\(\int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx\) [561]

Optimal result

Integrand size = 18, antiderivative size = 156 \[ \int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx=-\frac {(2 b c+a d) (a+b x)^{1+n}}{b^2 d^3 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d^2 (2+n)}-\frac {c^3 (a+b x)^{1+n}}{d^3 (b c-a d) (c+d x)}-\frac {c^2 (3 a d-b c (3+n)) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d^3 (b c-a d)^2 (1+n)} \] Output:

-(a*d+2*b*c)*(b*x+a)^(1+n)/b^2/d^3/(1+n)+(b*x+a)^(2+n)/b^2/d^2/(2+n)-c^3*( 
b*x+a)^(1+n)/d^3/(-a*d+b*c)/(d*x+c)-c^2*(3*a*d-b*c*(3+n))*(b*x+a)^(1+n)*hy 
pergeom([1, 1+n],[2+n],-d*(b*x+a)/(-a*d+b*c))/d^3/(-a*d+b*c)^2/(1+n)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.13 \[ \int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n} \left (\frac {x^2}{c+d x}+\frac {a^2 d^2 (c+d x)-b^2 c^2 (3+n) (c (2+n)+d x)+a b c d (c (3+2 n)+d (2+n) x)}{b d^2 (b c-a d) (1+n) (c+d x)}+\frac {b c^2 (2+n) (-3 a d+b c (3+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{d^2 (b c-a d)^2 (1+n)}\right )}{b d (2+n)} \] Input:

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

Output:

((a + b*x)^(1 + n)*(x^2/(c + d*x) + (a^2*d^2*(c + d*x) - b^2*c^2*(3 + n)*( 
c*(2 + n) + d*x) + a*b*c*d*(c*(3 + 2*n) + d*(2 + n)*x))/(b*d^2*(b*c - a*d) 
*(1 + n)*(c + d*x)) + (b*c^2*(2 + n)*(-3*a*d + b*c*(3 + n))*Hypergeometric 
2F1[1, 1 + n, 2 + n, (d*(a + b*x))/(-(b*c) + a*d)])/(d^2*(b*c - a*d)^2*(1 
+ n))))/(b*d*(2 + n))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.37, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {111, 25, 163, 78}

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.

\(\displaystyle \int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {\int -\frac {x (a+b x)^n (2 a c+(a d+b c (n+3)) x)}{(c+d x)^2}dx}{b d (n+2)}+\frac {x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)}-\frac {\int \frac {x (a+b x)^n (2 a c+(a d+b c (n+3)) x)}{(c+d x)^2}dx}{b d (n+2)}\)

\(\Big \downarrow \) 163

\(\displaystyle \frac {x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)}-\frac {\frac {b c^2 (n+2) (3 a d-b c (n+3)) \int \frac {(a+b x)^n}{c+d x}dx}{d^2 (b c-a d)}+\frac {(a+b x)^{n+1} (d x (b c-a d) (a d+b c (n+3))+c (b c (n+2) (a d+b c (n+3))-a d (a d+b c (3 n+5))))}{b d^2 (n+1) (c+d x) (b c-a d)}}{b d (n+2)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)}-\frac {\frac {b c^2 (n+2) (a+b x)^{n+1} (3 a d-b c (n+3)) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)^2}+\frac {(a+b x)^{n+1} (d x (b c-a d) (a d+b c (n+3))+c (b c (n+2) (a d+b c (n+3))-a d (a d+b c (3 n+5))))}{b d^2 (n+1) (c+d x) (b c-a d)}}{b d (n+2)}\)

Input:

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

Output:

(x^2*(a + b*x)^(1 + n))/(b*d*(2 + n)*(c + d*x)) - (((a + b*x)^(1 + n)*(c*( 
b*c*(2 + n)*(a*d + b*c*(3 + n)) - a*d*(a*d + b*c*(5 + 3*n))) + d*(b*c - a* 
d)*(a*d + b*c*(3 + n))*x))/(b*d^2*(b*c - a*d)*(1 + n)*(c + d*x)) + (b*c^2* 
(2 + n)*(3*a*d - b*c*(3 + n))*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n 
, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(d^2*(b*c - a*d)^2*(1 + n)))/(b*d* 
(2 + n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 78

