\(\int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx\) [548]

Optimal result

Integrand size = 18, antiderivative size = 127 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx=-\frac {c^2 (a+b x)^{1+n}}{3 a x^3}-\frac {d^2 (a+b x)^{1+n}}{b (1-n) x^2}+\frac {b \left (6 a^2 d^2-6 a b c d (1-n)+b^2 c^2 \left (2-3 n+n^2\right )\right ) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{3 a^4 \left (1-n^2\right )} \] Output:

-1/3*c^2*(b*x+a)^(1+n)/a/x^3-d^2*(b*x+a)^(1+n)/b/(1-n)/x^2+1/3*b*(6*a^2*d^ 
2-6*a*b*c*d*(1-n)+b^2*c^2*(n^2-3*n+2))*(b*x+a)^(1+n)*hypergeom([3, 1+n],[2 
+n],1+b*x/a)/a^4/(-n^2+1)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx=\frac {(a+b x)^{1+n} \left (-a^2 c (1+n) (b c (-2+n) x+2 a (c+3 d x))+b \left (6 a^2 d^2+6 a b c d (-1+n)+b^2 c^2 \left (2-3 n+n^2\right )\right ) x^3 \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )\right )}{6 a^4 (1+n) x^3} \] Input:

Integrate[((a + b*x)^n*(c + d*x)^2)/x^4,x]
 

Output:

((a + b*x)^(1 + n)*(-(a^2*c*(1 + n)*(b*c*(-2 + n)*x + 2*a*(c + 3*d*x))) + 
b*(6*a^2*d^2 + 6*a*b*c*d*(-1 + n) + b^2*c^2*(2 - 3*n + n^2))*x^3*Hypergeom 
etric2F1[2, 1 + n, 2 + n, 1 + (b*x)/a]))/(6*a^4*(1 + n)*x^3)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 87, 75}

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.

\(\displaystyle \int \frac {(c+d x)^2 (a+b x)^n}{x^4} \, dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {\int \frac {(a+b x)^n \left (3 a x d^2+c (6 a d-b c (2-n))\right )}{x^3}dx}{3 a}-\frac {c^2 (a+b x)^{n+1}}{3 a x^3}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {\frac {\left (6 a^2 d^2-b c (1-n) (6 a d-b c (2-n))\right ) \int \frac {(a+b x)^n}{x^2}dx}{2 a}-\frac {c (a+b x)^{n+1} (6 a d-b c (2-n))}{2 a x^2}}{3 a}-\frac {c^2 (a+b x)^{n+1}}{3 a x^3}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {\frac {b (a+b x)^{n+1} \left (6 a^2 d^2-b c (1-n) (6 a d-b c (2-n))\right ) \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {b x}{a}+1\right )}{2 a^3 (n+1)}-\frac {c (a+b x)^{n+1} (6 a d-b c (2-n))}{2 a x^2}}{3 a}-\frac {c^2 (a+b x)^{n+1}}{3 a x^3}\)

Input:

Int[((a + b*x)^n*(c + d*x)^2)/x^4,x]
 

Output:

-1/3*(c^2*(a + b*x)^(1 + n))/(a*x^3) + (-1/2*(c*(6*a*d - b*c*(2 - n))*(a + 
 b*x)^(1 + n))/(a*x^2) + (b*(6*a^2*d^2 - b*c*(6*a*d - b*c*(2 - n))*(1 - n) 
)*(a + b*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*x)/a])/(2*a^ 
3*(1 + n)))/(3*a)
 

Defintions of rubi rules used

rule 75

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

rule 87

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

rule 100

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

Maple [F]

\[\int \frac {\left (b x +a \right )^{n} \left (x d +c \right )^{2}}{x^{4}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x+a)^n*(d*x+c)^2/x^4,x, algorithm="fricas")
 

Output:

integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^4, x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3794 vs. \(2 (105) = 210\).

Time = 12.64 (sec) , antiderivative size = 3794, normalized size of antiderivative = 29.87 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**n*(d*x+c)**2/x**4,x)
 

Output:

-2*a**3*b**(n + 4)*c**2*n**4*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 
 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a** 
5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) 
+ 4*a**3*b**(n + 4)*c**2*n**3*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n 
+ 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a* 
*5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) 
 + a**3*b**(n + 4)*c**2*n**3*(a/b + x)**(n + 1)*gamma(n + 1)/(12*a**7*gamm 
a(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 
2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) + 2*a**3*b**(n + 4)*c**2*n**2* 
(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gam 
ma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 
 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 2*a**3*b**(n + 4)*c**2*n**2 
*(a/b + x)**(n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma 
(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)** 
3*gamma(n + 2)) - 4*a**3*b**(n + 4)*c**2*n*(a/b + x)**(n + 1)*lerchphi(1 + 
 b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n 
 + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3* 
gamma(n + 2)) - 3*a**3*b**(n + 4)*c**2*n*(a/b + x)**(n + 1)*gamma(n + 1)/( 
12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)** 
2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 3*a**2*b*b**(...
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((b*x+a)^n*(d*x+c)^2/x^4,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^2*(b*x + a)^n/x^4, x)
 

Giac [F]

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

integrate((b*x+a)^n*(d*x+c)^2/x^4,x, algorithm="giac")
 

Output:

integrate((d*x + c)^2*(b*x + a)^n/x^4, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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