\(\int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx\) [1598]

Optimal result

Integrand size = 28, antiderivative size = 538 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=-\frac {2 \left (b+\frac {a}{x^2}\right ) \sqrt {e x} \sqrt {c+d x}}{\sqrt {a} \sqrt {b c^2+a d^2} e^2 \left (1+\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b c^2+a d^2}}\right ) \sqrt {a+b x^2}}+\frac {2 \left (b c^2+a d^2\right )^{3/4} \left (1+\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b c^2+a d^2}}\right ) \sqrt {\frac {c^2 \left (b+\frac {a}{x^2}\right )}{\left (b c^2+a d^2\right ) \left (1+\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b c^2+a d^2}}\right )^2}} \sqrt {e x} \sqrt {c+d x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {d+\frac {c}{x}}}{\sqrt [4]{b c^2+a d^2}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}}\right )\right )}{a^{3/4} c^2 e^2 \sqrt {d+\frac {c}{x}} \sqrt {a+b x^2}}+\frac {\left (\sqrt {a} d-\sqrt {b c^2+a d^2}\right ) x \sqrt {c+d x} \sqrt {a+b x^2} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {d+\frac {c}{x}}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {a} d}{\sqrt {b c^2+a d^2}}\right )\right )}{a^{3/4} \sqrt [4]{b c^2+a d^2} \sqrt {d+\frac {c}{x}} (e x)^{3/2} \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (\sqrt {b c^2+a d^2} x+\sqrt {a} (c+d x)\right )^2}} \left (\sqrt {b c^2+a d^2} x+\sqrt {a} (c+d x)\right )} \] Output:

-2*(b+a/x^2)*(e*x)^(1/2)*(d*x+c)^(1/2)/a^(1/2)/(a*d^2+b*c^2)^(1/2)/e^2/(1+ 
a^(1/2)*(d+c/x)/(a*d^2+b*c^2)^(1/2))/(b*x^2+a)^(1/2)+2*(a*d^2+b*c^2)^(3/4) 
*(1+a^(1/2)*(d+c/x)/(a*d^2+b*c^2)^(1/2))*(c^2*(b+a/x^2)/(a*d^2+b*c^2)/(1+a 
^(1/2)*(d+c/x)/(a*d^2+b*c^2)^(1/2))^2)^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2)*Ell 
ipticE(sin(2*arctan(a^(1/4)*(d+c/x)^(1/2)/(a*d^2+b*c^2)^(1/4))),1/2*(2+2*a 
^(1/2)/(a*d^2+b*c^2)^(1/2)*d)^(1/2))/a^(3/4)/c^2/e^2/(d+c/x)^(1/2)/(b*x^2+ 
a)^(1/2)+(a^(1/2)*d-(a*d^2+b*c^2)^(1/2))*x*(d*x+c)^(1/2)*(b*x^2+a)^(1/2)*I 
nverseJacobiAM(2*arctan(a^(1/4)*(d+c/x)^(1/2)/(a*d^2+b*c^2)^(1/4)),1/2*(2+ 
2*a^(1/2)/(a*d^2+b*c^2)^(1/2)*d)^(1/2))/a^(3/4)/(a*d^2+b*c^2)^(1/4)/(d+c/x 
)^(1/2)/(e*x)^(3/2)/(c^2*(b*x^2+a)/(x*(a*d^2+b*c^2)^(1/2)+a^(1/2)*(d*x+c)) 
^2)^(1/2)/(x*(a*d^2+b*c^2)^(1/2)+a^(1/2)*(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.68 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\frac {2 x \sqrt {c+d x} \left (-\frac {a+b x^2}{a}+\frac {\sqrt {b} (c+d x) \sqrt {\frac {\left (b c^2+a d^2\right ) \left (a+b x^2\right )}{a b (c+d x)^2}} \left (\left (\sqrt {b} c+i \sqrt {a} d\right ) \sqrt {\frac {\left (\frac {i \sqrt {b} c}{\sqrt {a}}+d\right ) x}{c+d x}} E\left (\arcsin \left (\sqrt {\frac {c+\frac {i \sqrt {a} d}{\sqrt {b}}-\frac {i \sqrt {b} c x}{\sqrt {a}}+d x}{2 c+2 d x}}\right )|\frac {2 \sqrt {b} c}{\sqrt {b} c+i \sqrt {a} d}\right )+d \left (\sqrt {b} x \sqrt {\frac {\left (b c^2+a d^2\right ) \left (a+b x^2\right )}{a b (c+d x)^2}}-i \sqrt {a} \sqrt {\frac {\left (\frac {i \sqrt {b} c}{\sqrt {a}}+d\right ) x}{c+d x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c+\frac {i \sqrt {a} d}{\sqrt {b}}-\frac {i \sqrt {b} c x}{\sqrt {a}}+d x}{2 c+2 d x}}\right ),\frac {2 \sqrt {b} c}{\sqrt {b} c+i \sqrt {a} d}\right )\right )\right )}{b c^2+a d^2}\right )}{c (e x)^{3/2} \sqrt {a+b x^2}} \] Input:

