\(\int \frac {(c+d x)^n (a+b x^2)^2}{x^6} \, dx\) [202]

Optimal result

Integrand size = 20, antiderivative size = 207 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^6} \, dx=-\frac {a^2 (c+d x)^{1+n}}{5 c x^5}+\frac {a^2 d (4-n) (c+d x)^{1+n}}{20 c^2 x^4}+\frac {2 b^2 c (c+d x)^{1+n}}{d^2 (1-n) (2-n) x^3}-\frac {b^2 (c+d x)^{1+n}}{d (1-n) x^2}+\frac {d \left (120 b^2 c^4+a d^2 (1-n) (2-n) \left (40 b c^2+a d^2 \left (12-7 n+n^2\right )\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {d x}{c}\right )}{20 c^6 (1-n) \left (2+n-n^2\right )} \] Output:

-1/5*a^2*(d*x+c)^(1+n)/c/x^5+1/20*a^2*d*(4-n)*(d*x+c)^(1+n)/c^2/x^4+2*b^2* 
c*(d*x+c)^(1+n)/d^2/(1-n)/(2-n)/x^3-b^2*(d*x+c)^(1+n)/d/(1-n)/x^2+1/20*d*( 
120*b^2*c^4+a*d^2*(1-n)*(2-n)*(40*b*c^2+a*d^2*(n^2-7*n+12)))*(d*x+c)^(1+n) 
*hypergeom([4, 1+n],[2+n],1+d*x/c)/c^6/(1-n)/(-n^2+n+2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.44 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^6} \, dx=\frac {d (c+d x)^{1+n} \left (b^2 c^4 \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \left (2 b c^2 \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \operatorname {Hypergeometric2F1}\left (6,1+n,2+n,1+\frac {d x}{c}\right )\right )\right )}{c^6 (1+n)} \] Input:

Integrate[((c + d*x)^n*(a + b*x^2)^2)/x^6,x]
 

Output:

(d*(c + d*x)^(1 + n)*(b^2*c^4*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (d*x) 
/c] + a*d^2*(2*b*c^2*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (d*x)/c] + a*d 
^2*Hypergeometric2F1[6, 1 + n, 2 + n, 1 + (d*x)/c])))/(c^6*(1 + n))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {520, 2124, 520, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-5 b^2 c x^3-10 a b c x+a^2 d (4-n)\right )}{x^5}dx}{5 c}-\frac {a^2 (c+d x)^{n+1}}{5 c x^5}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (20 b^2 c^2 x^2+a \left (40 b c^2+a d^2 \left (n^2-7 n+12\right )\right )\right )}{x^4}dx}{4 c}-\frac {a^2 d (4-n) (c+d x)^{n+1}}{4 c x^4}}{5 c}-\frac {a^2 (c+d x)^{n+1}}{5 c x^5}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (a d (2-n) \left (40 b c^2+a d^2 \left (n^2-7 n+12\right )\right )-60 b^2 c^3 x\right ) (c+d x)^n}{x^3}dx}{3 c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 d (4-n) (c+d x)^{n+1}}{4 c x^4}}{5 c}-\frac {a^2 (c+d x)^{n+1}}{5 c x^5}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (a d^2 (1-n) (2-n) \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )+120 b^2 c^4\right ) \int \frac {(c+d x)^n}{x^2}dx}{2 c}-\frac {a d (2-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )}{2 c x^2}}{3 c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 d (4-n) (c+d x)^{n+1}}{4 c x^4}}{5 c}-\frac {a^2 (c+d x)^{n+1}}{5 c x^5}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {-\frac {a^2 d (4-n) (c+d x)^{n+1}}{4 c x^4}-\frac {-\frac {-\frac {d (c+d x)^{n+1} \left (a d^2 (1-n) (2-n) \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )+120 b^2 c^4\right ) \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {d x}{c}+1\right )}{2 c^3 (n+1)}-\frac {a d (2-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )}{2 c x^2}}{3 c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-7 n+12\right )+40 b c^2\right )}{3 c x^3}}{4 c}}{5 c}-\frac {a^2 (c+d x)^{n+1}}{5 c x^5}\)

Input:

Int[((c + d*x)^n*(a + b*x^2)^2)/x^6,x]
 

Output:

-1/5*(a^2*(c + d*x)^(1 + n))/(c*x^5) - (-1/4*(a^2*d*(4 - n)*(c + d*x)^(1 + 
 n))/(c*x^4) - (-1/3*(a*(40*b*c^2 + a*d^2*(12 - 7*n + n^2))*(c + d*x)^(1 + 
 n))/(c*x^3) - (-1/2*(a*d*(2 - n)*(40*b*c^2 + a*d^2*(12 - 7*n + n^2))*(c + 
 d*x)^(1 + n))/(c*x^2) - (d*(120*b^2*c^4 + a*d^2*(1 - n)*(2 - n)*(40*b*c^2 
 + a*d^2*(12 - 7*n + n^2)))*(c + d*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 
2 + n, 1 + (d*x)/c])/(2*c^3*(1 + n)))/(3*c))/(4*c))/(5*c)
 

