\(\int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx\) [1021]

Optimal result

Integrand size = 36, antiderivative size = 133 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\sqrt {e x} \sqrt {b c^2-b d^2 x^2}}{2 b c d (c+d x)^{3/2}}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {b c^2-b d^2 x^2}}\right )}{2 \sqrt {2} \sqrt {b} c d^{3/2}} \] Output:

-1/2*(e*x)^(1/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/c/d/(d*x+c)^(3/2)+1/4*e^(1/2)* 
arctan(1/e^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2)*2^(1/2)*b^(1/2)*d^(1/2)*(e*x)^(1 
/2)*(d*x+c)^(1/2))*2^(1/2)/b^(1/2)/c/d^(3/2)
 

Mathematica [A] (verified)

Time = 4.75 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {e x} \sqrt {c-d x} \left (-2 \sqrt {d} \sqrt {x} \sqrt {c-d x}+\sqrt {2} (c+d x) \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )}{4 c d^{3/2} \sqrt {x} \sqrt {c+d x} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

(Sqrt[e*x]*Sqrt[c - d*x]*(-2*Sqrt[d]*Sqrt[x]*Sqrt[c - d*x] + Sqrt[2]*(c + 
d*x)*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[x])/Sqrt[c - d*x]]))/(4*c*d^(3/2)*Sqrt[x 
]*Sqrt[c + d*x]*Sqrt[b*(c^2 - d^2*x^2)])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {586, 105, 104, 218}

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.

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

\(\Big \downarrow \) 586

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(-1/2*(Sqrt[e*x]*Sqrt[b*c - b*d*x])/(b*c* 
d*(c + d*x)) + (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[b]*Sqrt[d]*Sqrt[e*x])/(Sqrt[e 
]*Sqrt[b*c - b*d*x])])/(2*Sqrt[2]*Sqrt[b]*c*d^(3/2))))/Sqrt[b*c^2 - b*d^2* 
x^2]
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 586

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( 
b*x)/d)^FracPart[p])   Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(101)=202\).

Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.64

method	result	size
default	\(-\frac {\sqrt {e x}\, \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b c d e x +\ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d \right ) \sqrt {2}}{8 b c \left (d x +c \right )^{\frac {3}{2}} d^{2} \sqrt {\left (-d x +c \right ) b e x}\, \sqrt {-\frac {b \,c^{2} e}{d}}}\)	\(218\)

Input:

int((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/8*(e*x)^(1/2)*(b*(-d^2*x^2+c^2))^(1/2)/b/c*(ln((3*b*c*d*e*x-b*c^2*e+2*2 
^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/2)*d)/(d*x+c))*b*c*d*e*x+l 
n((3*b*c*d*e*x-b*c^2*e+2*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/ 
2)*d)/(d*x+c))*b*c^2*e+2*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/ 
2)*d)*2^(1/2)/(d*x+c)^(3/2)/d^2/((-d*x+c)*b*e*x)^(1/2)/(-1/d*b*c^2*e)^(1/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\left [\frac {\sqrt {\frac {1}{2}} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-\frac {e}{b d}} \log \left (-\frac {17 \, d^{3} e x^{3} + 3 \, c d^{2} e x^{2} - 13 \, c^{2} d e x + c^{3} e + 8 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (3 \, d^{2} x - c d\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {e}{b d}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 4 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x}}{8 \, {\left (b c d^{3} x^{2} + 2 \, b c^{2} d^{2} x + b c^{3} d\right )}}, -\frac {\sqrt {\frac {1}{2}} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {\frac {e}{b d}} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{b d}}}{3 \, d^{2} e x^{2} + 2 \, c d e x - c^{2} e}\right ) + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x}}{4 \, {\left (b c d^{3} x^{2} + 2 \, b c^{2} d^{2} x + b c^{3} d\right )}}\right ] \] Input:

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

Output:

[1/8*(sqrt(1/2)*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(-e/(b*d))*log(-(17*d^ 
3*e*x^3 + 3*c*d^2*e*x^2 - 13*c^2*d*e*x + c^3*e + 8*sqrt(1/2)*sqrt(-b*d^2*x 
^2 + b*c^2)*(3*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e/(b*d)))/(d^3*x 
^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 4*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x 
 + c)*sqrt(e*x))/(b*c*d^3*x^2 + 2*b*c^2*d^2*x + b*c^3*d), -1/4*(sqrt(1/2)* 
(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(e/(b*d))*arctan(4*sqrt(1/2)*sqrt(-b*d 
^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/(b*d))/(3*d^2*e*x^2 + 2*c 
*d*e*x - c^2*e)) + 2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x))/(b* 
c*d^3*x^2 + 2*b*c^2*d^2*x + b*c^3*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (101) = 202\).

Time = 0.23 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\frac {\sqrt {2} \sqrt {-b d e} e^{2} \log \left (\frac {{\left | -4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}{{\left | 4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}\right )}{d^{2} {\left | b \right |} {\left | c \right |} {\left | e \right |}} + \frac {8 \, {\left (\sqrt {-b d e} b c e^{6} - 3 \, \sqrt {-b d e} {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} e^{4}\right )}}{{\left (b^{2} c^{2} e^{4} - 6 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} b c e^{2} + {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{4}\right )} d^{2} {\left | e \right |}}}{8 \, e} \] Input:

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

Output:

-1/8*(sqrt(2)*sqrt(-b*d*e)*e^2*log(abs(-4*sqrt(2)*e^2*abs(b)*abs(c) - 6*b* 
c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2)/abs(4*s 
qrt(2)*e^2*abs(b)*abs(c) - 6*b*c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b 
*d*e^2*x + b*c*e^2))^2))/(d^2*abs(b)*abs(c)*abs(e)) + 8*(sqrt(-b*d*e)*b*c* 
e^6 - 3*sqrt(-b*d*e)*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2)) 
^2*e^4)/((b^2*c^2*e^4 - 6*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c* 
e^2))^2*b*c*e^2 + (sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^4) 
*d^2*abs(e)))/e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {e}\, \sqrt {b}\, \left (-4 \sqrt {x}\, \sqrt {-d x +c}\, d +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x \right )}{8 b c \,d^{2} \left (d x +c \right )} \] Input:

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

Output:

(sqrt(e)*sqrt(b)*( - 4*sqrt(x)*sqrt(c - d*x)*d + sqrt(d)*sqrt(2)*log((sqrt 
(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i + 
sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*s 
qrt(d)*i)/sqrt(c))*d*i*x - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sq 
rt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i - sqrt(d)*sqrt(2)*log((s 
qrt(c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i 
*x - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt 
(x)*sqrt(d)*i)/sqrt(c))*c*i - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c) 
*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x + sqrt(d)*sqrt(2)*l 
og((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c) 
)*c*i + sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + s 
qrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x))/(8*b*c*d**2*(c + d*x))