\(\int \frac {(e x)^{7/2}}{(a-b x^2) \sqrt {c-d x^2}} \, dx\) [1107]

Optimal result

Integrand size = 30, antiderivative size = 305 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\frac {2 e^3 \sqrt {e x} \sqrt {c-d x^2}}{3 b d}-\frac {2 \sqrt [4]{c} (b c+3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

2/3*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b/d-2/3*c^(1/4)*(3*a*d+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^2/d^(5 
/4)/(-d*x^2+c)^(1/2)+a*c^(1/4)*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^2/d^(1 
/4)/(-d*x^2+c)^(1/2)+a*c^(1/4)*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^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 = 11.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.48 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\frac {2 e^3 \sqrt {e x} \left (5 a \left (c-d x^2\right )-5 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(b c+3 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a b d \sqrt {c-d x^2}} \] Input:

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

Output:

(2*e^3*Sqrt[e*x]*(5*a*(c - d*x^2) - 5*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4 
, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (b*c + 3*a*d)*x^2*Sqrt[1 - (d*x^2)/ 
c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a*b*d*Sqrt[c - d 
*x^2])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 27, 979, 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.

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

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^6 x^4}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 979

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

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

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 979

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 
 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d 
*(m + n*(p + q) + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim 
p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x 
^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I 
GtQ[n, 0] && 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 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. \(483\) vs. \(2(227)=454\).

Time = 1.49 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.59

method	result	size
risch	\(\frac {2 \sqrt {-x^{2} d +c}\, x \,e^{4}}{3 b d \sqrt {e x}}-\frac {\left (\frac {\left (3 a d +b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {3 a^{2} d \left (\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{4} \sqrt {\left (-x^{2} d +c \right ) e x}}{3 d b \sqrt {e x}\, \sqrt {-x^{2} d +c}}\)	\(484\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {2 e^{3} \sqrt {-d e \,x^{3}+c e x}}{3 b d}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,e^{4}}{d \sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4} c}{3 d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}-\frac {a^{2} e^{4} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {a^{2} e^{4} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\)	\(558\)
default	\(-\frac {\left (3 \sqrt {2}\, \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^{2} b c \,d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}+3 \sqrt {2}\, \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^{2} d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-6 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}+4 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}+2 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-3 \sqrt {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 ) a^{2} b c \,d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}+3 \sqrt {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 ) a^{2} d^{2} \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}-4 a b \,d^{3} x^{3} \sqrt {a b}+4 b^{2} c \,d^{2} x^{3} \sqrt {a b}+4 a b c \,d^{2} x \sqrt {a b}-4 b^{2} c^{2} d x \sqrt {a b}\right ) e^{3} \sqrt {e x}}{6 b d \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 ) \sqrt {a b}}\)	\(842\)

Input:

int((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*(-d*x^2+c)^(1/2)*x/b/d*e^4/(e*x)^(1/2)-1/3/d/b*((3*a*d+b*c)/b/d*(c*d)^ 
(1/2)*((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2)*(-2*(x-1/d*(c*d)^(1/2))*d/ 
(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*Ellipti 
cF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+3*a^2*d/b*(1/2/( 
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))-1/2/(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))))*e^4*((-d*x^2+c)*e*x)^(1/2 
)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)
 

Fricas [F(-1)]

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

integrate((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(7/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)
 

Output:

-Integral((e*x)**(7/2)/(-a*sqrt(c - d*x**2) + b*x**2*sqrt(c - d*x**2)), x)
 

Maxima [F]

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

integrate((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((e*x)^(7/2)/((a - b*x^2)*(c - d*x^2)^(1/2)),x)
 

Output:

int((e*x)^(7/2)/((a - b*x^2)*(c - d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{3}}{b d \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}d x \right ) e^{3} \] Input:

int((e*x)^(7/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x)
 

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*x**3)/(a*c - a*d*x**2 - b*c*x**2 + b 
*d*x**4),x)*e**3