\(\int \frac {(e x)^{5/2}}{(a-b x^2) \sqrt {c-d x^2}} \, dx\) [1112]

Optimal result

Integrand size = 30, antiderivative size = 349 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

-2*c^(3/4)*e^(5/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4) 
/e^(1/2),I)/b/d^(3/4)/(-d*x^2+c)^(1/2)+2*c^(3/4)*e^(5/2)*(1-d*x^2/c)^(1/2) 
*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/b/d^(3/4)/(-d*x^2+c)^(1/ 
2)-a^(1/2)*c^(1/4)*e^(5/2)*(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^(3/2)/d^(1/4)/(-d* 
x^2+c)^(1/2)+a^(1/2)*c^(1/4)*e^(5/2)*(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^(3/2)/d^( 
1/4)/(-d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.20 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\frac {2 x (e x)^{5/2} \sqrt {\frac {c-d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{7 a \sqrt {c-d x^2}} \] Input:

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

Output:

(2*x*(e*x)^(5/2)*Sqrt[(c - d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c 
, (b*x^2)/a])/(7*a*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {368, 27, 983, 836, 27, 765, 762, 993, 1390, 1389, 327, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 983

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

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

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 983

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

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 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 [A] (verified)

Time = 0.87 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.32

method	result	size
default	\(-\frac {\left (4 \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d -4 \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}-2 \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +2 \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}\, \sqrt {a b}\, a d -\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}\, \sqrt {a b}\, a d -\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) a b c d -\operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b c d \right ) \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, e^{2} \sqrt {e x}}{2 \sqrt {-x^{2} d +c}\, x \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {a b}\, d +\sqrt {c d}\, b \right ) b}\)	\(461\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {2 e^{3} c \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}-\frac {e^{3} c \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}-\frac {a \,e^{3} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {a \,e^{3} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\)	\(509\)

Input:

int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(4*EllipticE(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c 
*d-4*EllipticE(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2- 
2*EllipticF(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d+2*E 
llipticF(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2+Ellipt 
icPi(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c 
*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*(a*b)^(1/2)*a*d-EllipticPi(((x*d+(c* 
d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1 
/2*2^(1/2))*(c*d)^(1/2)*(a*b)^(1/2)*a*d-EllipticPi(((x*d+(c*d)^(1/2))/(c*d 
)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*a* 
b*c*d-EllipticPi(((x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d 
)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*a*b*c*d)*(-x*d/(c*d)^(1/2))^(1/2)*(( 
-x*d+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((x*d+(c*d)^(1/2))/(c*d)^(1/2 
))^(1/2)*e^2*(e*x)^(1/2)/(-d*x^2+c)^(1/2)/x/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/ 
((a*b)^(1/2)*d+(c*d)^(1/2)*b)/b
 

Fricas [F(-1)]

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

integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(1/2)),x)
 

Output:

int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{b d \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}d x \right ) e^{2} \] Input:

int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x)
 

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a*c - a*d*x**2 - b*c*x**2 + b 
*d*x**4),x)*e**2