\(\int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [126]

Optimal result

Integrand size = 46, antiderivative size = 214 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {2 \left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (3+2 n) \sqrt {d+e x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} (f+g x)^n \left (c d e (f+g x)+\left (-2 a e^2 g (1+n)+c d (-e f+d g (3+2 n))\right ) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )\right )}{c^2 d^2 g \left (\frac {3}{2}+n\right ) \sqrt {d+e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1258, 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.

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

\(\Big \downarrow \) 1258

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

(2*e*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*g 
*(3 + 2*n)*Sqrt[d + e*x]) - (2*(2*a*e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2* 
n)))*(a*e + c*d*x)*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[1/2, -n, 3/ 
2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))])/(c^2*d^2*g*(3 + 2*n)*((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 1258

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(d + e*x)^(m - 2)*(f + g*x)^(n 
+ 1)*((a + b*x + c*x^2)^(p + 1)/(c*g*(n + p + 2))), x] - Simp[(b*e*g*(n + 1 
) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2))   Int[(d + e*x)^ 
(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, 
 g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p - 1, 0] && 
!LtQ[n, -1] && IntegerQ[2*p]
 

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]

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

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)^(1/2),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)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \] 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)^(1/2), 
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*d*x + a*e), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Timed out} \] 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)** 
(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \] 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)^(1/2), 
x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \] 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)^(1/2), 
x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] 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)^ 
(1/2),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)^ 
(1/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {too large to display} \] 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)^(1/2),x)
 

Output:

(2*( - 2*(f + g*x)**n*sqrt(a*e + c*d*x)*a*e**2*f + 2*(f + g*x)**n*sqrt(a*e 
 + c*d*x)*a*e**2*g*n*x + 2*(f + g*x)**n*sqrt(a*e + c*d*x)*c*d**2*f*n + 3*( 
f + g*x)**n*sqrt(a*e + c*d*x)*c*d**2*f + (f + g*x)**n*sqrt(a*e + c*d*x)*c* 
d*e*f*x - 8*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(4*a**2*e**2*f*g*n**2 + 
 6*a**2*e**2*f*g*n + 4*a**2*e**2*g**2*n**2*x + 6*a**2*e**2*g**2*n*x + 2*a* 
c*d*e*f**2*n + 3*a*c*d*e*f**2 + 4*a*c*d*e*f*g*n**2*x + 8*a*c*d*e*f*g*n*x + 
 3*a*c*d*e*f*g*x + 4*a*c*d*e*g**2*n**2*x**2 + 6*a*c*d*e*g**2*n*x**2 + 2*c* 
*2*d**2*f**2*n*x + 3*c**2*d**2*f**2*x + 2*c**2*d**2*f*g*n*x**2 + 3*c**2*d* 
*2*f*g*x**2),x)*a**3*e**4*g**3*n**4 - 20*int(((f + g*x)**n*sqrt(a*e + c*d* 
x)*x)/(4*a**2*e**2*f*g*n**2 + 6*a**2*e**2*f*g*n + 4*a**2*e**2*g**2*n**2*x 
+ 6*a**2*e**2*g**2*n*x + 2*a*c*d*e*f**2*n + 3*a*c*d*e*f**2 + 4*a*c*d*e*f*g 
*n**2*x + 8*a*c*d*e*f*g*n*x + 3*a*c*d*e*f*g*x + 4*a*c*d*e*g**2*n**2*x**2 + 
 6*a*c*d*e*g**2*n*x**2 + 2*c**2*d**2*f**2*n*x + 3*c**2*d**2*f**2*x + 2*c** 
2*d**2*f*g*n*x**2 + 3*c**2*d**2*f*g*x**2),x)*a**3*e**4*g**3*n**3 - 12*int( 
((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(4*a**2*e**2*f*g*n**2 + 6*a**2*e**2*f*g 
*n + 4*a**2*e**2*g**2*n**2*x + 6*a**2*e**2*g**2*n*x + 2*a*c*d*e*f**2*n + 3 
*a*c*d*e*f**2 + 4*a*c*d*e*f*g*n**2*x + 8*a*c*d*e*f*g*n*x + 3*a*c*d*e*f*g*x 
 + 4*a*c*d*e*g**2*n**2*x**2 + 6*a*c*d*e*g**2*n*x**2 + 2*c**2*d**2*f**2*n*x 
 + 3*c**2*d**2*f**2*x + 2*c**2*d**2*f*g*n*x**2 + 3*c**2*d**2*f*g*x**2),x)* 
a**3*e**4*g**3*n**2 + 8*int(((f + g*x)**n*sqrt(a*e + c*d*x)*x)/(4*a**2*...