\(\int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [1593]

Optimal result

Integrand size = 29, antiderivative size = 359 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {a-b x^2}}{3 a c e (e x)^{3/2}}-\frac {4 \sqrt {b} d \sqrt {\frac {b-\frac {a}{x^2}}{b}} \sqrt {e x} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} c^2 e^3 \sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} \left (b c^2+2 a d^2\right ) \sqrt {\frac {b-\frac {a}{x^2}}{b}} \sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b} c+\sqrt {a} d}} (e x)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{3 a^{3/2} c^2 e^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 15.37 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.52 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {x^{5/2} \sqrt {a-b x^2} \left (-\frac {2 (c-2 d x) (c+d x)}{a c^2 x^{3/2}}-\frac {2 \sqrt {x} \left (2 d \left (d+\frac {c}{x}\right )+\frac {b c^2 \sqrt {1+\frac {\sqrt {a}}{\sqrt {b} x}} \sqrt {\frac {\sqrt {a} (c+d x)}{\left (\sqrt {b} c+\sqrt {a} d\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 d \sqrt {\frac {c \left (-\sqrt {a}+\sqrt {b} x\right )}{\left (\sqrt {b} c+\sqrt {a} d\right ) x}} \sqrt {\frac {\sqrt {a} (c+d x)}{\left (-\sqrt {b} c+\sqrt {a} d\right ) x}} \left (\left (\sqrt {b} c+\sqrt {a} d\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right )|\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )-\sqrt {b} c \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right ),\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )\right )}{\left (-\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {\sqrt {a} c+\sqrt {b} c x}{\sqrt {b} c x-\sqrt {a} d x}}}\right )}{a c^2}\right )}{3 (e x)^{5/2} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(5/2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
 

Output:

(x^(5/2)*Sqrt[a - b*x^2]*((-2*(c - 2*d*x)*(c + d*x))/(a*c^2*x^(3/2)) - (2* 
Sqrt[x]*(2*d*(d + c/x) + (b*c^2*Sqrt[1 + Sqrt[a]/(Sqrt[b]*x)]*Sqrt[(Sqrt[a 
]*(c + d*x))/((Sqrt[b]*c + Sqrt[a]*d)*x)]*EllipticF[ArcSin[Sqrt[1 - Sqrt[a 
]/(Sqrt[b]*x)]/Sqrt[2]], (2*Sqrt[b]*c)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a]* 
Sqrt[1 - Sqrt[a]/(Sqrt[b]*x)]*(Sqrt[a] + Sqrt[b]*x)) + (2*d*Sqrt[(c*(-Sqrt 
[a] + Sqrt[b]*x))/((Sqrt[b]*c + Sqrt[a]*d)*x)]*Sqrt[(Sqrt[a]*(c + d*x))/(( 
-(Sqrt[b]*c) + Sqrt[a]*d)*x)]*((Sqrt[b]*c + Sqrt[a]*d)*EllipticE[ArcSin[Sq 
rt[(Sqrt[a]*(d + c/x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]], (-(Sqrt[b]*c) + Sqrt[ 
a]*d)/(Sqrt[b]*c + Sqrt[a]*d)] - Sqrt[b]*c*EllipticF[ArcSin[Sqrt[(Sqrt[a]* 
(d + c/x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]], (-(Sqrt[b]*c) + Sqrt[a]*d)/(Sqrt[ 
b]*c + Sqrt[a]*d)]))/((-Sqrt[a] + Sqrt[b]*x)*Sqrt[(Sqrt[a]*c + Sqrt[b]*c*x 
)/(Sqrt[b]*c*x - Sqrt[a]*d*x)])))/(a*c^2)))/(3*(e*x)^(5/2)*Sqrt[c + d*x])
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{(e x)^{5/2} \sqrt {a-b x^2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 636

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2352

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

\(\Big \downarrow \) 2354

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

\(\Big \downarrow \) 2261

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

Input:

Int[1/((e*x)^(5/2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 636

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/Sqrt[(a_) + (b_.)*(x_)^2], 
 x_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2] 
/(a*e*(m + 1))), x] - Simp[1/(2*a*e*(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + 
 d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2 
)*d*(2*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x 
^3 - 2*a*(m + 1)*(c + d*x)^(n + 1/2))/x, x], x], x] /; FreeQ[{a, b, c, d, e 
}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

rule 2261

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, 
p, q}, x] && PolyQ[Px, x]
 

rule 2352

Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x 
_)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m 
+ 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* 
(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 
2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 
2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, 
x] && LtQ[m, -1]
 

rule 2354

Int[(Px_)*((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[(Px /. 
 x -> x^k/e)*x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x 
], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolyQ[Px, x] 
&& FractionQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1335\) vs. \(2(289)=578\).

Time = 13.28 (sec) , antiderivative size = 1336, normalized size of antiderivative = 3.72

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1336\)
default	\(\text {Expression too large to display}\)	\(2748\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (6 \, \sqrt {a c e} a c d x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right ) + 3 \, \sqrt {-b x^{2} + a} \sqrt {d x + c} \sqrt {e x} a c^{2} + {\left (3 \, b c^{2} + 2 \, a d^{2}\right )} \sqrt {a c e} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right )}}{9 \, a^{2} c^{3} e^{3} x^{2}} \] Input:

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

Output:

-2/9*(6*sqrt(a*c*e)*a*c*d*x^2*weierstrassZeta(4/3*(3*b*c^2 + a*d^2)/(a*c^2 
), 8/27*(9*b*c^2*d - a*d^3)/(a*c^3), weierstrassPInverse(4/3*(3*b*c^2 + a* 
d^2)/(a*c^2), 8/27*(9*b*c^2*d - a*d^3)/(a*c^3), 1/3*(d*x + 3*c)/(c*x))) + 
3*sqrt(-b*x^2 + a)*sqrt(d*x + c)*sqrt(e*x)*a*c^2 + (3*b*c^2 + 2*a*d^2)*sqr 
t(a*c*e)*x^2*weierstrassPInverse(4/3*(3*b*c^2 + a*d^2)/(a*c^2), 8/27*(9*b* 
c^2*d - a*d^3)/(a*c^3), 1/3*(d*x + 3*c)/(c*x)))/(a^2*c^3*e^3*x^2)
 

Sympy [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{6}-b c \,x^{5}+a d \,x^{4}+a c \,x^{3}}d x \right )}{e^{3}} \] Input:

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

Output:

(sqrt(e)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c*x**3 + a*d*x**4 
 - b*c*x**5 - b*d*x**6),x))/e**3