\(\int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x^2)} \, dx\) [856]

Optimal result

Integrand size = 26, antiderivative size = 185 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {e}}-\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {e}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )} \, dx=\frac {d^{3/2} \sqrt {x} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+2 c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]}{\sqrt {e x}} \] Input:

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

Output:

(d^(3/2)*Sqrt[x]*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 
 - 4*b*c*#1^3 + b*#1^4 & , (c^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + 
 d*x] - #1] + 2*c*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 
 + Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^3 - 3* 
b*c^2*#1 - 8*a*d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ])/Sqrt[e*x]
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {610, 2009}

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.

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

\(\Big \downarrow \) 610

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {e}}-\frac {\sqrt {\sqrt {-a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {e}}\)

Input:

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

Output:

-((Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[ 
e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(3/4)*Sqrt[b]*Sqrt[e])) - 
 (Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[ 
e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(3/4)*Sqrt[b]*Sqrt[e])
 

Defintions of rubi rules used

rule 610

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs. \(2(133)=266\).

Time = 0.44 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.57

method	result	size
default	\(-\frac {\sqrt {d x +c}\, x e \left (\sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) \sqrt {-a b}\, c +\sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a d -\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, c +\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a d \right )}{2 \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a b \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\)	\(475\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (133) = 266\).

Time = 0.11 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.58 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )} \, dx=\frac {1}{2} \, \sqrt {-\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + d}{a b e}} \log \left (\frac {a^{2} b e^{2} x \sqrt {-\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + d}{a b e}} \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + \sqrt {d x + c} \sqrt {e x} c}{x}\right ) - \frac {1}{2} \, \sqrt {-\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + d}{a b e}} \log \left (-\frac {a^{2} b e^{2} x \sqrt {-\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + d}{a b e}} \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - \sqrt {d x + c} \sqrt {e x} c}{x}\right ) - \frac {1}{2} \, \sqrt {\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - d}{a b e}} \log \left (\frac {a^{2} b e^{2} x \sqrt {\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - d}{a b e}} \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} + \sqrt {d x + c} \sqrt {e x} c}{x}\right ) + \frac {1}{2} \, \sqrt {\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - d}{a b e}} \log \left (-\frac {a^{2} b e^{2} x \sqrt {\frac {a b e \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - d}{a b e}} \sqrt {-\frac {c^{2}}{a^{3} b e^{2}}} - \sqrt {d x + c} \sqrt {e x} c}{x}\right ) \] Input:

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

Output:

1/2*sqrt(-(a*b*e*sqrt(-c^2/(a^3*b*e^2)) + d)/(a*b*e))*log((a^2*b*e^2*x*sqr 
t(-(a*b*e*sqrt(-c^2/(a^3*b*e^2)) + d)/(a*b*e))*sqrt(-c^2/(a^3*b*e^2)) + sq 
rt(d*x + c)*sqrt(e*x)*c)/x) - 1/2*sqrt(-(a*b*e*sqrt(-c^2/(a^3*b*e^2)) + d) 
/(a*b*e))*log(-(a^2*b*e^2*x*sqrt(-(a*b*e*sqrt(-c^2/(a^3*b*e^2)) + d)/(a*b* 
e))*sqrt(-c^2/(a^3*b*e^2)) - sqrt(d*x + c)*sqrt(e*x)*c)/x) - 1/2*sqrt((a*b 
*e*sqrt(-c^2/(a^3*b*e^2)) - d)/(a*b*e))*log((a^2*b*e^2*x*sqrt((a*b*e*sqrt( 
-c^2/(a^3*b*e^2)) - d)/(a*b*e))*sqrt(-c^2/(a^3*b*e^2)) + sqrt(d*x + c)*sqr 
t(e*x)*c)/x) + 1/2*sqrt((a*b*e*sqrt(-c^2/(a^3*b*e^2)) - d)/(a*b*e))*log(-( 
a^2*b*e^2*x*sqrt((a*b*e*sqrt(-c^2/(a^3*b*e^2)) - d)/(a*b*e))*sqrt(-c^2/(a^ 
3*b*e^2)) - sqrt(d*x + c)*sqrt(e*x)*c)/x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 10.92 (sec) , antiderivative size = 669, normalized size of antiderivative = 3.62 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )} \, dx=\mathrm {atanh}\left (\frac {b\,c^{3/2}\,\sqrt {e\,x}\,\sqrt {-a^3\,b^3}\,\sqrt {\frac {c\,\sqrt {-a^3\,b^3}-a^2\,b\,d}{a^3\,b^2\,e}}-a^2\,b^2\,d\,\sqrt {e\,x}\,\sqrt {c+d\,x}\,\sqrt {\frac {c\,\sqrt {-a^3\,b^3}-a^2\,b\,d}{a^3\,b^2\,e}}-b\,c\,\sqrt {e\,x}\,\sqrt {-a^3\,b^3}\,\sqrt {c+d\,x}\,\sqrt {\frac {c\,\sqrt {-a^3\,b^3}-a^2\,b\,d}{a^3\,b^2\,e}}+a^2\,b^2\,\sqrt {c}\,d\,\sqrt {e\,x}\,\sqrt {\frac {c\,\sqrt {-a^3\,b^3}-a^2\,b\,d}{a^3\,b^2\,e}}}{a\,b^2\,c^2-c\,d\,\sqrt {-a^3\,b^3}-d^2\,x\,\sqrt {-a^3\,b^3}+\sqrt {c}\,d\,\sqrt {-a^3\,b^3}\,\sqrt {c+d\,x}-a\,b^2\,c^{3/2}\,\sqrt {c+d\,x}+a\,b^2\,c\,d\,x}\right )\,\sqrt {\frac {c\,\sqrt {-a^3\,b^3}-a^2\,b\,d}{a^3\,b^2\,e}}-\mathrm {atanh}\left (\frac {b\,c^{3/2}\,\sqrt {e\,x}\,\sqrt {-a^3\,b^3}\,\sqrt {-\frac {c\,\sqrt {-a^3\,b^3}+a^2\,b\,d}{a^3\,b^2\,e}}+a^2\,b^2\,d\,\sqrt {e\,x}\,\sqrt {c+d\,x}\,\sqrt {-\frac {c\,\sqrt {-a^3\,b^3}+a^2\,b\,d}{a^3\,b^2\,e}}-b\,c\,\sqrt {e\,x}\,\sqrt {-a^3\,b^3}\,\sqrt {c+d\,x}\,\sqrt {-\frac {c\,\sqrt {-a^3\,b^3}+a^2\,b\,d}{a^3\,b^2\,e}}-a^2\,b^2\,\sqrt {c}\,d\,\sqrt {e\,x}\,\sqrt {-\frac {c\,\sqrt {-a^3\,b^3}+a^2\,b\,d}{a^3\,b^2\,e}}}{a\,b^2\,c^2+c\,d\,\sqrt {-a^3\,b^3}+d^2\,x\,\sqrt {-a^3\,b^3}-\sqrt {c}\,d\,\sqrt {-a^3\,b^3}\,\sqrt {c+d\,x}-a\,b^2\,c^{3/2}\,\sqrt {c+d\,x}+a\,b^2\,c\,d\,x}\right )\,\sqrt {-\frac {c\,\sqrt {-a^3\,b^3}+a^2\,b\,d}{a^3\,b^2\,e}} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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