\(\int \frac {(d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m}}{\sqrt {f+g x}} \, dx\) [121]

Optimal result

Integrand size = 46, antiderivative size = 103 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m,\frac {3}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\frac {2 \left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} \sqrt {f+g x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m,\frac {3}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \] Input:

Integrate[(d + e*x)^m/(Sqrt[f + g*x]*(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*Sqrt[f + g*x]*Hype 
rgeometric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*((a*e + c* 
d*x)*(d + e*x))^m)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 103, 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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[(d + e*x)^m/(Sqrt[f + g*x]*(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*Sqrt[f + g*x]*Hype 
rgeometric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*(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]

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

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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