\(\int \frac {(e x)^{11/2}}{(a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [1147]

Optimal result

Integrand size = 30, antiderivative size = 443 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {c (2 b c+a d) e^5 \sqrt {e x}}{2 b d (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{5/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right ) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 d^{5/4} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 a \sqrt [4]{c} (3 b c-a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 a \sqrt [4]{c} (3 b c-a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \] Output:

1/2*c*(a*d+2*b*c)*e^5*(e*x)^(1/2)/b/d/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*a* 
e^3*(e*x)^(5/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(1/2)-1/2*c^(1/4)*(3*a^ 
2*d^2-8*a*b*c*d+2*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^2/d^(5/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)-3/4 
*a*c^(1/4)*(-a*d+3*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^2/d^(1/4)/(- 
a*d+b*c)^2/(-d*x^2+c)^(1/2)-3/4*a*c^(1/4)*(-a*d+3*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^2/d^(1/4)/(-a*d+b*c)^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 = 11.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {e^5 \sqrt {e x} \left (5 a \left (-2 a b c^2+2 b^2 c^2 x^2+a^2 d \left (-c+d x^2\right )\right )+5 a c (2 b c+a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a b d (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[(e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {368, 27, 970, 1048, 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.

\(\displaystyle \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^{10} x^6}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 970

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

\(\Big \downarrow \) 1048

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {e^2 \int -\frac {2 \left (a c (2 b c+a d) e^2-\left (2 b^2 c^2-8 a b d c+3 a^2 d^2\right ) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \int \frac {a c (2 b c+a d) e^2-\left (2 b^2 c^2-8 a b d c+3 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {\left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}+\frac {3 a^2 d e^2 (3 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}+\frac {3 a^2 d e^2 (3 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a^2 d e^2 (3 b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a^2 d e^2 (3 b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a^2 d e^2 (3 b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {3 a^2 d e^2 (3 b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {3 a^2 d e^2 (3 b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}\right )}{d (b c-a d)}-\frac {c e^2 \sqrt {e x} (a d+2 b c)}{d \sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\)

Input:

Int[(e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

2*e^3*((a*e^2*(e*x)^(5/2))/(4*b*(b*c - a*d)*Sqrt[c - d*x^2]*(a*e^2 - b*e^2 
*x^2)) - (-((c*(2*b*c + a*d)*e^2*Sqrt[e*x])/(d*(b*c - a*d)*Sqrt[c - d*x^2] 
)) + (e^2*((c^(1/4)*(2*b^2*c^2 - 8*a*b*c*d + 3*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*(3*b*c - 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]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*S 
qrt[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))/(d*(b*c - a*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 1048

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

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. \(1071\) vs. \(2(355)=710\).

Time = 2.02 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.42

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1072\)
default	\(\text {Expression too large to display}\)	\(2789\)

Input:

int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),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/b*a^2/(a*d- 
b*c)^2*e^5*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+1/d*e^6*x*c^2/(a*d-b*c)^2/(-( 
x^2-c/d)*d*e*x)^(1/2)-1/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)*Ellipt 
icF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*e^6/b^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))*a^2/b^2*e^6/(a*d-b*c)^2+1/2/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))*c^2*e^6/(a*d-b*c)^2-3/8*e^6*a^3/b^2/(a*d-b*c)^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))+9/8*e^6*a^2/b/(a*d-b*c)^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+3/8*e^6*a^3/b 
^2/(a*d-b*c)^2/(a*b)^(1/2)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(11/2)/(-b*x**2+a)**2/(-d*x**2+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="maxima" 
)
 

Output:

integrate((e*x)^(11/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((e*x)^(11/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{11/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:

int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x)
 

Output:

int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x)
 

Output:

(sqrt(e)*e**5*(10*sqrt(x)*sqrt(c - d*x**2)*a*c - 2*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**2*d - 2*sqrt(x)*a**3*c*d**2*x**2 + sqrt(x)*a**3*d**3*x**4 
 + 3*sqrt(x)*a**2*b*c**3 - 8*sqrt(x)*a**2*b*c**2*d*x**2 + 7*sqrt(x)*a**2*b 
*c*d**2*x**4 - 2*sqrt(x)*a**2*b*d**3*x**6 - 6*sqrt(x)*a*b**2*c**3*x**2 + 1 
3*sqrt(x)*a*b**2*c**2*d*x**4 - 8*sqrt(x)*a*b**2*c*d**2*x**6 + sqrt(x)*a*b* 
*2*d**3*x**8 + 3*sqrt(x)*b**3*c**3*x**4 - 6*sqrt(x)*b**3*c**2*d*x**6 + 3*s 
qrt(x)*b**3*c*d**2*x**8),x)*a**4*c**3*d + 5*int(sqrt(c - d*x**2)/(sqrt(x)* 
a**3*c**2*d - 2*sqrt(x)*a**3*c*d**2*x**2 + sqrt(x)*a**3*d**3*x**4 + 3*sqrt 
(x)*a**2*b*c**3 - 8*sqrt(x)*a**2*b*c**2*d*x**2 + 7*sqrt(x)*a**2*b*c*d**2*x 
**4 - 2*sqrt(x)*a**2*b*d**3*x**6 - 6*sqrt(x)*a*b**2*c**3*x**2 + 13*sqrt(x) 
*a*b**2*c**2*d*x**4 - 8*sqrt(x)*a*b**2*c*d**2*x**6 + sqrt(x)*a*b**2*d**3*x 
**8 + 3*sqrt(x)*b**3*c**3*x**4 - 6*sqrt(x)*b**3*c**2*d*x**6 + 3*sqrt(x)*b* 
*3*c*d**2*x**8),x)*a**4*c**2*d**2*x**2 - 15*int(sqrt(c - d*x**2)/(sqrt(x)* 
a**3*c**2*d - 2*sqrt(x)*a**3*c*d**2*x**2 + sqrt(x)*a**3*d**3*x**4 + 3*sqrt 
(x)*a**2*b*c**3 - 8*sqrt(x)*a**2*b*c**2*d*x**2 + 7*sqrt(x)*a**2*b*c*d**2*x 
**4 - 2*sqrt(x)*a**2*b*d**3*x**6 - 6*sqrt(x)*a*b**2*c**3*x**2 + 13*sqrt(x) 
*a*b**2*c**2*d*x**4 - 8*sqrt(x)*a*b**2*c*d**2*x**6 + sqrt(x)*a*b**2*d**3*x 
**8 + 3*sqrt(x)*b**3*c**3*x**4 - 6*sqrt(x)*b**3*c**2*d*x**6 + 3*sqrt(x)*b* 
*3*c*d**2*x**8),x)*a**3*b*c**4 + 20*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*...