\(\int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [96]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 48, antiderivative size = 192 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \] Output:

-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2)-8/3*g*(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)^(3/2)-16/3*c*d*g*(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)^(1/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.55 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (-a^2 e^2 g^2+2 a c d e g (3 f+2 g x)+c^2 d^2 \left (3 f^2+12 f g x+8 g^2 x^2\right )\right )}{3 (c d f-a e g)^3 \sqrt {(a e+c d x) (d+e x)} (f+g x)^{3/2}} \] Input:

Integrate[(d + e*x)^(3/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2)^(3/2)),x]
 

Output:

(-2*Sqrt[d + e*x]*(-(a^2*e^2*g^2) + 2*a*c*d*e*g*(3*f + 2*g*x) + c^2*d^2*(3 
*f^2 + 12*f*g*x + 8*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*Sqrt[(a*e + c*d*x)*(d 
+ e*x)]*(f + g*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1252, 1254, 1248}

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.

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

\(\Big \downarrow \) 1252

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1248

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

Input:

Int[(d + e*x)^(3/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2)^(3/2)),x]
 

Output:

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

Defintions of rubi rules used

rule 1248
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] / 
; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
 

rule 1252
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)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x] + Si 
mp[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g)))   Int[(d + e*x)^(m 
 - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e 
, f, g, n}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[p, 
-1] && RationalQ[n]
 

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]
 
Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62

method result size
default \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 f^{2} c^{2} d^{2}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (a e g -d f c \right )^{3}}\) \(120\)
gosper \(-\frac {2 \left (c d x +a e \right ) \left (-8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) \(168\)
orering \(-\frac {2 \left (-8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (g x +f \right )^{\frac {3}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) \(169\)

Input:

int((e*x+d)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x, 
method=_RETURNVERBOSE)
 

Output:

-2/3/(e*x+d)^(1/2)/(g*x+f)^(3/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-8*c^2*d^2*g 
^2*x^2-4*a*c*d*e*g^2*x-12*c^2*d^2*f*g*x+a^2*e^2*g^2-6*a*c*d*e*f*g-3*c^2*d^ 
2*f^2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (170) = 340\).

Time = 0.20 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.38 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 3 \, c^{2} d^{2} f^{2} + 6 \, a c d e f g - a^{2} e^{2} g^{2} + 4 \, {\left (3 \, c^{2} d^{2} f g + a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a c^{3} d^{4} e f^{5} - 3 \, a^{2} c^{2} d^{3} e^{2} f^{4} g + 3 \, a^{3} c d^{2} e^{3} f^{3} g^{2} - a^{4} d e^{4} f^{2} g^{3} + {\left (c^{4} d^{4} e f^{3} g^{2} - 3 \, a c^{3} d^{3} e^{2} f^{2} g^{3} + 3 \, a^{2} c^{2} d^{2} e^{3} f g^{4} - a^{3} c d e^{4} g^{5}\right )} x^{4} + {\left (2 \, c^{4} d^{4} e f^{4} g + {\left (c^{4} d^{5} - 5 \, a c^{3} d^{3} e^{2}\right )} f^{3} g^{2} - 3 \, {\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g^{3} + {\left (3 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g^{4} - {\left (a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{5}\right )} x^{3} + {\left (c^{4} d^{4} e f^{5} - a^{4} d e^{4} g^{5} + {\left (2 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} f^{4} g - {\left (5 \, a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} f^{2} g^{3} + {\left (a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5}\right )} f g^{4}\right )} x^{2} - {\left (2 \, a^{4} d e^{4} f g^{4} - {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} f^{5} + {\left (a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g + 3 \, {\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{3} g^{2} - {\left (5 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{2} g^{3}\right )} x\right )}} \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 
/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(3/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 
2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 
/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*( 
g*x + f)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (170) = 340\).

Time = 0.33 (sec) , antiderivative size = 672, normalized size of antiderivative = 3.50 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2}{3} \, {\left (\frac {3 \, \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} c^{2} d^{2} g^{2}}{{\left (c^{3} d^{3} e^{3} f^{3} {\left | g \right |} - 3 \, a c^{2} d^{2} e^{4} f^{2} g {\left | g \right |} + 3 \, a^{2} c d e^{5} f g^{2} {\left | g \right |} - a^{3} e^{6} g^{3} {\left | g \right |}\right )} {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )}} - \frac {2 \, {\left (5 \, \sqrt {c d g} c^{3} d^{3} e^{4} f^{2} g^{4} - 10 \, \sqrt {c d g} a c^{2} d^{2} e^{5} f g^{5} + 5 \, \sqrt {c d g} a^{2} c d e^{6} g^{6} + 12 \, \sqrt {c d g} {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2} c^{2} d^{2} e^{2} f g^{3} - 12 \, \sqrt {c d g} {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2} a c d e^{3} g^{4} + 3 \, \sqrt {c d g} {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{4} c d g^{2}\right )}}{{\left (c^{2} d^{2} e f^{2} {\left | g \right |} - 2 \, a c d e^{2} f g {\left | g \right |} + a^{2} e^{3} g^{2} {\left | g \right |}\right )} {\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )}^{3}}\right )} e^{3} \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 
/2),x, algorithm="giac")
 

Output:

