\(\int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [103]

Optimal result

Integrand size = 48, antiderivative size = 193 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2}}{(c d f-a e g) \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {8 c d (d+e x)^{3/2} \sqrt {f+g x}}{3 (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c d g \sqrt {d+e x} \sqrt {f+g x}}{3 (c d f-a e g)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.53 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (3 a^2 e^2 g^2+6 a c d e g (f+2 g x)+c^2 d^2 \left (-f^2+4 f g x+8 g^2 x^2\right )\right )}{3 (c d f-a e g)^3 ((a e+c d x) (d+e x))^{3/2} \sqrt {f+g x}} \] Input:

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

Output:

(2*(d + e*x)^(3/2)*(3*a^2*e^2*g^2 + 6*a*c*d*e*g*(f + 2*g*x) + c^2*d^2*(-f^ 
2 + 4*f*g*x + 8*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*((a*e + c*d*x)*(d + e*x))^ 
(3/2)*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1252, 1252, 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.

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

\(\Big \downarrow \) 1252

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

\(\Big \downarrow \) 1252

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

\(\Big \downarrow \) 1248

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (8 g^{2} x^{2} d^{2} c^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -f^{2} c^{2} d^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right )^{2} \left (a e g -d f c \right )^{3}}\)	\(121\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} d^{2} c^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \sqrt {g x +f}\, \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(169\)
orering	\(-\frac {2 \left (8 g^{2} x^{2} d^{2} c^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \sqrt {g x +f}\, {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\)	\(170\)

Input:

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

Output:

-2/3/(e*x+d)^(1/2)/(g*x+f)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(8*c^2*d^2*g^ 
2*x^2+12*a*c*d*e*g^2*x+4*c^2*d^2*f*g*x+3*a^2*e^2*g^2+6*a*c*d*e*f*g-c^2*d^2 
*f^2)/(c*d*x+a*e)^2/(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. 667 vs. \(2 (171) = 342\).

Time = 0.30 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.46 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} - c^{2} d^{2} f^{2} + 6 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \, {\left (c^{2} d^{2} f g + 3 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a^{2} c^{3} d^{4} e^{2} f^{4} - 3 \, a^{3} c^{2} d^{3} e^{3} f^{3} g + 3 \, a^{4} c d^{2} e^{4} f^{2} g^{2} - a^{5} d e^{5} f g^{3} + {\left (c^{5} d^{5} e f^{3} g - 3 \, a c^{4} d^{4} e^{2} f^{2} g^{2} + 3 \, a^{2} c^{3} d^{3} e^{3} f g^{3} - a^{3} c^{2} d^{2} e^{4} g^{4}\right )} x^{4} + {\left (c^{5} d^{5} e f^{4} + {\left (c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} f^{3} g - 3 \, {\left (a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} + {\left (3 \, a^{2} c^{3} d^{4} e^{2} + 5 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} - {\left (a^{3} c^{2} d^{3} e^{3} + 2 \, a^{4} c d e^{5}\right )} g^{4}\right )} x^{3} + {\left ({\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} - {\left (a c^{4} d^{5} e + 5 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g - 3 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} + {\left (5 \, a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f g^{3} - {\left (2 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} g^{4}\right )} x^{2} - {\left (a^{5} d e^{5} g^{4} - {\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{4} + {\left (5 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{3} g - 3 \, {\left (a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f^{2} g^{2} - {\left (a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f g^{3}\right )} x\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (171) = 342\).

