\(\int \frac {(c+d x)^n (a+b x^2)^2}{x^3} \, dx\) [199]

Optimal result

Integrand size = 20, antiderivative size = 149 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^3} \, dx=-\frac {b^2 c (c+d x)^{1+n}}{d^2 (1+n)}-\frac {a^2 (c+d x)^{1+n}}{2 c x^2}+\frac {2 a b (c+d x)^{1+n}}{d n x}+\frac {b^2 (c+d x)^{2+n}}{d^2 (2+n)}+\frac {a \left (4 b c^2-a d^2 (1-n) n\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )}{2 c^3 n (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^3} \, dx=-\frac {(c+d x)^{1+n} \left (b^2 c^3 (c-d (1+n) x)+2 a b c^2 d^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {d x}{c}\right )+a^2 d^4 (2+n) \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )\right )}{c^3 d^2 (1+n) (2+n)} \] Input:

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

Output:

-(((c + d*x)^(1 + n)*(b^2*c^3*(c - d*(1 + n)*x) + 2*a*b*c^2*d^2*(2 + n)*Hy 
pergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x)/c] + a^2*d^4*(2 + n)*Hypergeome 
tric2F1[3, 1 + n, 2 + n, 1 + (d*x)/c]))/(c^3*d^2*(1 + n)*(2 + n)))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {520, 2124, 522, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-2 b^2 c x^3-4 a b c x+a^2 d (1-n)\right )}{x^2}dx}{2 c}-\frac {a^2 (c+d x)^{n+1}}{2 c x^2}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (2 b^2 c^2 x^2+a \left (4 b c^2-a d^2 (1-n) n\right )\right )}{x}dx}{c}-\frac {a^2 d (1-n) (c+d x)^{n+1}}{c x}}{2 c}-\frac {a^2 (c+d x)^{n+1}}{2 c x^2}\)

\(\Big \downarrow \) 522

\(\displaystyle -\frac {-\frac {\int \left (-\frac {2 b^2 c^3 (c+d x)^n}{d}+\frac {a \left (4 b c^2-a d^2 (1-n) n\right ) (c+d x)^n}{x}+\frac {2 b^2 c^2 (c+d x)^{n+1}}{d}\right )dx}{c}-\frac {a^2 d (1-n) (c+d x)^{n+1}}{c x}}{2 c}-\frac {a^2 (c+d x)^{n+1}}{2 c x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {a^2 d (1-n) (c+d x)^{n+1}}{c x}-\frac {-\frac {a (c+d x)^{n+1} \left (4 b c^2-a d^2 (1-n) n\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}-\frac {2 b^2 c^3 (c+d x)^{n+1}}{d^2 (n+1)}+\frac {2 b^2 c^2 (c+d x)^{n+2}}{d^2 (n+2)}}{c}}{2 c}-\frac {a^2 (c+d x)^{n+1}}{2 c x^2}\)

Input:

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

Output:

-1/2*(a^2*(c + d*x)^(1 + n))/(c*x^2) - (-((a^2*d*(1 - n)*(c + d*x)^(1 + n) 
)/(c*x)) - ((-2*b^2*c^3*(c + d*x)^(1 + n))/(d^2*(1 + n)) + (2*b^2*c^2*(c + 
 d*x)^(2 + n))/(d^2*(2 + n)) - (a*(4*b*c^2 - a*d^2*(1 - n)*n)*(c + d*x)^(1 
 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x)/c])/(c*(1 + n)))/c)/(2* 
c)
 

Defintions of rubi rules used

rule 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

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

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x + c)^n/x^3, x)
 

Sympy [A] (verification not implemented)

