\(\int \frac {(a+b x)^n (c+d x)^2}{x} \, dx\) [545]

Optimal result

Integrand size = 18, antiderivative size = 88 \[ \int \frac {(a+b x)^n (c+d x)^2}{x} \, dx=\frac {d (2 b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac {d^2 (a+b x)^{2+n}}{b^2 (2+n)}-\frac {c^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^n (c+d x)^2}{x} \, dx=-\frac {(a+b x)^{1+n} \left (a d (a d-b (2 c (2+n)+d (1+n) x))+b^2 c^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a b^2 (1+n) (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 88, 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.

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (71) = 142\).

Time = 2.68 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.59 \[ \int \frac {(a+b x)^n (c+d x)^2}{x} \, dx=2 c d \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + d^{2} \left (\begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases}\right ) - \frac {b^{n + 1} c^{2} n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 1} c^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \] Input:

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

Output:

2*c*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)) + 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)) - b**(n + 1)*c**2*n*(a/b + x)**(n + 1)*lerchphi(1 + 
b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**(n + 1)*c**2*(a/b + x) 
**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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