\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx\) [60]

Optimal result

Integrand size = 48, antiderivative size = 134 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx=-\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g \sqrt {d+e x} \sqrt {f+g x}}+\frac {2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{g^{3/2}} \] Output:

-2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(e*x+d)^(1/2)/(g*x+f)^(1/2)+2 
*c^(1/2)*d^(1/2)*arctanh(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c 
^(1/2)/d^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(1/2))/g^(3/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (-\sqrt {g} \sqrt {a e+c d x}+\sqrt {c} \sqrt {d} \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{g^{3/2} \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \] Input:

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

Output:

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-(Sqrt[g]*Sqrt[a*e + c*d*x]) + Sqrt[c] 
*Sqrt[d]*Sqrt[f + g*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sq 
rt[a*e + c*d*x])]))/(g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1249, 1268, 66, 221}

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.

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e+c d x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} \sqrt {f+g x}}\)

Input:

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

Output:

(-2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*Sqrt[d + e*x]*Sqrt[f + 
 g*x]) + (2*Sqrt[c]*Sqrt[d]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[ 
g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(g^(3/2)*Sqrt[a*d* 
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
 

Defintions of rubi rules used

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 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 1249

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(f + g*x)^(n + 1)*((a + 
 b*x + c*x^2)^p/(g*(n + 1))), x] + Simp[c*(m/(e*g*(n + 1)))   Int[(d + e*x) 
^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && G 
tQ[p, 0] && LtQ[n, -1] &&  !(IntegerQ[n + p] && LeQ[n + p + 2, 0])
 

rule 1268

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

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.40

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (\ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c d g x +\ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c d f -2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\right )}{\sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, g \sqrt {g x +f}\, \sqrt {e x +d}}\)	\(187\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 521, normalized size of antiderivative = 3.89 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx=\left [\frac {{\left (e g x^{2} + d f + {\left (e f + d g\right )} x\right )} \sqrt {\frac {c d}{g}} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, {\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {\frac {c d}{g}} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e g^{2} x^{2} + d f g + {\left (e f g + d g^{2}\right )} x\right )}}, -\frac {{\left (e g x^{2} + d f + {\left (e f + d g\right )} x\right )} \sqrt {-\frac {c d}{g}} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {-\frac {c d}{g}} g}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{e g^{2} x^{2} + d f g + {\left (e f g + d g^{2}\right )} x}\right ] \] Input:

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

Output:

[1/2*((e*g*x^2 + d*f + (e*f + d*g)*x)*sqrt(c*d/g)*log(-(8*c^2*d^2*e*g^2*x^ 
3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 4*(2*c*d*g^2*x + c*d*f 
*g + a*e*g^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sq 
rt(g*x + f)*sqrt(c*d/g) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^ 
2 + (c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2* 
e^3)*g^2)*x)/(e*x + d)) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq 
rt(e*x + d)*sqrt(g*x + f))/(e*g^2*x^2 + d*f*g + (e*f*g + d*g^2)*x), -((e*g 
*x^2 + d*f + (e*f + d*g)*x)*sqrt(-c*d/g)*arctan(2*sqrt(c*d*e*x^2 + a*d*e + 
 (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-c*d/g)*g/(2*c*d*e*g* 
x^2 + c*d^2*f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)) + 2*sqrt(c*d 
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e*g^2*x^ 
2 + d*f*g + (e*f*g + d*g^2)*x)]
 

Sympy [F]

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

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

Output:

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(sqrt(d + e*x)*(f + g*x)**(3/2)), x 
)
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(3 
/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x 
+ f)^(3/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {e^{2} {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \right |}\right )}{\sqrt {c d g} g {\left | e \right |}} + \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} e^{2} {\left | c \right |} {\left | d \right |}}{\sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} g {\left | e \right |}}\right )} {\left | e \right |}}{e^{2}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(3 
/2),x, algorithm="giac")
 

Output:

-2*(e^2*abs(c)*abs(d)*log(abs(-sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*sqr 
t(c*d*g) + sqrt(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + 
 a*e^3)*c*d*g)))/(sqrt(c*d*g)*g*abs(e)) + sqrt((e*x + d)*c*d*e - c*d^2*e + 
 a*e^3)*e^2*abs(c)*abs(d)/(sqrt(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x + d)*c 
*d*e - c*d^2*e + a*e^3)*c*d*g)*g*abs(e)))*abs(e)/e^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx=\frac {-2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, g +2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) f +2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) g x -2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, f -2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, g x}{g^{2} \left (g x +f \right )} \] Input:

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

Output:

(2*( - sqrt(f + g*x)*sqrt(a*e + c*d*x)*g + sqrt(g)*sqrt(d)*sqrt(c)*log((sq 
rt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d* 
f))*f + sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*s 
qrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*g*x - sqrt(g)*sqrt(d)*sqrt(c)*f 
 - sqrt(g)*sqrt(d)*sqrt(c)*g*x))/(g**2*(f + g*x))