sqrt[c - d*x^2]) - (3*a^2*d*(9*b*c - 7*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)*Sqr 
t[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)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2) 
*Sqrt[c - d*x^2])))/b))/(3*b*d))/(4*b*(b*c - a*d)))
 

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 970

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) 
^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) 
/(b*n*(b*c - a*d)*(p + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d 
*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ 
n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ 
n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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 1052

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m 
- n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 
 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1))   Int[(g*x)^(m - n)*(a + 
 b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( 
f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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. \(1050\) vs. \(2(355)=710\).

Time = 2.94 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.37

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1051\)
risch	\(\text {Expression too large to display}\)	\(1262\)
default	\(\text {Expression too large to display}\)	\(2845\)

Input:

int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(-1/2*e^5*a^2/(a 
*d-b*c)/b^2*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2/3/b^2*e^5/d*(-d*e*x^3+c*e* 
x)^(1/2)+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*e^6/b^3-1/4*(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))*a^2/b^3*e^6/(a*d-b*c)+1/3/d^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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2) 
)/b^2*e^6*c+7/8*a^3*e^6/b^3/(a*d-b*c)/(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))-9/8*a^2*e^6/b^2/(a*d-b*c)/(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-7/8*a^3*e^6/b^3/(a*d-b*c)/(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/(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(11/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((e*x)^(11/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate((e*x)^(11/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{11/2}}{{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \] Input:

int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)),x)
 

Output:

int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {too large to display} \] Input:

int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)
 

Output:

(sqrt(e)*e**5*(10*sqrt(x)*sqrt(c - d*x**2)*a*c + 6*sqrt(x)*sqrt(c - d*x**2 
)*a*d*x**2 - 6*sqrt(x)*sqrt(c - d*x**2)*b*c*x**2 - 5*int(sqrt(c - d*x**2)/ 
(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x) 
*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 
 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2* 
x**4 + sqrt(x)*b**3*c*d*x**6),x)*a**4*c**2*d + 5*int(sqrt(c - d*x**2)/(sqr 
t(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a** 
2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - s 
qrt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 
 + sqrt(x)*b**3*c*d*x**6),x)*a**3*b*c**3 + 5*int(sqrt(c - d*x**2)/(sqrt(x) 
*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b* 
c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt( 
x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + s 
qrt(x)*b**3*c*d*x**6),x)*a**3*b*c**2*d*x**2 - 5*int(sqrt(c - d*x**2)/(sqrt 
(x)*a**3*c*d - sqrt(x)*a**3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2 
*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sq 
rt(x)*a*b**2*c*d*x**4 - sqrt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 
+ sqrt(x)*b**3*c*d*x**6),x)*a**2*b**2*c**3*x**2 + 21*int((sqrt(x)*sqrt(c - 
 d*x**2)*x**3)/(a**3*c*d - a**3*d**2*x**2 - a**2*b*c**2 - a**2*b*c*d*x**2 
+ 2*a**2*b*d**2*x**4 + 2*a*b**2*c**2*x**2 - a*b**2*c*d*x**4 - a*b**2*d*...