\(\int \frac {(d+e x)^{3/2} (f+g x)^n}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [108]

Optimal result

Integrand size = 46, antiderivative size = 112 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-n,-\frac {1}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )}{c d \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98, 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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*((f + g*x)**n*sqrt(a*e + c*d*x)*f + 2*int(((f + g*x)**n*sqrt(a*e + c*d* 
x)*x)/(2*a**3*e**3*f*g*n + 2*a**3*e**3*g**2*n*x - a**2*c*d*e**2*f**2 + 4*a 
**2*c*d*e**2*f*g*n*x - a**2*c*d*e**2*f*g*x + 4*a**2*c*d*e**2*g**2*n*x**2 - 
 2*a*c**2*d**2*e*f**2*x + 2*a*c**2*d**2*e*f*g*n*x**2 - 2*a*c**2*d**2*e*f*g 
*x**2 + 2*a*c**2*d**2*e*g**2*n*x**3 - c**3*d**3*f**2*x**2 - c**3*d**3*f*g* 
x**3),x)*a**3*e**3*g**3*n**2 - 2*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(2 
*a**3*e**3*f*g*n + 2*a**3*e**3*g**2*n*x - a**2*c*d*e**2*f**2 + 4*a**2*c*d* 
e**2*f*g*n*x - a**2*c*d*e**2*f*g*x + 4*a**2*c*d*e**2*g**2*n*x**2 - 2*a*c** 
2*d**2*e*f**2*x + 2*a*c**2*d**2*e*f*g*n*x**2 - 2*a*c**2*d**2*e*f*g*x**2 + 
2*a*c**2*d**2*e*g**2*n*x**3 - c**3*d**3*f**2*x**2 - c**3*d**3*f*g*x**3),x) 
*a**2*c*d*e**2*f*g**2*n**2 - int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(2*a** 
3*e**3*f*g*n + 2*a**3*e**3*g**2*n*x - a**2*c*d*e**2*f**2 + 4*a**2*c*d*e**2 
*f*g*n*x - a**2*c*d*e**2*f*g*x + 4*a**2*c*d*e**2*g**2*n*x**2 - 2*a*c**2*d* 
*2*e*f**2*x + 2*a*c**2*d**2*e*f*g*n*x**2 - 2*a*c**2*d**2*e*f*g*x**2 + 2*a* 
c**2*d**2*e*g**2*n*x**3 - c**3*d**3*f**2*x**2 - c**3*d**3*f*g*x**3),x)*a** 
2*c*d*e**2*f*g**2*n + 2*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(2*a**3*e** 
3*f*g*n + 2*a**3*e**3*g**2*n*x - a**2*c*d*e**2*f**2 + 4*a**2*c*d*e**2*f*g* 
n*x - a**2*c*d*e**2*f*g*x + 4*a**2*c*d*e**2*g**2*n*x**2 - 2*a*c**2*d**2*e* 
f**2*x + 2*a*c**2*d**2*e*f*g*n*x**2 - 2*a*c**2*d**2*e*f*g*x**2 + 2*a*c**2* 
d**2*e*g**2*n*x**3 - c**3*d**3*f**2*x**2 - c**3*d**3*f*g*x**3),x)*a**2*...