\(\int \frac {(d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m}}{(f+g x)^{5/2}} \, dx\) [123]

Optimal result

Integrand size = 46, antiderivative size = 105 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^{5/2}} \, dx=-\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},m,-\frac {1}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}} \] Output:

-2/3*(-g*(c*d*x+a*e)/(-a*e*g+c*d*f))^m*(e*x+d)^m*hypergeom([-3/2, m],[-1/2 
],c*d*(g*x+f)/(-a*e*g+c*d*f))/g/(g*x+f)^(3/2)/((a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^m)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^{5/2}} \, dx=-\frac {2 \left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},m,-\frac {1}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}} \] Input:

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

Output:

(-2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*Hypergeometric2F1 
[-3/2, m, -1/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(3*g*((a*e + c*d*x)*(d + 
 e*x))^m*(f + g*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1268, 80, 79}

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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2 (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},m,-\frac {1}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}}\)

Input:

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

Output:

(-2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*Hypergeometric2F1 
[-3/2, m, -1/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(3*g*(f + g*x)^(3/2)*(a* 
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m)
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 1268

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

Maple [F]

\[\int \frac {\left (e x +d \right )^{m} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{-m}}{\left (g x +f \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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