\(\int \frac {(e x)^{7/2}}{(a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [1148]

Optimal result

Integrand size = 30, antiderivative size = 420 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {(2 b c+a d) e^3 \sqrt {e x}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 \sqrt {e x}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (2 b c+a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c+a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c+a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \] Output:

1/2*(a*d+2*b*c)*e^3*(e*x)^(1/2)/b/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*a*e^3* 
(e*x)^(1/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(1/2)+1/2*c^(1/4)*(a*d+2*b* 
c)*e^(7/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2) 
,I)/b/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(a*d+5*b*c)*e^(7/2 
)*(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/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)-1/4*c 
^(1/4)*(a*d+5*b*c)*e^(7/2)*(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/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 = 11.18 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e^3 \sqrt {e x} \left (5 a \left (3 a c-2 b c x^2-a d x^2\right )+15 a c \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(2 b c+a d) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

-1/10*(e^3*Sqrt[e*x]*(5*a*(3*a*c - 2*b*c*x^2 - a*d*x^2) + 15*a*c*(-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*c + a*d)*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]))/(a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2] 
)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.01, 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, 1024, 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.

\(\displaystyle \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 970

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

\(\Big \downarrow \) 1024

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

Int[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

2*e^3*((a*e^2*Sqrt[e*x])/(4*b*(b*c - a*d)*Sqrt[c - d*x^2]*(a*e^2 - b*e^2*x 
^2)) - (-(((2*b*c + a*d)*Sqrt[e*x])/((b*c - a*d)*Sqrt[c - d*x^2])) + (b*(- 
((c^(1/4)*(2*b*c + 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*d^(1/4)*Sqrt[c - d*x^2])) + (a* 
(5*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]*El 
lipticPi[(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))/(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 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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. \(955\) vs. \(2(332)=664\).

Time = 2.12 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.28

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {a \,e^{3} \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right )^{2} \left (-b \,x^{2}+a \right )}+\frac {e^{4} x c}{\left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d 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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4} a}{4 \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )^{2} b}+\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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c \,e^{4}}{2 d \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )^{2}}+\frac {a^{2} e^{4} \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 )}{8 b \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {5 a \,e^{4} \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 ) c}{8 \left (a d -b c \right )^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {a^{2} e^{4} \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 )}{8 b \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {5 a \,e^{4} \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 ) c}{8 \left (a d -b c \right )^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\)	\(956\)
default	\(\text {Expression too large to display}\)	\(2518\)

Input:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(7/2)/(-b*x**2+a)**2/(-d*x**2+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate((e*x)^(7/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((e*x)^(7/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:

int((e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x)
 

Output:

int((e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x)
 

Output:

(sqrt(e)*e**3*(sqrt(c - d*x**2)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 
2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 
 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x* 
*4 - 2*sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a**2*c - sqrt(c 
- d*x**2)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x** 
2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x 
**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c* 
d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a*b*c*x**2 + sqrt(c - d*x**2)*int((sqr 
t(c - d*x**2)*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x) 
*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqr 
t(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sq 
rt(x)*b**2*d**2*x**8),x)*a**2*d + 3*sqrt(c - d*x**2)*int((sqrt(c - d*x**2) 
*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x* 
*4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2 
*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d* 
*2*x**8),x)*a*b*c - sqrt(c - d*x**2)*int((sqrt(c - d*x**2)*x**2)/(sqrt(x)* 
a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a 
*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)* 
b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a*b* 
d*x**2 - 3*sqrt(c - d*x**2)*int((sqrt(c - d*x**2)*x**2)/(sqrt(x)*a**2*c...