Time = 0.27 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.28 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2}{3} \, {\left (\frac {6 \, \sqrt {c d g} g^{3}}{{\left (c^{2} d^{2} e f^{2} {\left | g \right |} - 2 \, a c d e^{2} f g {\left | g \right |} + a^{2} e^{3} g^{2} {\left | g \right |}\right )} {\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )}} + \frac {\sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left (\frac {5 \, {\left (c^{5} d^{5} e f^{2} g^{4} {\left | g \right |} - 2 \, a c^{4} d^{4} e^{2} f g^{5} {\left | g \right |} + a^{2} c^{3} d^{3} e^{3} g^{6} {\left | g \right |}\right )} {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )}}{c^{6} d^{6} e^{4} f^{5} g^{2} - 5 \, a c^{5} d^{5} e^{5} f^{4} g^{3} + 10 \, a^{2} c^{4} d^{4} e^{6} f^{3} g^{4} - 10 \, a^{3} c^{3} d^{3} e^{7} f^{2} g^{5} + 5 \, a^{4} c^{2} d^{2} e^{8} f g^{6} - a^{5} c d e^{9} g^{7}} - \frac {6 \, {\left (c^{5} d^{5} e^{3} f^{3} g^{4} {\left | g \right |} - 3 \, a c^{4} d^{4} e^{4} f^{2} g^{5} {\left | g \right |} + 3 \, a^{2} c^{3} d^{3} e^{5} f g^{6} {\left | g \right |} - a^{3} c^{2} d^{2} e^{6} g^{7} {\left | g \right |}\right )}}{c^{6} d^{6} e^{4} f^{5} g^{2} - 5 \, a c^{5} d^{5} e^{5} f^{4} g^{3} + 10 \, a^{2} c^{4} d^{4} e^{6} f^{3} g^{4} - 10 \, a^{3} c^{3} d^{3} e^{7} f^{2} g^{5} + 5 \, a^{4} c^{2} d^{2} e^{8} f g^{6} - a^{5} c d e^{9} g^{7}}\right )}}{{\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )}^{2}}\right )} e^{3} \] Input:

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

Output:

2/3*(6*sqrt(c*d*g)*g^3/((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)) + 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)*(5*(c^5*d^5*e* 
f^2*g^4*abs(g) - 2*a*c^4*d^4*e^2*f*g^5*abs(g) + a^2*c^3*d^3*e^3*g^6*abs(g) 
)*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^6*d^6*e^4*f^5*g^2 - 5*a*c^5*d^5*e^5*f 
^4*g^3 + 10*a^2*c^4*d^4*e^6*f^3*g^4 - 10*a^3*c^3*d^3*e^7*f^2*g^5 + 5*a^4*c 
^2*d^2*e^8*f*g^6 - a^5*c*d*e^9*g^7) - 6*(c^5*d^5*e^3*f^3*g^4*abs(g) - 3*a* 
c^4*d^4*e^4*f^2*g^5*abs(g) + 3*a^2*c^3*d^3*e^5*f*g^6*abs(g) - a^3*c^2*d^2* 
e^6*g^7*abs(g))/(c^6*d^6*e^4*f^5*g^2 - 5*a*c^5*d^5*e^5*f^4*g^3 + 10*a^2*c^ 
4*d^4*e^6*f^3*g^4 - 10*a^3*c^3*d^3*e^7*f^2*g^5 + 5*a^4*c^2*d^2*e^8*f*g^6 - 
 a^5*c*d*e^9*g^7))/(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)*e^3
 

Mupad [B] (verification not implemented)

Time = 7.56 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g-2\,c^2\,d^2\,f^2\right )}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {8\,g\,x\,\left (3\,a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^3\,\sqrt {f+g\,x}+\frac {a^2\,e\,\sqrt {f+g\,x}}{c^2\,d}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a e f g}{3}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a e \,g^{2} x}{3}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d f g x}{3}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d \,g^{2} x^{2}}{3}-2 \sqrt {g x +f}\, a^{2} e^{2} g^{2}-4 \sqrt {g x +f}\, a c d e f g -8 \sqrt {g x +f}\, a c d e \,g^{2} x +\frac {2 \sqrt {g x +f}\, c^{2} d^{2} f^{2}}{3}-\frac {8 \sqrt {g x +f}\, c^{2} d^{2} f g x}{3}-\frac {16 \sqrt {g x +f}\, c^{2} d^{2} g^{2} x^{2}}{3}}{\sqrt {c d x +a e}\, \left (a^{3} c d \,e^{3} g^{4} x^{2}-3 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x^{2}+3 a \,c^{3} d^{3} e \,f^{2} g^{2} x^{2}-c^{4} d^{4} f^{3} g \,x^{2}+a^{4} e^{4} g^{4} x -2 a^{3} c d \,e^{3} f \,g^{3} x +2 a \,c^{3} d^{3} e \,f^{3} g x -c^{4} d^{4} f^{4} x +a^{4} e^{4} f \,g^{3}-3 a^{3} c d \,e^{3} f^{2} g^{2}+3 a^{2} c^{2} d^{2} e^{2} f^{3} g -a \,c^{3} d^{3} e \,f^{4}\right )} \] Input:

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

Output:

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