2/3*(3*sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d 
*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*c^2*d^2*g^2/((c^3*d^3*e^3*f^3*abs( 
g) - 3*a*c^2*d^2*e^4*f^2*g*abs(g) + 3*a^2*c*d*e^5*f*g^2*abs(g) - a^3*e^6*g 
^3*abs(g))*(c*d*e^2*f*g - a*e^3*g^2 - (e^2*f + (e*x + d)*e*g - d*e*g)*c*d* 
g)) - 2*(5*sqrt(c*d*g)*c^3*d^3*e^4*f^2*g^4 - 10*sqrt(c*d*g)*a*c^2*d^2*e^5* 
f*g^5 + 5*sqrt(c*d*g)*a^2*c*d*e^6*g^6 + 12*sqrt(c*d*g)*(sqrt(e^2*f + (e*x 
+ d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + ( 
e*x + d)*e*g - d*e*g)*c*d*g))^2*c^2*d^2*e^2*f*g^3 - 12*sqrt(c*d*g)*(sqrt(e 
^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt(-c*d*e^2*f*g + a*e^3*g^2 
+ (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g))^2*a*c*d*e^3*g^4 + 3*sqrt(c*d*g)* 
(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt(-c*d*e^2*f*g + a*e 
^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g))^4*c*d*g^2)/((c^2*d^2*e*f^ 
2*abs(g) - 2*a*c*d*e^2*f*g*abs(g) + a^2*e^3*g^2*abs(g))*(c*d*e^2*f*g - a*e 
^3*g^2 + (sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt(-c*d*e^2* 
f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g))^2)^3))*e^3
 

Mupad [B] (verification not implemented)

Time = 7.88 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {8\,x\,\left (a\,e\,g+3\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (-2\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g+6\,c^2\,d^2\,f^2\right )}{3\,c\,d\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c\,d\,g\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3\,\sqrt {f+g\,x}+\frac {a\,f\,\sqrt {f+g\,x}}{c\,g}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}} \] Input:

int((d + e*x)^(3/2)/((f + g*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^ 
2)^(3/2)),x)
 

Output:

(((8*x*(a*e*g + 3*c*d*f)*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^3) + ((d + 
e*x)^(1/2)*(6*c^2*d^2*f^2 - 2*a^2*e^2*g^2 + 12*a*c*d*e*f*g))/(3*c*d*e*g*(a 
*e*g - c*d*f)^3) + (16*c*d*g*x^2*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^3)) 
*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(x^3*(f + g*x)^(1/2) + (a* 
f*(f + g*x)^(1/2))/(c*g) + (x*(f + g*x)^(1/2)*(a*e^2*f + c*d^2*f + a*d*e*g 
))/(c*d*e*g) + (x^2*(f + g*x)^(1/2)*(a*e^2*g + c*d^2*g + c*d*e*f))/(c*d*e* 
g))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d \,f^{2}}{3}-\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d f g x}{3}-\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d \,g^{2} x^{2}}{3}-\frac {2 \sqrt {g x +f}\, a^{2} e^{2} g^{2}}{3}+4 \sqrt {g x +f}\, a c d e f g +\frac {8 \sqrt {g x +f}\, a c d e \,g^{2} x}{3}+2 \sqrt {g x +f}\, c^{2} d^{2} f^{2}+8 \sqrt {g x +f}\, c^{2} d^{2} f g x +\frac {16 \sqrt {g x +f}\, c^{2} d^{2} g^{2} x^{2}}{3}}{\sqrt {c d x +a e}\, \left (a^{3} e^{3} g^{5} x^{2}-3 a^{2} c d \,e^{2} f \,g^{4} x^{2}+3 a \,c^{2} d^{2} e \,f^{2} g^{3} x^{2}-c^{3} d^{3} f^{3} g^{2} x^{2}+2 a^{3} e^{3} f \,g^{4} x -6 a^{2} c d \,e^{2} f^{2} g^{3} x +6 a \,c^{2} d^{2} e \,f^{3} g^{2} x -2 c^{3} d^{3} f^{4} g x +a^{3} e^{3} f^{2} g^{3}-3 a^{2} c d \,e^{2} f^{3} g^{2}+3 a \,c^{2} d^{2} e \,f^{4} g -c^{3} d^{3} f^{5}\right )} \] Input:

int((e*x+d)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

(2*( - 8*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c*d*f**2 - 16*sqrt(g)*s 
qrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c*d*f*g*x - 8*sqrt(g)*sqrt(d)*sqrt(c)*sqr 
t(a*e + c*d*x)*c*d*g**2*x**2 - sqrt(f + g*x)*a**2*e**2*g**2 + 6*sqrt(f + g 
*x)*a*c*d*e*f*g + 4*sqrt(f + g*x)*a*c*d*e*g**2*x + 3*sqrt(f + g*x)*c**2*d* 
*2*f**2 + 12*sqrt(f + g*x)*c**2*d**2*f*g*x + 8*sqrt(f + g*x)*c**2*d**2*g** 
2*x**2))/(3*sqrt(a*e + c*d*x)*(a**3*e**3*f**2*g**3 + 2*a**3*e**3*f*g**4*x 
+ a**3*e**3*g**5*x**2 - 3*a**2*c*d*e**2*f**3*g**2 - 6*a**2*c*d*e**2*f**2*g 
**3*x - 3*a**2*c*d*e**2*f*g**4*x**2 + 3*a*c**2*d**2*e*f**4*g + 6*a*c**2*d* 
*2*e*f**3*g**2*x + 3*a*c**2*d**2*e*f**2*g**3*x**2 - c**3*d**3*f**5 - 2*c** 
3*d**3*f**4*g*x - c**3*d**3*f**3*g**2*x**2))