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 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 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]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{2}}{x^{6}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((d*x+c)^n*(b*x^2+a)^2/x^6,x, algorithm="fricas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x + c)^n/x^6, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^6} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**n*(b*x**2+a)**2/x**6,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((d*x+c)^n*(b*x^2+a)^2/x^6,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n/x^6, x)
 

Giac [F]

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

integrate((d*x+c)^n*(b*x^2+a)^2/x^6,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n/x^6, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^6} \, dx=\frac {120 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) b^{2} c^{4} d n \,x^{5}+40 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{3} n^{3} x^{5}-120 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{3} n^{2} x^{5}+80 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{3} n \,x^{5}-24 \left (d x +c \right )^{n} a^{2} c^{4}-10 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{5} n^{4} x^{5}+35 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{5} n^{3} x^{5}-50 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{5} n^{2} x^{5}+24 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{5} n \,x^{5}+\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{5} n^{5} x^{5}+9 \left (d x +c \right )^{n} a^{2} d^{4} n^{3} x^{4}-26 \left (d x +c \right )^{n} a^{2} d^{4} n^{2} x^{4}+24 \left (d x +c \right )^{n} a^{2} d^{4} n \,x^{4}-80 \left (d x +c \right )^{n} a b \,c^{4} x^{2}-120 \left (d x +c \right )^{n} b^{2} c^{4} x^{4}-\left (d x +c \right )^{n} a^{2} c \,d^{3} n^{3} x^{3}-\left (d x +c \right )^{n} a^{2} d^{4} n^{4} x^{4}-6 \left (d x +c \right )^{n} a^{2} c^{3} d n x -2 \left (d x +c \right )^{n} a^{2} c^{2} d^{2} n^{2} x^{2}+8 \left (d x +c \right )^{n} a^{2} c^{2} d^{2} n \,x^{2}+7 \left (d x +c \right )^{n} a^{2} c \,d^{3} n^{2} x^{3}-12 \left (d x +c \right )^{n} a^{2} c \,d^{3} n \,x^{3}-40 \left (d x +c \right )^{n} a b \,c^{3} d n \,x^{3}-40 \left (d x +c \right )^{n} a b \,c^{2} d^{2} n^{2} x^{4}+80 \left (d x +c \right )^{n} a b \,c^{2} d^{2} n \,x^{4}}{120 c^{4} x^{5}} \] Input:

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

Output:

( - 24*(c + d*x)**n*a**2*c**4 - 6*(c + d*x)**n*a**2*c**3*d*n*x - 2*(c + d* 
x)**n*a**2*c**2*d**2*n**2*x**2 + 8*(c + d*x)**n*a**2*c**2*d**2*n*x**2 - (c 
 + d*x)**n*a**2*c*d**3*n**3*x**3 + 7*(c + d*x)**n*a**2*c*d**3*n**2*x**3 - 
12*(c + d*x)**n*a**2*c*d**3*n*x**3 - (c + d*x)**n*a**2*d**4*n**4*x**4 + 9* 
(c + d*x)**n*a**2*d**4*n**3*x**4 - 26*(c + d*x)**n*a**2*d**4*n**2*x**4 + 2 
4*(c + d*x)**n*a**2*d**4*n*x**4 - 80*(c + d*x)**n*a*b*c**4*x**2 - 40*(c + 
d*x)**n*a*b*c**3*d*n*x**3 - 40*(c + d*x)**n*a*b*c**2*d**2*n**2*x**4 + 80*( 
c + d*x)**n*a*b*c**2*d**2*n*x**4 - 120*(c + d*x)**n*b**2*c**4*x**4 + int(( 
c + d*x)**n/(c*x + d*x**2),x)*a**2*d**5*n**5*x**5 - 10*int((c + d*x)**n/(c 
*x + d*x**2),x)*a**2*d**5*n**4*x**5 + 35*int((c + d*x)**n/(c*x + d*x**2),x 
)*a**2*d**5*n**3*x**5 - 50*int((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**5*n* 
*2*x**5 + 24*int((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**5*n*x**5 + 40*int( 
(c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**3*n**3*x**5 - 120*int((c + d*x) 
**n/(c*x + d*x**2),x)*a*b*c**2*d**3*n**2*x**5 + 80*int((c + d*x)**n/(c*x + 
 d*x**2),x)*a*b*c**2*d**3*n*x**5 + 120*int((c + d*x)**n/(c*x + d*x**2),x)* 
b**2*c**4*d*n*x**5)/(120*c**4*x**5)