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

rule 111

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

rule 163

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n 
+ 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* 
(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f 
*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* 
d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* 
d*(b*c - a*d)*(m + 1)*(m + n + 3))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 
1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
 

Maple [F]

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

Input:

int(x^3*(b*x+a)^n/(d*x+c)^2,x)
 

Output:

int(x^3*(b*x+a)^n/(d*x+c)^2,x)
 

Fricas [F]

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

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

Output:

integral((b*x + a)^n*x^3/(d^2*x^2 + 2*c*d*x + c^2), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**3*(b*x+a)**n/(d*x+c)**2,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((b*x + a)^n*x^3/(d*x + c)^2, x)
 

Giac [F]

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

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

Output:

integrate((b*x + a)^n*x^3/(d*x + c)^2, x)
 

Mupad [F(-1)]

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

int((x^3*(a + b*x)^n)/(c + d*x)^2,x)
 

Output:

int((x^3*(a + b*x)^n)/(c + d*x)^2, x)
 

Reduce [F]

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

int(x^3*(b*x+a)^n/(d*x+c)^2,x)
 

Output:

( - (a + b*x)**n*a**3*c*d**2*n - (a + b*x)**n*a**3*d**3*n*x - (a + b*x)**n 
*a**2*b*c**2*d*n**2 - 4*(a + b*x)**n*a**2*b*c**2*d*n - 4*(a + b*x)**n*a**2 
*b*c*d**2*n*x + (a + b*x)**n*a**2*b*d**3*n**2*x**2 + (a + b*x)**n*a*b**2*c 
**3*n**2 + 5*(a + b*x)**n*a*b**2*c**3*n + 6*(a + b*x)**n*a*b**2*c**3 + (a 
+ b*x)**n*a*b**2*c**2*d*n**3*x + 5*(a + b*x)**n*a*b**2*c**2*d*n**2*x + 5*( 
a + b*x)**n*a*b**2*c**2*d*n*x + 6*(a + b*x)**n*a*b**2*c**2*d*x - (a + b*x) 
**n*a*b**2*c*d**2*n**3*x**2 - (a + b*x)**n*a*b**2*c*d**2*n**2*x**2 - 3*(a 
+ b*x)**n*a*b**2*c*d**2*n*x**2 + (a + b*x)**n*a*b**2*d**3*n**2*x**3 + (a + 
 b*x)**n*a*b**2*d**3*n*x**3 - (a + b*x)**n*b**3*c**3*n**3*x - 5*(a + b*x)* 
*n*b**3*c**3*n**2*x - 6*(a + b*x)**n*b**3*c**3*n*x + (a + b*x)**n*b**3*c** 
2*d*n**3*x**2 + 3*(a + b*x)**n*b**3*c**2*d*n**2*x**2 - (a + b*x)**n*b**3*c 
*d**2*n**3*x**3 - (a + b*x)**n*b**3*c*d**2*n**2*x**3 + 3*int(((a + b*x)**n 
*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c 
**2*d*n*x + a*b*c**2*d*x - a*b*c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d** 
3*x**3 - b**2*c**3*n*x - 2*b**2*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a** 
3*b**2*c**3*d**3*n**3 + 9*int(((a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d** 
2*x + a**2*d**3*x**2 - a*b*c**3*n - 2*a*b*c**2*d*n*x + a*b*c**2*d*x - a*b* 
c*d**2*n*x**2 + 2*a*b*c*d**2*x**2 + a*b*d**3*x**3 - b**2*c**3*n*x - 2*b**2 
*c**2*d*n*x**2 - b**2*c*d**2*n*x**3),x)*a**3*b**2*c**3*d**3*n**2 + 6*int(( 
(a + b*x)**n*x)/(a**2*c**2*d + 2*a**2*c*d**2*x + a**2*d**3*x**2 - a*b*c...