\(\int (d+e x)^m (f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\) [119]

Optimal result

Integrand size = 46, antiderivative size = 105 \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \] Output:

2/5*(-g*(c*d*x+a*e)/(-a*e*g+c*d*f))^m*(e*x+d)^m*(g*x+f)^(5/2)*hypergeom([5 
/2, m],[7/2],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 = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {2 \left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} (f+g x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \] Input:

Integrate[((d + e*x)^m*(f + g*x)^(3/2))/(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*(f + g*x)^(5/2)*Hy 
pergeometric2F1[5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(5*g*((a*e 
+ c*d*x)*(d + e*x))^m)
 

Rubi [A] (verified)

Time = 0.43 (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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g}\)

Input:

Int[((d + e*x)^m*(f + g*x)^(3/2))/(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*(f + g*x)^(5/2)*Hy 
pergeometric2F1[5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(5*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]

\[\int \left (e x +d \right )^{m} \left (g x +f \right )^{\frac {3}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{-m}d x\]

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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