\(\int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [41]

Optimal result

Integrand size = 46, antiderivative size = 280 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {5 c^3 d^3 \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {g} (c d f-a e g)^{7/2}} \] Output:

1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(e*x+d)^(1/2)/( 
g*x+f)^3+5/12*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2 
/(e*x+d)^(1/2)/(g*x+f)^2+5/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 
2)/(-a*e*g+c*d*f)^3/(e*x+d)^(1/2)/(g*x+f)-5/8*c^3*d^3*arctan(1/g^(1/2)/(a* 
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e*g+c*d*f)^(1/2)*(e*x+d)^(1/2))/g 
^(1/2)/(-a*e*g+c*d*f)^(7/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {c^3 d^3 \sqrt {d+e x} \left (\frac {(a e+c d x) \left (8 a^2 e^2 g^2-2 a c d e g (13 f+5 g x)+c^2 d^2 \left (33 f^2+40 f g x+15 g^2 x^2\right )\right )}{c^3 d^3 (c d f-a e g)^3 (f+g x)^3}+\frac {15 \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{7/2}}\right )}{24 \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2]),x]
 

Output:

(c^3*d^3*Sqrt[d + e*x]*(((a*e + c*d*x)*(8*a^2*e^2*g^2 - 2*a*c*d*e*g*(13*f 
+ 5*g*x) + c^2*d^2*(33*f^2 + 40*f*g*x + 15*g^2*x^2)))/(c^3*d^3*(c*d*f - a* 
e*g)^3*(f + g*x)^3) + (15*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d 
*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[g]*(c*d*f - a*e*g)^(7/2))))/(24*Sqrt[(a*e 
 + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1254, 1254, 1254, 1255, 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.

\(\displaystyle \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\)

\(\Big \downarrow \) 1255

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\)

Input:

Int[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] 
),x]
 

Output:

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e* 
x]*(f + g*x)^3) + (5*c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2*( 
c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c*d*(Sqrt[a*d*e + (c*d^2 + 
a*e^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c*d*Arc 
Tan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a* 
e*g]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))))/(4*(c*d*f - a*e*g) 
)))/(6*(c*d*f - a*e*g))
 

Defintions of rubi rules used

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 1254

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

rule 1255

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + 
 (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2   Subst[Int[1/(c*(e*f + d*g) - b*e 
*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ 
a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} g^{3} x^{3}+45 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+45 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} f^{3}-15 c^{2} d^{2} g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+10 a c d e \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-40 c^{2} d^{2} f g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-8 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+26 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a c d e f g -33 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -d f c \right )^{3} \left (g x +f \right )^{3} \sqrt {\left (a e g -d f c \right ) g}}\)	\(440\)

Input:

int((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

1/24*((e*x+d)*(c*d*x+a*e))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*c^3*d^3*g^3*x^3+45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d 
*f)*g)^(1/2))*c^3*d^3*f*g^2*x^2+45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d 
*f)*g)^(1/2))*c^3*d^3*f^2*g*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f 
)*g)^(1/2))*c^3*d^3*f^3-15*c^2*d^2*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f 
)*g)^(1/2)+10*a*c*d*e*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-40*c 
^2*d^2*f*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*((a*e*g-c*d*f)*g) 
^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g^2+26*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e 
)^(1/2)*a*c*d*e*f*g-33*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*f 
^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e*g-c*d*f)^3/(g*x+f)^3/((a*e*g-c*d* 
f)*g)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (248) = 496\).

Time = 0.49 (sec) , antiderivative size = 2028, normalized size of antiderivative = 7.24 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="fricas")
 

Output:

