\(\int \frac {(a+b x)^n (c+d x)^3}{x} \, dx\) [552]

Optimal result

Integrand size = 18, antiderivative size = 131 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c^3 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=(a+b x)^{1+n} \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)}{b^3 (2+n)}+\frac {d^3 (a+b x)^2}{b^3 (3+n)}-\frac {c^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b x}{a}\right )}{a+a n}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 2009}

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.

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 99

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (116) = 232\).

Time = 3.38 (sec) , antiderivative size = 923, normalized size of antiderivative = 7.05 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

3*c**2*d*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 
 1), Ne(n, -1)), (log(a + b*x), True))/b, True)) + 3*c*d**2*Piecewise((a** 
n*x**2/2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x) + a/(a*b**2 + b**3* 
x) + b*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(n, -2)), (-a*log(a/b + x)/b**2 
 + x/b, Eq(n, -1)), (-a**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + 
a*b*n*x*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*n*x**2*(a + b* 
x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*x**2*(a + b*x)**n/(b**2*n**2 
+ 3*b**2*n + 2*b**2), True)) + d**3*Piecewise((a**n*x**3/3, Eq(b, 0)), (2* 
a**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 3*a**2/(2*a** 
2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x*log(a/b + x)/(2*a**2*b**3 + 4 
*a*b**4*x + 2*b**5*x**2) + 4*a*b*x/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2 
) + 2*b**2*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2), Eq( 
n, -3)), (-2*a**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2/(a*b**3 + b**4*x 
) - 2*a*b*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*x**2/(a*b**3 + b**4*x), 
Eq(n, -2)), (a**2*log(a/b + x)/b**3 - a*x/b**2 + x**2/(2*b), Eq(n, -1)), ( 
2*a**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a** 
2*b*n*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b* 
*2*n**2*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 
 a*b**2*n*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) 
 + b**3*n**2*x**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6...
 

Maxima [F]

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

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

Output:

3*(b*x + a)^(n + 1)*c^2*d/(b*(n + 1)) + integrate((d^3*x^3 + 3*c*d^2*x^2 + 
 c^3)*(b*x + a)^n/x, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=\frac {\left (b x +a \right )^{n} b^{3} c^{3} n^{3}+2 \left (b x +a \right )^{n} a^{3} d^{3} n +6 \left (b x +a \right )^{n} b^{3} c^{3} n^{2}+11 \left (b x +a \right )^{n} b^{3} c^{3} n +\left (b x +a \right )^{n} a \,b^{2} d^{3} n^{3} x^{2}+\left (b x +a \right )^{n} a \,b^{2} d^{3} n^{2} x^{2}+\left (b x +a \right )^{n} b^{3} d^{3} n^{3} x^{3}-3 \left (b x +a \right )^{n} a^{2} b c \,d^{2} n^{2}-9 \left (b x +a \right )^{n} a^{2} b c \,d^{2} n -2 \left (b x +a \right )^{n} a^{2} b \,d^{3} n^{2} x +3 \left (b x +a \right )^{n} a \,b^{2} c^{2} d \,n^{3}+15 \left (b x +a \right )^{n} a \,b^{2} c^{2} d \,n^{2}+18 \left (b x +a \right )^{n} a \,b^{2} c^{2} d n +3 \left (b x +a \right )^{n} b^{3} c^{2} d \,n^{3} x +15 \left (b x +a \right )^{n} b^{3} c^{2} d \,n^{2} x +18 \left (b x +a \right )^{n} b^{3} c^{2} d n x +3 \left (b x +a \right )^{n} b^{3} c \,d^{2} n^{3} x^{2}+12 \left (b x +a \right )^{n} b^{3} c \,d^{2} n^{2} x^{2}+9 \left (b x +a \right )^{n} b^{3} c \,d^{2} n \,x^{2}+\left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a \,b^{3} c^{3} n^{4}+6 \left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a \,b^{3} c^{3} n^{3}+11 \left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a \,b^{3} c^{3} n^{2}+6 \left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a \,b^{3} c^{3} n +3 \left (b x +a \right )^{n} b^{3} d^{3} n^{2} x^{3}+2 \left (b x +a \right )^{n} b^{3} d^{3} n \,x^{3}+6 \left (b x +a \right )^{n} b^{3} c^{3}+3 \left (b x +a \right )^{n} a \,b^{2} c \,d^{2} n^{3} x +9 \left (b x +a \right )^{n} a \,b^{2} c \,d^{2} n^{2} x}{b^{3} n \left (n^{3}+6 n^{2}+11 n +6\right )} \] Input:

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

Output:

(2*(a + b*x)**n*a**3*d**3*n - 3*(a + b*x)**n*a**2*b*c*d**2*n**2 - 9*(a + b 
*x)**n*a**2*b*c*d**2*n - 2*(a + b*x)**n*a**2*b*d**3*n**2*x + 3*(a + b*x)** 
n*a*b**2*c**2*d*n**3 + 15*(a + b*x)**n*a*b**2*c**2*d*n**2 + 18*(a + b*x)** 
n*a*b**2*c**2*d*n + 3*(a + b*x)**n*a*b**2*c*d**2*n**3*x + 9*(a + b*x)**n*a 
*b**2*c*d**2*n**2*x + (a + b*x)**n*a*b**2*d**3*n**3*x**2 + (a + b*x)**n*a* 
b**2*d**3*n**2*x**2 + (a + b*x)**n*b**3*c**3*n**3 + 6*(a + b*x)**n*b**3*c* 
*3*n**2 + 11*(a + b*x)**n*b**3*c**3*n + 6*(a + b*x)**n*b**3*c**3 + 3*(a + 
b*x)**n*b**3*c**2*d*n**3*x + 15*(a + b*x)**n*b**3*c**2*d*n**2*x + 18*(a + 
b*x)**n*b**3*c**2*d*n*x + 3*(a + b*x)**n*b**3*c*d**2*n**3*x**2 + 12*(a + b 
*x)**n*b**3*c*d**2*n**2*x**2 + 9*(a + b*x)**n*b**3*c*d**2*n*x**2 + (a + b* 
x)**n*b**3*d**3*n**3*x**3 + 3*(a + b*x)**n*b**3*d**3*n**2*x**3 + 2*(a + b* 
x)**n*b**3*d**3*n*x**3 + int((a + b*x)**n/(a*x + b*x**2),x)*a*b**3*c**3*n* 
*4 + 6*int((a + b*x)**n/(a*x + b*x**2),x)*a*b**3*c**3*n**3 + 11*int((a + b 
*x)**n/(a*x + b*x**2),x)*a*b**3*c**3*n**2 + 6*int((a + b*x)**n/(a*x + b*x* 
*2),x)*a*b**3*c**3*n)/(b**3*n*(n**3 + 6*n**2 + 11*n + 6))