\(\int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx\) [122]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=-20 7^{-1-n} \sqrt {10} \sqrt {5-7 x} (e+f x)^n \left (\frac {e+f x}{7 e+5 f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-n,\frac {3}{2},\frac {1}{10} (5-7 x),\frac {f (5-7 x)}{7 e+5 f}\right ) \] Output:

-20*7^(-1-n)*10^(1/2)*(5-7*x)^(1/2)*(f*x+e)^n*AppellF1(1/2,-n,-3/2,3/2,f*( 
5-7*x)/(7*e+5*f),1/2-7/10*x)/(((f*x+e)/(7*e+5*f))^n)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(87)=174\).

Time = 3.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.18 \[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=-\frac {2\ 7^{-1-n} \sqrt {50-70 x} (5+7 x) (e+f x)^n \left (\frac {e+f x}{7 e-5 f}\right )^{-n} \left (30 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-n,\frac {3}{2},\frac {1}{10} (5+7 x),-\frac {f (5+7 x)}{7 e-5 f}\right )-30 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{10} (5+7 x),-\frac {f (5+7 x)}{7 e-5 f}\right )+(5+7 x) \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{2},-n,\frac {5}{2},\frac {1}{10} (5+7 x),-\frac {f (5+7 x)}{7 e-5 f}\right )\right )}{3 \sqrt {25-49 x^2}} \] Input:

Integrate[((5 + 7*x)^2*(e + f*x)^n)/Sqrt[25 - 49*x^2],x]
 

Output:

(-2*7^(-1 - n)*Sqrt[50 - 70*x]*(5 + 7*x)*(e + f*x)^n*(30*AppellF1[1/2, -1/ 
2, -n, 3/2, (5 + 7*x)/10, -((f*(5 + 7*x))/(7*e - 5*f))] - 30*AppellF1[1/2, 
 1/2, -n, 3/2, (5 + 7*x)/10, -((f*(5 + 7*x))/(7*e - 5*f))] + (5 + 7*x)*App 
ellF1[3/2, -1/2, -n, 5/2, (5 + 7*x)/10, -((f*(5 + 7*x))/(7*e - 5*f))]))/(3 
*((e + f*x)/(7*e - 5*f))^n*Sqrt[25 - 49*x^2])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {717, 156, 155}

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.

\(\displaystyle \int \frac {(7 x+5)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx\)

\(\Big \downarrow \) 717

\(\displaystyle \int \frac {(7 x+5)^{3/2} (e+f x)^n}{\sqrt {5-7 x}}dx\)

\(\Big \downarrow \) 156

\(\displaystyle 7^{-n} (e+f x)^n \left (\frac {e+f x}{7 e+5 f}\right )^{-n} \int \frac {(7 x+5)^{3/2} \left (\frac {7 e}{7 e+5 f}+\frac {7 f x}{7 e+5 f}\right )^n}{\sqrt {5-7 x}}dx\)

\(\Big \downarrow \) 155

\(\displaystyle -20 \sqrt {10} 7^{-n-1} \sqrt {5-7 x} (e+f x)^n \left (\frac {e+f x}{7 e+5 f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-n,\frac {3}{2},\frac {1}{10} (5-7 x),\frac {f (5-7 x)}{7 e+5 f}\right )\)

Input:

Int[((5 + 7*x)^2*(e + f*x)^n)/Sqrt[25 - 49*x^2],x]
 

Output:

(-20*7^(-1 - n)*Sqrt[10]*Sqrt[5 - 7*x]*(e + f*x)^n*AppellF1[1/2, -3/2, -n, 
 3/2, (5 - 7*x)/10, (f*(5 - 7*x))/(7*e + 5*f)])/((e + f*x)/(7*e + 5*f))^n
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 717

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

Maple [F]

\[\int \frac {\left (5+7 x \right )^{2} \left (f x +e \right )^{n}}{\sqrt {-49 x^{2}+25}}d x\]

Input:

int((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x)
 

Output:

int((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n} {\left (7 \, x + 5\right )}^{2}}{\sqrt {-49 \, x^{2} + 25}} \,d x } \] Input:

integrate((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-49*x^2 + 25)*(f*x + e)^n*(7*x + 5)/(7*x - 5), x)
 

Sympy [F]

\[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=\int \frac {\left (e + f x\right )^{n} \left (7 x + 5\right )^{2}}{\sqrt {- \left (7 x - 5\right ) \left (7 x + 5\right )}}\, dx \] Input:

integrate((5+7*x)**2*(f*x+e)**n/(-49*x**2+25)**(1/2),x)
 

Output:

Integral((e + f*x)**n*(7*x + 5)**2/sqrt(-(7*x - 5)*(7*x + 5)), x)
 

Maxima [F]

\[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n} {\left (7 \, x + 5\right )}^{2}}{\sqrt {-49 \, x^{2} + 25}} \,d x } \] Input:

integrate((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x + e)^n*(7*x + 5)^2/sqrt(-49*x^2 + 25), x)
 

Giac [F]

\[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n} {\left (7 \, x + 5\right )}^{2}}{\sqrt {-49 \, x^{2} + 25}} \,d x } \] Input:

integrate((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x, algorithm="giac")
 

Output:

integrate((f*x + e)^n*(7*x + 5)^2/sqrt(-49*x^2 + 25), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=\int \frac {{\left (e+f\,x\right )}^n\,{\left (7\,x+5\right )}^2}{\sqrt {25-49\,x^2}} \,d x \] Input:

int(((e + f*x)^n*(7*x + 5)^2)/(25 - 49*x^2)^(1/2),x)
 

Output:

int(((e + f*x)^n*(7*x + 5)^2)/(25 - 49*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(5+7 x)^2 (e+f x)^n}{\sqrt {25-49 x^2}} \, dx=25 \left (\int \frac {\left (f x +e \right )^{n}}{\sqrt {-49 x^{2}+25}}d x \right )+49 \left (\int \frac {\left (f x +e \right )^{n} x^{2}}{\sqrt {-49 x^{2}+25}}d x \right )+70 \left (\int \frac {\left (f x +e \right )^{n} x}{\sqrt {-49 x^{2}+25}}d x \right ) \] Input:

int((5+7*x)^2*(f*x+e)^n/(-49*x^2+25)^(1/2),x)
 

Output:

25*int((e + f*x)**n/sqrt( - 49*x**2 + 25),x) + 49*int(((e + f*x)**n*x**2)/ 
sqrt( - 49*x**2 + 25),x) + 70*int(((e + f*x)**n*x)/sqrt( - 49*x**2 + 25),x 
)