[1/48*(15*(c^3*d^3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4* 
g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^4*f*g^2)*x^2 + (c^3*d^3*e*f^3 + 3*c^ 
3*d^4*f^2*g)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a 
*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x + 2*sqrt(c*d*e*x^2 + a*d*e + (c 
*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + 
(e*f + d*g)*x)) + 2*(33*c^3*d^3*f^3*g - 59*a*c^2*d^2*e*f^2*g^2 + 34*a^2*c* 
d*e^2*f*g^3 - 8*a^3*e^3*g^4 + 15*(c^3*d^3*f*g^3 - a*c^2*d^2*e*g^4)*x^2 + 1 
0*(4*c^3*d^3*f^2*g^2 - 5*a*c^2*d^2*e*f*g^3 + a^2*c*d*e^2*g^4)*x)*sqrt(c*d* 
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g - 4*a*c^3 
*d^4*e*f^6*g^2 + 6*a^2*c^2*d^3*e^2*f^5*g^3 - 4*a^3*c*d^2*e^3*f^4*g^4 + a^4 
*d*e^4*f^3*g^5 + (c^4*d^4*e*f^4*g^4 - 4*a*c^3*d^3*e^2*f^3*g^5 + 6*a^2*c^2* 
d^2*e^3*f^2*g^6 - 4*a^3*c*d*e^4*f*g^7 + a^4*e^5*g^8)*x^4 + (3*c^4*d^4*e*f^ 
5*g^3 + a^4*d*e^4*g^8 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^4*g^4 - 2*(2*a*c^3* 
d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^5 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f 
^2*g^6 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^7)*x^3 + 3*(c^4*d^4*e*f^6*g^2 + 
 a^4*d*e^4*f*g^7 + (c^4*d^5 - 4*a*c^3*d^3*e^2)*f^5*g^3 - 2*(2*a*c^3*d^4*e 
- 3*a^2*c^2*d^2*e^3)*f^4*g^4 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g 
^5 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^6)*x^2 + (c^4*d^4*e*f^7*g + 3*a^4*d 
*e^4*f^2*g^6 + (3*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^6*g^2 - 6*(2*a*c^3*d^4*e - 
a^2*c^2*d^2*e^3)*f^5*g^3 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^4*g^...
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{4}}\, dx \] Input:

integrate((e*x+d)**(1/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)** 
(1/2),x)
 

Output:

Integral(sqrt(d + e*x)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**4), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}} \,d x } \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x 
+ f)^4), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{24} \, {\left (\frac {15 \, c^{3} d^{3} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c^{3} d^{3} e^{2} f^{3} {\left | e \right |} - 3 \, a c^{2} d^{2} e^{3} f^{2} g {\left | e \right |} + 3 \, a^{2} c d e^{4} f g^{2} {\left | e \right |} - a^{3} e^{5} g^{3} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} + \frac {33 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{5} e^{4} f^{2} - 66 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{4} e^{5} f g + 33 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{6} g^{2} + 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4} e^{2} f g - 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{3} g^{2} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} g^{2}}{{\left (c^{3} d^{3} e^{2} f^{3} {\left | e \right |} - 3 \, a c^{2} d^{2} e^{3} f^{2} g {\left | e \right |} + 3 \, a^{2} c d e^{4} f g^{2} {\left | e \right |} - a^{3} e^{5} g^{3} {\left | e \right |}\right )} {\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{3}}\right )} e^{4} \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="giac")
 

Output:

1/24*(15*c^3*d^3*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c* 
d*f*g - a*e*g^2)*e))/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs( 
e) + 3*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^5*g^3*abs(e))*sqrt(c*d*f*g - a*e*g 
^2)*e) + (33*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^5*d^5*e^4*f^2 - 66* 
sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^4*d^4*e^5*f*g + 33*sqrt((e*x + 
 d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^3*d^3*e^6*g^2 + 40*((e*x + d)*c*d*e - c 
*d^2*e + a*e^3)^(3/2)*c^4*d^4*e^2*f*g - 40*((e*x + d)*c*d*e - c*d^2*e + a* 
e^3)^(3/2)*a*c^3*d^3*e^3*g^2 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2 
)*c^3*d^3*g^2)/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs(e) + 3 
*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^5*g^3*abs(e))*(c*d*e^2*f - a*e^3*g + ((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^3))*e^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:

int((d + e*x)^(1/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 
1/2)),x)
 

Output:

