\(\int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [134]

Optimal result

Integrand size = 46, antiderivative size = 261 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {c d \left (4 a e^2 g-c d (e f+3 d g)\right ) \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}} \] Output:

-1/2*(-d*g+e*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)/( 
e*x+d)^(1/2)/(g*x+f)^2-1/4*(4*a*e^2*g-c*d*(3*d*g+e*f))*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)^2/(e*x+d)^(1/2)/(g*x+f)+1/4*c*d*(4*a 
*e^2*g-c*d*(3*d*g+e*f))*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^(3/2)/(-a*e*g+c*d*f)^(5/2)
 

Mathematica [A] (verified)

Time = 1.46 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {c d \sqrt {d+e x} \left (\frac {\sqrt {g} (a e+c d x) (-2 a e g (d g+e (f+2 g x))+c d (e f (-f+g x)+d g (5 f+3 g x)))}{c d (c d f-a e g)^2 (f+g x)^2}+\frac {\left (-4 a e^2 g+c d (e f+3 d g)\right ) \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}}\right )}{4 g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1257, 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.

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

\(\Big \downarrow \) 1257

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1255

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

-1/2*((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*(c*d*f - 
 a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) - ((4*a*e^2*g - c*d*(e*f + 3*d*g))*(Sqr 
t[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*ArcTan[(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*g*(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]
 

rule 1257

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(e*f - d*g)*(d + e*x)^(m - 2)*( 
f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g - b*e 
*g))), x] - Simp[e*((b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/( 
g*(n + 1)*(c*e*f + c*d*g - b*e*g)))   Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && Integer 
Q[2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(235)=470\).

Time = 2.97 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.54

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (4 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a c d \,e^{2} g^{3} x^{2}-3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{3} g^{3} x^{2}-\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{2} e f \,g^{2} x^{2}+8 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a c d \,e^{2} f \,g^{2} x -6 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{3} f \,g^{2} x -2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{2} e \,f^{2} g x +4 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a c d \,e^{2} f^{2} g -3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{3} f^{2} g -\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{2} d^{2} e \,f^{3}-4 a \,e^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+3 c \,d^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+c d e f g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-2 a d e \,g^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-2 a \,e^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+5 c \,d^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-c d e \,f^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}\right )}{4 \sqrt {e x +d}\, \sqrt {\left (a e g -d f c \right ) g}\, \left (g x +f \right )^{2} g \left (a e g -d f c \right )^{2} \sqrt {c d x +a e}}\)	\(663\)

Input:

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

Output:

1/4*((e*x+d)*(c*d*x+a*e))^(1/2)*(4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d 
*f)*g)^(1/2))*a*c*d*e^2*g^3*x^2-3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d* 
f)*g)^(1/2))*c^2*d^3*g^3*x^2-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g) 
^(1/2))*c^2*d^2*e*f*g^2*x^2+8*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g 
)^(1/2))*a*c*d*e^2*f*g^2*x-6*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g) 
^(1/2))*c^2*d^3*f*g^2*x-2*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1 
/2))*c^2*d^2*e*f^2*g*x+4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/ 
2))*a*c*d*e^2*f^2*g-3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)) 
*c^2*d^3*f^2*g-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^ 
2*e*f^3-4*a*e^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*c*d^2*g^ 
2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+c*d*e*f*g*x*(c*d*x+a*e)^(1/2 
)*((a*e*g-c*d*f)*g)^(1/2)-2*a*d*e*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^ 
(1/2)-2*a*e^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+5*c*d^2*f*g*(c 
*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-c*d*e*f^2*(c*d*x+a*e)^(1/2)*((a*e* 
g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^2/g/(a*e* 
g-c*d*f)^2/(c*d*x+a*e)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (235) = 470\).

