\(\int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx\) [855]

Optimal result

Integrand size = 26, antiderivative size = 223 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx=\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} b}+\frac {2 \sqrt {d} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b}-\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} b} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.21 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx=\frac {\sqrt {d} \sqrt {e x} \left (-2 \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )+\text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-4 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{b \sqrt {x}} \] Input:

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

Output:

(Sqrt[d]*Sqrt[e*x]*(-2*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d*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 & , 
 (b*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b*c^2*Lo 
g[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 4*a*d^2*Log[c + 2 
*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b*c*Log[c + 2*d*x - 2*Sq 
rt[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) & ]))/(b*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {609, 65, 221, 2353, 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.

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

\(\Big \downarrow \) 609

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {d} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b}-\frac {e \left (\frac {\sqrt {\sqrt {-a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \sqrt {e}}-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \sqrt {e}}\right )}{b}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 221

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

rule 609

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

rule 2009

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

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(163)=326\).

Time = 0.45 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.63

method	result	size
default	\(\frac {\sqrt {e x}\, \sqrt {d x +c}\, e \left (2 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, d -\sqrt {d e}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) \sqrt {-a b}\, c -\sqrt {d e}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a d -\sqrt {d e}\, \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, c +\sqrt {d e}\, \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a d \right )}{2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-a b}\, b \sqrt {d e}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\)	\(587\)

Input:

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

Output:

1/2*(e*x)^(1/2)*(d*x+c)^(1/2)*e*(2*ln(1/2*(2*d*e*x+2*((d*x+c)*e*x)^(1/2)*( 
d*e)^(1/2)+c*e)/(d*e)^(1/2))*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*(-e*(a*d+c* 
(-a*b)^(1/2))/b)^(1/2)*(-a*b)^(1/2)*d-(d*e)^(1/2)*(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-(d*e)^(1/2)*(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-(d*e)^(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)))*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*(-a*b)^( 
1/2)*c+(d*e)^(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)))* 
(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*a*d)/((d*x+c)*e*x)^(1/2)/(-a*b)^(1/2)/b/ 
(d*e)^(1/2)/(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. 394 vs. \(2 (163) = 326\).

Time = 0.12 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.53 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx=\left [-\frac {b \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} \log \left (\frac {b x \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} + \sqrt {d x + c} \sqrt {e x}}{x}\right ) - b \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} \log \left (-\frac {b x \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} - \sqrt {d x + c} \sqrt {e x}}{x}\right ) + b \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} \log \left (\frac {b x \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} + \sqrt {d x + c} \sqrt {e x}}{x}\right ) - b \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} \log \left (-\frac {b x \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} - \sqrt {d x + c} \sqrt {e x}}{x}\right ) - 2 \, \sqrt {d e} \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right )}{2 \, b}, -\frac {b \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} \log \left (\frac {b x \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} + \sqrt {d x + c} \sqrt {e x}}{x}\right ) - b \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} \log \left (-\frac {b x \sqrt {\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} + d e}{b^{2}}} - \sqrt {d x + c} \sqrt {e x}}{x}\right ) + b \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} \log \left (\frac {b x \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} + \sqrt {d x + c} \sqrt {e x}}{x}\right ) - b \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} \log \left (-\frac {b x \sqrt {-\frac {b^{2} \sqrt {-\frac {c^{2} e^{2}}{a b^{3}}} - d e}{b^{2}}} - \sqrt {d x + c} \sqrt {e x}}{x}\right ) + 4 \, \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right )}{2 \, b}\right ] \] Input:

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

Output:

