\(\int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx\) [1599]

Optimal result

Integrand size = 28, antiderivative size = 697 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {a+b x^2}}{3 a c e (e x)^{3/2}}+\frac {4 \sqrt {-a} d \left (\sqrt {b} c+\sqrt {-a} d\right ) \sqrt {\sqrt {-a} \sqrt {b} c+a d} x \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (b c^2+a d^2\right ) x^2}} \sqrt {1+\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c-a d\right ) x}} \sqrt {1-\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c+a d\right ) x}} E\left (\arcsin \left (\frac {\sqrt {a} \sqrt {e} \sqrt {c+d x}}{\sqrt {\sqrt {-a} \sqrt {b} c+a d} \sqrt {e x}}\right )|-\frac {\sqrt {-a} \sqrt {b} c+a d}{\sqrt {-a} \sqrt {b} c-a d}\right )}{3 a^{3/2} c^3 e^{5/2} \sqrt {a+b x^2} \sqrt {1-\frac {2 a d (c+d x)}{\left (b c^2+a d^2\right ) x}+\frac {a (c+d x)^2}{\left (b c^2+a d^2\right ) x^2}}}+\frac {2 \sqrt {b} \left (\sqrt {b} c-2 \sqrt {-a} d\right ) \sqrt {\sqrt {-a} \sqrt {b} c+a d} x \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (b c^2+a d^2\right ) x^2}} \sqrt {1+\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c-a d\right ) x}} \sqrt {1-\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c+a d\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {e} \sqrt {c+d x}}{\sqrt {\sqrt {-a} \sqrt {b} c+a d} \sqrt {e x}}\right ),-\frac {\sqrt {-a} \sqrt {b} c+a d}{\sqrt {-a} \sqrt {b} c-a d}\right )}{3 a^{3/2} c^2 e^{5/2} \sqrt {a+b x^2} \sqrt {1-\frac {2 a d (c+d x)}{\left (b c^2+a d^2\right ) x}+\frac {a (c+d x)^2}{\left (b c^2+a d^2\right ) x^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.25 (sec) , antiderivative size = 529, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\frac {2 x \left (-2 d x (c+d x) \left (a+b x^2\right )-(c-2 d x) (c+d x) \left (a+b x^2\right )+\frac {\left (i \sqrt {b}+\frac {\sqrt {a}}{x}\right ) x^3 \sqrt {\frac {\sqrt {a} (c+d x)}{\left (-i \sqrt {b} c+\sqrt {a} d\right ) x}} \left (b c^2 \sqrt {2+\frac {2 i \sqrt {a}}{\sqrt {b} x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c \left (\sqrt {b}-\frac {i \sqrt {a}}{x}\right )}{\sqrt {b} c+i \sqrt {a} d}}\right ),\frac {c+\frac {i \sqrt {a} d}{\sqrt {b}}}{2 c}\right )+4 \sqrt {a} d \sqrt {\frac {i \sqrt {a} c+\sqrt {b} c x}{\sqrt {b} c x-i \sqrt {a} d x}} \left (\left (i \sqrt {b} c+\sqrt {a} d\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-i \sqrt {b} c+\sqrt {a} d}}\right )|\frac {-i \sqrt {b} c+\sqrt {a} d}{i \sqrt {b} c+\sqrt {a} d}\right )-i \sqrt {b} c \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-i \sqrt {b} c+\sqrt {a} d}}\right ),\frac {-i \sqrt {b} c+\sqrt {a} d}{i \sqrt {b} c+\sqrt {a} d}\right )\right )\right )}{2 \sqrt {a} \sqrt {\frac {c \left (-i \sqrt {a}+\sqrt {b} x\right )}{\left (\sqrt {b} c+i \sqrt {a} d\right ) x}}}\right )}{3 a c^2 (e x)^{5/2} \sqrt {c+d x} \sqrt {a+b x^2}} \] Input:

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

Output:

(2*x*(-2*d*x*(c + d*x)*(a + b*x^2) - (c - 2*d*x)*(c + d*x)*(a + b*x^2) + ( 
(I*Sqrt[b] + Sqrt[a]/x)*x^3*Sqrt[(Sqrt[a]*(c + d*x))/(((-I)*Sqrt[b]*c + Sq 
rt[a]*d)*x)]*(b*c^2*Sqrt[2 + ((2*I)*Sqrt[a])/(Sqrt[b]*x)]*EllipticF[ArcSin 
[Sqrt[(c*(Sqrt[b] - (I*Sqrt[a])/x))/(Sqrt[b]*c + I*Sqrt[a]*d)]], (c + (I*S 
qrt[a]*d)/Sqrt[b])/(2*c)] + 4*Sqrt[a]*d*Sqrt[(I*Sqrt[a]*c + Sqrt[b]*c*x)/( 
Sqrt[b]*c*x - I*Sqrt[a]*d*x)]*((I*Sqrt[b]*c + Sqrt[a]*d)*EllipticE[ArcSin[ 
Sqrt[(Sqrt[a]*(d + c/x))/((-I)*Sqrt[b]*c + Sqrt[a]*d)]], ((-I)*Sqrt[b]*c + 
 Sqrt[a]*d)/(I*Sqrt[b]*c + Sqrt[a]*d)] - I*Sqrt[b]*c*EllipticF[ArcSin[Sqrt 
[(Sqrt[a]*(d + c/x))/((-I)*Sqrt[b]*c + Sqrt[a]*d)]], ((-I)*Sqrt[b]*c + Sqr 
t[a]*d)/(I*Sqrt[b]*c + Sqrt[a]*d)])))/(2*Sqrt[a]*Sqrt[(c*((-I)*Sqrt[a] + S 
qrt[b]*x))/((Sqrt[b]*c + I*Sqrt[a]*d)*x)])))/(3*a*c^2*(e*x)^(5/2)*Sqrt[c + 
 d*x]*Sqrt[a + b*x^2])
 

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2352

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2354

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

\(\Big \downarrow \) 2261

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

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. \(1312\) vs. \(2(575)=1150\).

Time = 18.76 (sec) , antiderivative size = 1313, normalized size of antiderivative = 1.88

method	result	size
risch	\(\text {Expression too large to display}\)	\(1313\)
elliptic	\(\text {Expression too large to display}\)	\(1388\)
default	\(\text {Expression too large to display}\)	\(2907\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left (6 \, \sqrt {a c e} a c d x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right ) + 3 \, \sqrt {b x^{2} + a} \sqrt {d x + c} \sqrt {e x} a c^{2} - {\left (3 \, b c^{2} - 2 \, a d^{2}\right )} \sqrt {a c e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right )}}{9 \, a^{2} c^{3} e^{3} x^{2}} \] Input:

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

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)* 
sqrt(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="maxima 
")
 

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