\(\int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx\) [567]

Optimal result

Integrand size = 22, antiderivative size = 192 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=-\frac {2 f (9 c f-2 d e (11+2 n)) (b x)^{7/2} (c+d x)^{1+n}}{b d^2 (9+2 n) (11+2 n)}+\frac {2 f^2 (b x)^{9/2} (c+d x)^{1+n}}{b^2 d (11+2 n)}+\frac {2 \left (d^2 e^2 (9+2 n) (11+2 n)+7 c f (9 c f-2 d e (11+2 n))\right ) (b x)^{7/2} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )}{7 b d^2 (9+2 n) (11+2 n)} \] Output:

-2*f*(9*c*f-2*d*e*(11+2*n))*(b*x)^(7/2)*(d*x+c)^(1+n)/b/d^2/(9+2*n)/(11+2* 
n)+2*f^2*(b*x)^(9/2)*(d*x+c)^(1+n)/b^2/d/(11+2*n)+2/7*(d^2*e^2*(9+2*n)*(11 
+2*n)+7*c*f*(9*c*f-2*d*e*(11+2*n)))*(b*x)^(7/2)*(d*x+c)^n*hypergeom([7/2, 
-n],[9/2],-d*x/c)/b/d^2/(9+2*n)/(11+2*n)/(((d*x+c)/c)^n)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.76 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\frac {2 x (b x)^{5/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (-7 f (c+d x) \left (1+\frac {d x}{c}\right )^n (9 c f-d (e (22+4 n)+f (9+2 n) x))+\left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )\right )}{7 d^2 (9+2 n) (11+2 n)} \] Input:

Integrate[(b*x)^(5/2)*(c + d*x)^n*(e + f*x)^2,x]
 

Output:

(2*x*(b*x)^(5/2)*(c + d*x)^n*(-7*f*(c + d*x)*(1 + (d*x)/c)^n*(9*c*f - d*(e 
*(22 + 4*n) + f*(9 + 2*n)*x)) + (63*c^2*f^2 - 14*c*d*e*f*(11 + 2*n) + d^2* 
e^2*(99 + 40*n + 4*n^2))*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)]))/(7* 
d^2*(9 + 2*n)*(11 + 2*n)*(1 + (d*x)/c)^n)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {101, 27, 90, 76, 74}

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.

\(\displaystyle \int (b x)^{5/2} (e+f x)^2 (c+d x)^n \, dx\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {2 \int -\frac {1}{2} b (b x)^{5/2} (c+d x)^n (e (7 c f-d e (2 n+11))+f (9 c f-d e (2 n+13)) x)dx}{b d (2 n+11)}+\frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)}-\frac {\int (b x)^{5/2} (c+d x)^n (e (7 c f-d e (2 n+11))+f (9 c f-d e (2 n+13)) x)dx}{d (2 n+11)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)}-\frac {\frac {2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d (2 n+9)}-\frac {\left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \int (b x)^{5/2} (c+d x)^ndx}{d (2 n+9)}}{d (2 n+11)}\)

\(\Big \downarrow \) 76

\(\displaystyle \frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)}-\frac {\frac {2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d (2 n+9)}-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \int (b x)^{5/2} \left (\frac {d x}{c}+1\right )^ndx}{d (2 n+9)}}{d (2 n+11)}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)}-\frac {\frac {2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d (2 n+9)}-\frac {2 (b x)^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )}{7 b d (2 n+9)}}{d (2 n+11)}\)

Input:

Int[(b*x)^(5/2)*(c + d*x)^n*(e + f*x)^2,x]
 

Output:

(2*f*(b*x)^(7/2)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(11 + 2*n)) - ((2*f*(9* 
c*f - d*e*(13 + 2*n))*(b*x)^(7/2)*(c + d*x)^(1 + n))/(b*d*(9 + 2*n)) - (2* 
(63*c^2*f^2 - 14*c*d*e*f*(11 + 2*n) + d^2*e^2*(99 + 40*n + 4*n^2))*(b*x)^( 
7/2)*(c + d*x)^n*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)])/(7*b*d*(9 + 
2*n)*(1 + (d*x)/c)^n))/(d*(11 + 2*n))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

Maple [F]

\[\int \left (b x \right )^{\frac {5}{2}} \left (x d +c \right )^{n} \left (f x +e \right )^{2}d x\]

Input:

int((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x)
 

Output:

int((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x)
 

Fricas [F]

\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x, algorithm="fricas")
 

