\(\int \frac {(e x)^{3/2}}{(a-b x^2)^2 \sqrt {c-d x^2}} \, dx\) [1140]

Optimal result

Integrand size = 30, antiderivative size = 363 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \] Output:

1/2*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/(-a*d+b*c)/(-b*x^2+a)+1/2*c^(1/4)*d^(3/ 
4)*e^(3/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2) 
,I)/b/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(a*d+b*c)*e^(3/2)*(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)/a/b/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(a*d+ 
b*c)*e^(3/2)*(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)/a/b/d^(1/4)/(-a*d+b*c)/(-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 = 169, normalized size of antiderivative = 0.47 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {e \sqrt {e x} \left (5 a \left (c-d x^2\right )+5 c \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a (-b c+a d) \left (a-b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {368, 27, 971, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

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

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 971

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* 
((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) 
*(p + 1))   Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - 
 n + 1) + d*(m + n*(p + q + 1) + 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] && GeQ[n, m - n + 
 1] && GtQ[m - n + 1, 0] && 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. \(812\) vs. \(2(281)=562\).

Time = 1.61 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.24

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {e \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right ) \left (-b \,x^{2}+a \right )}-\frac {e^{2} \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 )}{4 \left (a d -b c \right ) b \sqrt {-d e \,x^{3}+c e x}}-\frac {e^{2} \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 ) a}{8 \left (a d -b c \right ) b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{2} \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 ) \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 {e^{2} \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 ) a}{8 \left (a d -b c \right ) b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {e^{2} \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 ) \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}}\)	\(813\)
default	\(\text {Expression too large to display}\)	\(2246\)

Input:

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

Fricas [F(-1)]

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

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)),x)
 

Output:

int((e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) e \] Input:

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

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*x)/(a**2*c - a**2*d*x**2 - 2*a*b*c*x 
**2 + 2*a*b*d*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*e