int((d + e*x)^(1/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 
1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.77 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{3} d^{3} f^{3}+45 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{3} d^{3} f^{2} g x +45 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{3} d^{3} g^{3} x^{3}-8 \sqrt {c d x +a e}\, a^{3} e^{3} g^{4}+34 \sqrt {c d x +a e}\, a^{2} c d \,e^{2} f \,g^{3}+10 \sqrt {c d x +a e}\, a^{2} c d \,e^{2} g^{4} x -59 \sqrt {c d x +a e}\, a \,c^{2} d^{2} e \,f^{2} g^{2}-50 \sqrt {c d x +a e}\, a \,c^{2} d^{2} e f \,g^{3} x -15 \sqrt {c d x +a e}\, a \,c^{2} d^{2} e \,g^{4} x^{2}+33 \sqrt {c d x +a e}\, c^{3} d^{3} f^{3} g +40 \sqrt {c d x +a e}\, c^{3} d^{3} f^{2} g^{2} x +15 \sqrt {c d x +a e}\, c^{3} d^{3} f \,g^{3} x^{2}}{24 g \left (a^{4} e^{4} g^{7} x^{3}-4 a^{3} c d \,e^{3} f \,g^{6} x^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{5} x^{3}-4 a \,c^{3} d^{3} e \,f^{3} g^{4} x^{3}+c^{4} d^{4} f^{4} g^{3} x^{3}+3 a^{4} e^{4} f \,g^{6} x^{2}-12 a^{3} c d \,e^{3} f^{2} g^{5} x^{2}+18 a^{2} c^{2} d^{2} e^{2} f^{3} g^{4} x^{2}-12 a \,c^{3} d^{3} e \,f^{4} g^{3} x^{2}+3 c^{4} d^{4} f^{5} g^{2} x^{2}+3 a^{4} e^{4} f^{2} g^{5} x -12 a^{3} c d \,e^{3} f^{3} g^{4} x +18 a^{2} c^{2} d^{2} e^{2} f^{4} g^{3} x -12 a \,c^{3} d^{3} e \,f^{5} g^{2} x +3 c^{4} d^{4} f^{6} g x +a^{4} e^{4} f^{3} g^{4}-4 a^{3} c d \,e^{3} f^{4} g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{5} g^{2}-4 a \,c^{3} d^{3} e \,f^{6} g +c^{4} d^{4} f^{7}\right )} \] Input:

int((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

(15*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqr 
t( - a*e*g + c*d*f)))*c**3*d**3*f**3 + 45*sqrt(g)*sqrt( - a*e*g + c*d*f)*a 
tan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**3*d**3*f**2 
*g*x + 45*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt( 
g)*sqrt( - a*e*g + c*d*f)))*c**3*d**3*f*g**2*x**2 + 15*sqrt(g)*sqrt( - a*e 
*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c 
**3*d**3*g**3*x**3 - 8*sqrt(a*e + c*d*x)*a**3*e**3*g**4 + 34*sqrt(a*e + c* 
d*x)*a**2*c*d*e**2*f*g**3 + 10*sqrt(a*e + c*d*x)*a**2*c*d*e**2*g**4*x - 59 
*sqrt(a*e + c*d*x)*a*c**2*d**2*e*f**2*g**2 - 50*sqrt(a*e + c*d*x)*a*c**2*d 
**2*e*f*g**3*x - 15*sqrt(a*e + c*d*x)*a*c**2*d**2*e*g**4*x**2 + 33*sqrt(a* 
e + c*d*x)*c**3*d**3*f**3*g + 40*sqrt(a*e + c*d*x)*c**3*d**3*f**2*g**2*x + 
 15*sqrt(a*e + c*d*x)*c**3*d**3*f*g**3*x**2)/(24*g*(a**4*e**4*f**3*g**4 + 
3*a**4*e**4*f**2*g**5*x + 3*a**4*e**4*f*g**6*x**2 + a**4*e**4*g**7*x**3 - 
4*a**3*c*d*e**3*f**4*g**3 - 12*a**3*c*d*e**3*f**3*g**4*x - 12*a**3*c*d*e** 
3*f**2*g**5*x**2 - 4*a**3*c*d*e**3*f*g**6*x**3 + 6*a**2*c**2*d**2*e**2*f** 
5*g**2 + 18*a**2*c**2*d**2*e**2*f**4*g**3*x + 18*a**2*c**2*d**2*e**2*f**3* 
g**4*x**2 + 6*a**2*c**2*d**2*e**2*f**2*g**5*x**3 - 4*a*c**3*d**3*e*f**6*g 
- 12*a*c**3*d**3*e*f**5*g**2*x - 12*a*c**3*d**3*e*f**4*g**3*x**2 - 4*a*c** 
3*d**3*e*f**3*g**4*x**3 + c**4*d**4*f**7 + 3*c**4*d**4*f**6*g*x + 3*c**4*d 
**4*f**5*g**2*x**2 + c**4*d**4*f**4*g**3*x**3))