\(\int \frac {(d+e x)^{5/2} (f+g x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [49]

Optimal result

Integrand size = 46, antiderivative size = 230 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (c d f-a e g)^3 (d+e x)^{3/2}}{3 c^4 d^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {6 g (c d f-a e g)^2 \sqrt {d+e x}}{c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {6 g^2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^4 d^4 \sqrt {d+e x}}+\frac {2 g^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^4 d^4 (d+e x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-6 a c^2 d^2 e g \left (f^2-6 f g x+g^2 x^2\right )+c^3 d^3 \left (-f^3-9 f^2 g x+9 f g^2 x^2+g^3 x^3\right )\right )}{3 c^4 d^4 ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[((d + e*x)^(5/2)*(f + g*x)^3)/(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)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(f - g*x) - 6*a*c 
^2*d^2*e*g*(f^2 - 6*f*g*x + g^2*x^2) + c^3*d^3*(-f^3 - 9*f^2*g*x + 9*f*g^2 
*x^2 + g^3*x^3)))/(3*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1251, 1251, 1221, 1122}

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.

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1122

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

Input:

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

Defintions of rubi rules used

rule 1122

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), 
 x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
EqQ[m + p, 0]
 

rule 1221

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

rule 1251

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

Maple [A] (verified)

Time = 2.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-x^{3} g^{3} d^{3} c^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} c^{4} d^{4}}\)	\(179\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-x^{3} g^{3} d^{3} c^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 d^{4} c^{4} \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(187\)
orering	\(-\frac {2 \left (-x^{3} g^{3} d^{3} c^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 d^{4} c^{4} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\)	\(188\)

Input:

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

Output:

-2/3/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-c^3*d^3*g^3*x^3+6*a*c^2*d 
^2*e*g^3*x^2-9*c^3*d^3*f*g^2*x^2+24*a^2*c*d*e^2*g^3*x-36*a*c^2*d^2*e*f*g^2 
*x+9*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-24*a^2*c*d*e^2*f*g^2+6*a*c^2*d^2*e*f^2 
*g+c^3*d^3*f^3)/(c*d*x+a*e)^2/c^4/d^4
 

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} g^{3} x^{3} - c^{3} d^{3} f^{3} - 6 \, a c^{2} d^{2} e f^{2} g + 24 \, a^{2} c d e^{2} f g^{2} - 16 \, a^{3} e^{3} g^{3} + 3 \, {\left (3 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} - 3 \, {\left (3 \, c^{3} d^{3} f^{2} g - 12 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{6} d^{6} e x^{3} + a^{2} c^{4} d^{5} e^{2} + {\left (c^{6} d^{7} + 2 \, a c^{5} d^{5} e^{2}\right )} x^{2} + {\left (2 \, a c^{5} d^{6} e + a^{2} c^{4} d^{4} e^{3}\right )} x\right )}} \] Input:

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

Output:

2/3*(c^3*d^3*g^3*x^3 - c^3*d^3*f^3 - 6*a*c^2*d^2*e*f^2*g + 24*a^2*c*d*e^2* 
f*g^2 - 16*a^3*e^3*g^3 + 3*(3*c^3*d^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 - 3*( 
3*c^3*d^3*f^2*g - 12*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)*x)*sqrt(c*d*e* 
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^6*d^6*e*x^3 + a^2*c^4*d^ 
5*e^2 + (c^6*d^7 + 2*a*c^5*d^5*e^2)*x^2 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3 
)*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, c d x + 2 \, a e\right )} f^{2} g}{{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d e x + 8 \, a^{2} e^{2}\right )} f g^{2}}{{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{2} e x^{2} - 24 \, a^{2} c d e^{2} x - 16 \, a^{3} e^{3}\right )} g^{3}}{3 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )} \sqrt {c d x + a e}} - \frac {2 \, f^{3}}{3 \, {\left (c^{2} d^{2} x + a c d e\right )} \sqrt {c d x + a e}} \] Input:

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

Output:

-2*(3*c*d*x + 2*a*e)*f^2*g/((c^3*d^3*x + a*c^2*d^2*e)*sqrt(c*d*x + a*e)) + 
 2*(3*c^2*d^2*x^2 + 12*a*c*d*e*x + 8*a^2*e^2)*f*g^2/((c^4*d^4*x + a*c^3*d^ 
3*e)*sqrt(c*d*x + a*e)) + 2/3*(c^3*d^3*x^3 - 6*a*c^2*d^2*e*x^2 - 24*a^2*c* 
d*e^2*x - 16*a^3*e^3)*g^3/((c^5*d^5*x + a*c^4*d^4*e)*sqrt(c*d*x + a*e)) - 
2/3*f^3/((c^2*d^2*x + a*c*d*e)*sqrt(c*d*x + a*e))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (c^{3} d^{3} e^{4} f^{3} - 3 \, a c^{2} d^{2} e^{5} f^{2} g + 3 \, a^{2} c d e^{6} f g^{2} - a^{3} e^{7} g^{3} + 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} e^{2} f^{2} g - 18 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c d e^{3} f g^{2} + 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} e^{4} g^{3}\right )}}{3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4} {\left | e \right |}} + \frac {2 \, {\left (9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{9} d^{9} e^{8} f g^{2} - 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{8} d^{8} e^{9} g^{3} + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{8} d^{8} e^{6} g^{3}\right )}}{3 \, c^{12} d^{12} e^{8} {\left | e \right |}} \] Input:

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

Output:

-2/3*(c^3*d^3*e^4*f^3 - 3*a*c^2*d^2*e^5*f^2*g + 3*a^2*c*d*e^6*f*g^2 - a^3* 
e^7*g^3 + 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^2*e^2*f^2*g - 18*((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c*d*e^3*f*g^2 + 9*((e*x + d)*c*d*e - c* 
d^2*e + a*e^3)*a^2*e^4*g^3)/(((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^4 
*d^4*abs(e)) + 2/3*(9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^9*d^9*e^8* 
f*g^2 - 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^8*d^8*e^9*g^3 + ((e* 
x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^8*d^8*e^6*g^3)/(c^12*d^12*e^8*abs( 
e))
 

Mupad [B] (verification not implemented)

Time = 6.51 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (\frac {32\,a^3\,e^3\,g^3}{3}-16\,a^2\,c\,d\,e^2\,f\,g^2+4\,a\,c^2\,d^2\,e\,f^2\,g+\frac {2\,c^3\,d^3\,f^3}{3}\right )}{c^6\,d^6\,e}-\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,d^3\,e}+\frac {g^2\,x^2\,\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-12\,a\,c\,d\,e\,f\,g+3\,c^2\,d^2\,f^2\right )}{c^5\,d^5\,e}\right )}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^6\,d^7+2\,a\,c^5\,d^5\,e^2\right )}{c^6\,d^6\,e}} \] Input:

int(((f + g*x)^3*(d + e*x)^(5/2))/(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)*(((d + e*x)^(1/2)*((32*a^3 
*e^3*g^3)/3 + (2*c^3*d^3*f^3)/3 + 4*a*c^2*d^2*e*f^2*g - 16*a^2*c*d*e^2*f*g 
^2))/(c^6*d^6*e) - (2*g^3*x^3*(d + e*x)^(1/2))/(3*c^3*d^3*e) + (g^2*x^2*(4 
*a*e*g - 6*c*d*f)*(d + e*x)^(1/2))/(c^4*d^4*e) + (2*g*x*(d + e*x)^(1/2)*(8 
*a^2*e^2*g^2 + 3*c^2*d^2*f^2 - 12*a*c*d*e*f*g))/(c^5*d^5*e)))/(x^3 + (a^2* 
e)/(c^2*d) + (a*x*(a*e^2 + 2*c*d^2))/(c^2*d^2) + (x^2*(c^6*d^7 + 2*a*c^5*d 
^5*e^2))/(c^6*d^6*e))
 

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {2}{3} c^{3} d^{3} g^{3} x^{3}-4 a \,c^{2} d^{2} e \,g^{3} x^{2}+6 c^{3} d^{3} f \,g^{2} x^{2}-16 a^{2} c d \,e^{2} g^{3} x +24 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x -\frac {32}{3} a^{3} e^{3} g^{3}+16 a^{2} c d \,e^{2} f \,g^{2}-4 a \,c^{2} d^{2} e \,f^{2} g -\frac {2}{3} c^{3} d^{3} f^{3}}{\sqrt {c d x +a e}\, c^{4} d^{4} \left (c d x +a e \right )} \] Input:

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

Output:

(2*( - 16*a**3*e**3*g**3 + 24*a**2*c*d*e**2*f*g**2 - 24*a**2*c*d*e**2*g**3 
*x - 6*a*c**2*d**2*e*f**2*g + 36*a*c**2*d**2*e*f*g**2*x - 6*a*c**2*d**2*e* 
g**3*x**2 - c**3*d**3*f**3 - 9*c**3*d**3*f**2*g*x + 9*c**3*d**3*f*g**2*x** 
2 + c**3*d**3*g**3*x**3))/(3*sqrt(a*e + c*d*x)*c**4*d**4*(a*e + c*d*x))