Time = 0.16 (sec) , antiderivative size = 1705, normalized size of antiderivative = 6.53 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*((c^2*d^3*e*f^3 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f* 
g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^3)*x^3 + (2*c^2*d^2*e^2*f^2*g + (7*c^2 
*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*g^3)*x^2 + (c^2* 
d^2*e^2*f^3 + (5*c^2*d^3*e - 4*a*c*d*e^3)*f^2*g + 2*(3*c^2*d^4 - 4*a*c*d^2 
*e^2)*f*g^2)*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*(c^2*d^2*e*f^3*g - 2*a^2*d*e^2*g^4 - (5*c^2*d^3 - a*c* 
d*e^2)*f^2*g^2 + (7*a*c*d^2*e - 2*a^2*e^3)*f*g^3 - (c^2*d^2*e*f^2*g^2 + (3 
*c^2*d^3 - 5*a*c*d*e^2)*f*g^3 - (3*a*c*d^2*e - 4*a^2*e^3)*g^4)*x)*sqrt(c*d 
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^4*f^5*g^2 - 3*a* 
c^2*d^3*e*f^4*g^3 + 3*a^2*c*d^2*e^2*f^3*g^4 - a^3*d*e^3*f^2*g^5 + (c^3*d^3 
*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)* 
x^3 + (2*c^3*d^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f 
^3*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^2*g^5 + (3*a^2*c*d^2*e^2 - 2*a^ 
3*e^4)*f*g^6)*x^2 + (c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 
3*a*c^2*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^4 + (6*a^ 
2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x), -1/4*((c^2*d^3*e*f^3 + (3*c^2*d^4 - 4* 
a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f*g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^3) 
*x^3 + (2*c^2*d^2*e^2*f^2*g + (7*c^2*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (235) = 470\).

Time = 0.14 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.04 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {e^{2} {\left (\frac {{\left (c^{3} d^{3} e f + 3 \, c^{3} d^{4} g - 4 \, a c^{2} d^{2} e^{2} g\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c^{2} d^{2} f^{2} g {\left | e \right |} - 2 \, a c d e f g^{2} {\left | e \right |} + a^{2} e^{2} g^{3} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{4} e^{3} f^{2} - 5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} f g + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} f g + 5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{4} e^{3} g^{2} - 4 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{2} d^{2} e^{5} g^{2} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e f g - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{4} g^{2} + 4 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{2} g^{2}}{{\left (c^{2} d^{2} f^{2} g {\left | e \right |} - 2 \, a c d e f g^{2} {\left | e \right |} + a^{2} e^{2} g^{3} {\left | e \right |}\right )} {\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{2}}\right )}}{4 \, c d} \] Input:

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

Output:

1/4*e^2*((c^3*d^3*e*f + 3*c^3*d^4*g - 4*a*c^2*d^2*e^2*g)*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^2*d^2*f^2 
*g*abs(e) - 2*a*c*d*e*f*g^2*abs(e) + a^2*e^2*g^3*abs(e))*sqrt(c*d*f*g - a* 
e*g^2)*e) - (sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^4*e^3*f^2 - 5*s 
qrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^5*e^2*f*g + 3*sqrt((e*x + d)* 
c*d*e - c*d^2*e + a*e^3)*a*c^3*d^3*e^4*f*g + 5*sqrt((e*x + d)*c*d*e - c*d^ 
2*e + a*e^3)*a*c^3*d^4*e^3*g^2 - 4*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3) 
*a^2*c^2*d^2*e^5*g^2 - ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*e 
*f*g - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^4*g^2 + 4*((e*x + 
 d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^2*d^2*e^2*g^2)/((c^2*d^2*f^2*g*abs( 
e) - 2*a*c*d*e*f*g^2*abs(e) + a^2*e^2*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)^2))/(c*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 927, normalized size of antiderivative = 3.55 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*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)))*a*c*d*e**2*f**2*g + 8*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)))*a*c*d*e**2*f* 
g**2*x + 4*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)))*a*c*d*e**2*g**3*x**2 - 3*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* 
*2*d**3*f**2*g - 6*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**2*d**3*f*g**2*x - 3*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**2*d**3*g**3*x**2 - 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**2*d**2*e*f**3 - 2*sqrt(g)*s 
qrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + 
c*d*f)))*c**2*d**2*e*f**2*g*x - 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**2*d**2*e*f*g**2*x**2 
- 2*sqrt(a*e + c*d*x)*a**2*d*e**2*g**4 - 2*sqrt(a*e + c*d*x)*a**2*e**3*f*g 
**3 - 4*sqrt(a*e + c*d*x)*a**2*e**3*g**4*x + 7*sqrt(a*e + c*d*x)*a*c*d**2* 
e*f*g**3 + 3*sqrt(a*e + c*d*x)*a*c*d**2*e*g**4*x + sqrt(a*e + c*d*x)*a*c*d 
*e**2*f**2*g**2 + 5*sqrt(a*e + c*d*x)*a*c*d*e**2*f*g**3*x - 5*sqrt(a*e + c 
*d*x)*c**2*d**3*f**2*g**2 - 3*sqrt(a*e + c*d*x)*c**2*d**3*f*g**3*x + sqrt( 
a*e + c*d*x)*c**2*d**2*e*f**3*g - sqrt(a*e + c*d*x)*c**2*d**2*e*f**2*g*...