Output:

integral((b^2*f^2*x^4 + 2*b^2*e*f*x^3 + b^2*e^2*x^2)*sqrt(b*x)*(d*x + c)^n 
, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 99.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.57 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\frac {2 b^{\frac {5}{2}} c^{n} e^{2} x^{\frac {7}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - n \\ \frac {9}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{7} + \frac {4 b^{\frac {5}{2}} c^{n} e f x^{\frac {9}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{2}, - n \\ \frac {11}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{9} + \frac {2 b^{\frac {5}{2}} c^{n} f^{2} x^{\frac {11}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {11}{2}, - n \\ \frac {13}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{11} \] Input:

integrate((b*x)**(5/2)*(d*x+c)**n*(f*x+e)**2,x)
 

Output:

2*b**(5/2)*c**n*e**2*x**(7/2)*hyper((7/2, -n), (9/2,), d*x*exp_polar(I*pi) 
/c)/7 + 4*b**(5/2)*c**n*e*f*x**(9/2)*hyper((9/2, -n), (11/2,), d*x*exp_pol 
ar(I*pi)/c)/9 + 2*b**(5/2)*c**n*f**2*x**(11/2)*hyper((11/2, -n), (13/2,), 
d*x*exp_polar(I*pi)/c)/11
 

Maxima [F]

\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x, algorithm="maxima")
 

Output:

integrate((b*x)^(5/2)*(f*x + e)^2*(d*x + c)^n, x)
 

Giac [F]

\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*x)^(5/2)*(f*x + e)^2*(d*x + c)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^n \,d x \] Input:

int((e + f*x)^2*(b*x)^(5/2)*(c + d*x)^n,x)
 

Output:

int((e + f*x)^2*(b*x)^(5/2)*(c + d*x)^n, x)
 

Reduce [F]

\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\text {too large to display} \] Input:

int((b*x)^(5/2)*(d*x+c)^n*(f*x+e)^2,x)
 

Output:

(2*sqrt(b)*b**2*(1890*sqrt(x)*(c + d*x)**n*c**5*f**2*n - 840*sqrt(x)*(c + 
d*x)**n*c**4*d*e*f*n**2 - 4620*sqrt(x)*(c + d*x)**n*c**4*d*e*f*n - 1260*sq 
rt(x)*(c + d*x)**n*c**4*d*f**2*n**2*x - 630*sqrt(x)*(c + d*x)**n*c**4*d*f* 
*2*n*x + 120*sqrt(x)*(c + d*x)**n*c**3*d**2*e**2*n**3 + 1200*sqrt(x)*(c + 
d*x)**n*c**3*d**2*e**2*n**2 + 2970*sqrt(x)*(c + d*x)**n*c**3*d**2*e**2*n + 
 560*sqrt(x)*(c + d*x)**n*c**3*d**2*e*f*n**3*x + 3360*sqrt(x)*(c + d*x)**n 
*c**3*d**2*e*f*n**2*x + 1540*sqrt(x)*(c + d*x)**n*c**3*d**2*e*f*n*x + 504* 
sqrt(x)*(c + d*x)**n*c**3*d**2*f**2*n**3*x**2 + 1008*sqrt(x)*(c + d*x)**n* 
c**3*d**2*f**2*n**2*x**2 + 378*sqrt(x)*(c + d*x)**n*c**3*d**2*f**2*n*x**2 
- 80*sqrt(x)*(c + d*x)**n*c**2*d**3*e**2*n**4*x - 840*sqrt(x)*(c + d*x)**n 
*c**2*d**3*e**2*n**3*x - 2380*sqrt(x)*(c + d*x)**n*c**2*d**3*e**2*n**2*x - 
 990*sqrt(x)*(c + d*x)**n*c**2*d**3*e**2*n*x - 224*sqrt(x)*(c + d*x)**n*c* 
*2*d**3*e*f*n**4*x**2 - 1680*sqrt(x)*(c + d*x)**n*c**2*d**3*e*f*n**3*x**2 
- 2632*sqrt(x)*(c + d*x)**n*c**2*d**3*e*f*n**2*x**2 - 924*sqrt(x)*(c + d*x 
)**n*c**2*d**3*e*f*n*x**2 - 144*sqrt(x)*(c + d*x)**n*c**2*d**3*f**2*n**4*x 
**3 - 648*sqrt(x)*(c + d*x)**n*c**2*d**3*f**2*n**3*x**3 - 828*sqrt(x)*(c + 
 d*x)**n*c**2*d**3*f**2*n**2*x**3 - 270*sqrt(x)*(c + d*x)**n*c**2*d**3*f** 
2*n*x**3 + 32*sqrt(x)*(c + d*x)**n*c*d**4*e**2*n**5*x**2 + 384*sqrt(x)*(c 
+ d*x)**n*c*d**4*e**2*n**4*x**2 + 1456*sqrt(x)*(c + d*x)**n*c*d**4*e**2*n* 
*3*x**2 + 1824*sqrt(x)*(c + d*x)**n*c*d**4*e**2*n**2*x**2 + 594*sqrt(x)...