[-1/2*(b*sqrt((b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/b^2)*log((b*x*sqrt((b^2*s 
qrt(-c^2*e^2/(a*b^3)) + d*e)/b^2) + sqrt(d*x + c)*sqrt(e*x))/x) - b*sqrt(( 
b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/b^2)*log(-(b*x*sqrt((b^2*sqrt(-c^2*e^2/( 
a*b^3)) + d*e)/b^2) - sqrt(d*x + c)*sqrt(e*x))/x) + b*sqrt(-(b^2*sqrt(-c^2 
*e^2/(a*b^3)) - d*e)/b^2)*log((b*x*sqrt(-(b^2*sqrt(-c^2*e^2/(a*b^3)) - d*e 
)/b^2) + sqrt(d*x + c)*sqrt(e*x))/x) - b*sqrt(-(b^2*sqrt(-c^2*e^2/(a*b^3)) 
 - d*e)/b^2)*log(-(b*x*sqrt(-(b^2*sqrt(-c^2*e^2/(a*b^3)) - d*e)/b^2) - sqr 
t(d*x + c)*sqrt(e*x))/x) - 2*sqrt(d*e)*log(2*d*e*x + c*e + 2*sqrt(d*e)*sqr 
t(d*x + c)*sqrt(e*x)))/b, -1/2*(b*sqrt((b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/ 
b^2)*log((b*x*sqrt((b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/b^2) + sqrt(d*x + c) 
*sqrt(e*x))/x) - b*sqrt((b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/b^2)*log(-(b*x* 
sqrt((b^2*sqrt(-c^2*e^2/(a*b^3)) + d*e)/b^2) - sqrt(d*x + c)*sqrt(e*x))/x) 
 + b*sqrt(-(b^2*sqrt(-c^2*e^2/(a*b^3)) - d*e)/b^2)*log((b*x*sqrt(-(b^2*sqr 
t(-c^2*e^2/(a*b^3)) - d*e)/b^2) + sqrt(d*x + c)*sqrt(e*x))/x) - b*sqrt(-(b 
^2*sqrt(-c^2*e^2/(a*b^3)) - d*e)/b^2)*log(-(b*x*sqrt(-(b^2*sqrt(-c^2*e^2/( 
a*b^3)) - d*e)/b^2) - sqrt(d*x + c)*sqrt(e*x))/x) + 4*sqrt(-d*e)*arctan(sq 
rt(-d*e)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e)))/b]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 10.51 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{a+b x^2} \, dx=\frac {4\,\sqrt {d}\,\sqrt {e}\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {e\,x}}{\sqrt {e}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{b}-\mathrm {atanh}\left (\frac {2\,b\,\sqrt {e\,x}\,\sqrt {c+d\,x}\,\sqrt {-\frac {e\,\left (c\,\sqrt {-a\,b^5}-a\,b^2\,d\right )}{a\,b^4}}-2\,b\,\sqrt {c}\,\sqrt {e\,x}\,\sqrt {-\frac {e\,\left (c\,\sqrt {-a\,b^5}-a\,b^2\,d\right )}{a\,b^4}}}{2\,c\,e+2\,d\,e\,x-2\,\sqrt {c}\,e\,\sqrt {c+d\,x}}\right )\,\sqrt {-\frac {e\,\left (c\,\sqrt {-a\,b^5}-a\,b^2\,d\right )}{a\,b^4}}-\mathrm {atanh}\left (\frac {2\,b\,\sqrt {e\,x}\,\sqrt {c+d\,x}\,\sqrt {\frac {e\,\left (c\,\sqrt {-a\,b^5}+a\,b^2\,d\right )}{a\,b^4}}-2\,b\,\sqrt {c}\,\sqrt {e\,x}\,\sqrt {\frac {e\,\left (c\,\sqrt {-a\,b^5}+a\,b^2\,d\right )}{a\,b^4}}}{2\,c\,e+2\,d\,e\,x-2\,\sqrt {c}\,e\,\sqrt {c+d\,x}}\right )\,\sqrt {\frac {e\,\left (c\,\sqrt {-a\,b^5}+a\,b^2\,d\right )}{a\,b^4}} \] Input:

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

Output:

(4*d^(1/2)*e^(1/2)*atanh((d^(1/2)*(e*x)^(1/2))/(e^(1/2)*((c + d*x)^(1/2) - 
 c^(1/2)))))/b - atanh((2*b*(e*x)^(1/2)*(c + d*x)^(1/2)*(-(e*(c*(-a*b^5)^( 
1/2) - a*b^2*d))/(a*b^4))^(1/2) - 2*b*c^(1/2)*(e*x)^(1/2)*(-(e*(c*(-a*b^5) 
^(1/2) - a*b^2*d))/(a*b^4))^(1/2))/(2*c*e + 2*d*e*x - 2*c^(1/2)*e*(c + d*x 
)^(1/2)))*(-(e*(c*(-a*b^5)^(1/2) - a*b^2*d))/(a*b^4))^(1/2) - atanh((2*b*( 
e*x)^(1/2)*(c + d*x)^(1/2)*((e*(c*(-a*b^5)^(1/2) + a*b^2*d))/(a*b^4))^(1/2 
) - 2*b*c^(1/2)*(e*x)^(1/2)*((e*(c*(-a*b^5)^(1/2) + a*b^2*d))/(a*b^4))^(1/ 
2))/(2*c*e + 2*d*e*x - 2*c^(1/2)*e*(c + d*x)^(1/2)))*((e*(c*(-a*b^5)^(1/2) 
 + a*b^2*d))/(a*b^4))^(1/2)
 

Reduce [F]

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

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

Output:

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