Integrate[1/((e*x)^(3/2)*Sqrt[c + d*x]*Sqrt[a + b*x^2]),x]
 

Output:

(2*x*Sqrt[c + d*x]*(-((a + b*x^2)/a) + (Sqrt[b]*(c + d*x)*Sqrt[((b*c^2 + a 
*d^2)*(a + b*x^2))/(a*b*(c + d*x)^2)]*((Sqrt[b]*c + I*Sqrt[a]*d)*Sqrt[(((I 
*Sqrt[b]*c)/Sqrt[a] + d)*x)/(c + d*x)]*EllipticE[ArcSin[Sqrt[(c + (I*Sqrt[ 
a]*d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[b] 
*c)/(Sqrt[b]*c + I*Sqrt[a]*d)] + d*(Sqrt[b]*x*Sqrt[((b*c^2 + a*d^2)*(a + b 
*x^2))/(a*b*(c + d*x)^2)] - I*Sqrt[a]*Sqrt[(((I*Sqrt[b]*c)/Sqrt[a] + d)*x) 
/(c + d*x)]*EllipticF[ArcSin[Sqrt[(c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sqrt[b]* 
c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[b]*c)/(Sqrt[b]*c + I*Sqrt[a]* 
d)])))/(b*c^2 + a*d^2)))/(c*(e*x)^(3/2)*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.

\(\displaystyle \int \frac {1}{(e x)^{3/2} \sqrt {a+b x^2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 636

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2354

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2251

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

Input:

Int[1/((e*x)^(3/2)*Sqrt[c + d*x]*Sqrt[a + b*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, 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 2251

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

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. \(1028\) vs. \(2(459)=918\).

Time = 15.46 (sec) , antiderivative size = 1029, normalized size of antiderivative = 1.91

method	result	size
risch	\(\text {Expression too large to display}\)	\(1029\)
elliptic	\(\text {Expression too large to display}\)	\(1065\)
default	\(\text {Expression too large to display}\)	\(2136\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.35 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (\sqrt {a c e} d {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right ) + 3 \, \sqrt {a c e} c {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, b c^{2} - a d^{2}\right )}}{3 \, a c^{2}}, -\frac {8 \, {\left (9 \, b c^{2} d + a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right )\right )}}{3 \, a c^{2} e^{2}} \] Input:

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

Output:

2/3*(sqrt(a*c*e)*d*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(a*c*e)*c*w 
eierstrassZeta(-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))))/(a*c^2*e^2)
 

Sympy [F]

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

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

Output:

Integral(1/((e*x)**(3/2)*sqrt(a + b*x**2)*sqrt(c + d*x)), x)
 

Maxima [F]

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

integrate(1/(e*x)^(3/2)/(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)^(3/2)), x)
 

Giac [F]

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

integrate(1/(e*x)^(3/2)/(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)^(3/2)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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