Time = 3.76 (sec) , antiderivative size = 1090, normalized size of antiderivative = 7.32 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

a**2*c**2*d**(n + 3)*n**3*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1) 
*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d** 
2*(c/d + x)**2*gamma(n + 2)) - a**2*c**2*d**(n + 3)*n*(c/d + x)**(n + 1)*l 
erchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d 
*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**2*c**2*d**(n 
 + 3)*n*(c/d + x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x 
*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**2*c**2*d**(n + 
 3)*(c/d + x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gam 
ma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) + 2*a**2*c*d*d**(n + 3) 
*n**3*x*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2* 
c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gam 
ma(n + 2)) - a**2*c*d*d**(n + 3)*n**2*x*(c/d + x)**(n + 1)*gamma(n + 1)/(- 
2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*g 
amma(n + 2)) - 2*a**2*c*d*d**(n + 3)*n*x*(c/d + x)**(n + 1)*lerchphi(1 + d 
*x/c, 1, n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 
2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) + a**2*c*d*d**(n + 3)*x*(c/d + 
 x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) 
+ 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**2*d**2*d**(n + 3)*n**3*(c/d 
+ x)**2*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2* 
c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^3} \, dx=\frac {-\left (d x +c \right )^{n} a^{2} c \,d^{2} n^{3}-3 \left (d x +c \right )^{n} a^{2} c \,d^{2} n^{2}-2 \left (d x +c \right )^{n} a^{2} c \,d^{2} n -\left (d x +c \right )^{n} a^{2} d^{3} n^{4} x -3 \left (d x +c \right )^{n} a^{2} d^{3} n^{3} x -2 \left (d x +c \right )^{n} a^{2} d^{3} n^{2} x +4 \left (d x +c \right )^{n} a b c \,d^{2} n^{2} x^{2}+12 \left (d x +c \right )^{n} a b c \,d^{2} n \,x^{2}+8 \left (d x +c \right )^{n} a b c \,d^{2} x^{2}-2 \left (d x +c \right )^{n} b^{2} c^{3} n \,x^{2}+2 \left (d x +c \right )^{n} b^{2} c^{2} d \,n^{2} x^{3}+2 \left (d x +c \right )^{n} b^{2} c \,d^{2} n^{2} x^{4}+2 \left (d x +c \right )^{n} b^{2} c \,d^{2} n \,x^{4}+\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{5} x^{2}+2 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{4} x^{2}-\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{3} x^{2}-2 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{2} x^{2}+4 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{2} n^{3} x^{2}+12 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{2} n^{2} x^{2}+8 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{2} n \,x^{2}}{2 c \,d^{2} n \,x^{2} \left (n^{2}+3 n +2\right )} \] Input:

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

Output:

( - (c + d*x)**n*a**2*c*d**2*n**3 - 3*(c + d*x)**n*a**2*c*d**2*n**2 - 2*(c 
 + d*x)**n*a**2*c*d**2*n - (c + d*x)**n*a**2*d**3*n**4*x - 3*(c + d*x)**n* 
a**2*d**3*n**3*x - 2*(c + d*x)**n*a**2*d**3*n**2*x + 4*(c + d*x)**n*a*b*c* 
d**2*n**2*x**2 + 12*(c + d*x)**n*a*b*c*d**2*n*x**2 + 8*(c + d*x)**n*a*b*c* 
d**2*x**2 - 2*(c + d*x)**n*b**2*c**3*n*x**2 + 2*(c + d*x)**n*b**2*c**2*d*n 
**2*x**3 + 2*(c + d*x)**n*b**2*c*d**2*n**2*x**4 + 2*(c + d*x)**n*b**2*c*d* 
*2*n*x**4 + int((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**4*n**5*x**2 + 2*int 
((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**4*n**4*x**2 - int((c + d*x)**n/(c* 
x + d*x**2),x)*a**2*d**4*n**3*x**2 - 2*int((c + d*x)**n/(c*x + d*x**2),x)* 
a**2*d**4*n**2*x**2 + 4*int((c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**2*n 
**3*x**2 + 12*int((c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**2*n**2*x**2 + 
 8*int((c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**2*n*x**2)/(2*c*d**2*n*x* 
*2*(n**2 + 3*n + 2))