\(\int \frac {(c+d x)^n (a+b x^2)^2}{x^4} \, dx\) [200]

Optimal result

Integrand size = 20, antiderivative size = 133 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^4} \, dx=\frac {b^2 (c+d x)^{1+n}}{d (1+n)}-\frac {a^2 (c+d x)^{1+n}}{3 c x^3}-\frac {2 a b (c+d x)^{1+n}}{d (1-n) x^2}+\frac {a d \left (12 b c^2+a d^2 \left (2-3 n+n^2\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )}{3 c^4 \left (1-n^2\right )} \] Output:

b^2*(d*x+c)^(1+n)/d/(1+n)-1/3*a^2*(d*x+c)^(1+n)/c/x^3-2*a*b*(d*x+c)^(1+n)/ 
d/(1-n)/x^2+1/3*a*d*(12*b*c^2+a*d^2*(n^2-3*n+2))*(d*x+c)^(1+n)*hypergeom([ 
3, 1+n],[2+n],1+d*x/c)/c^4/(-n^2+1)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^4} \, dx=\frac {(c+d x)^{1+n} \left (b^2 c^4+2 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )+a^2 d^4 \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {d x}{c}\right )\right )}{c^4 d (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.45, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {520, 2124, 520, 25, 90, 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.

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (6 b^2 c^2 x^2+a \left (12 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right )}{x^2}dx}{2 c}-\frac {a^2 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^2 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {\int -\frac {\left (6 b^2 x c^3+a d n \left (12 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right ) (c+d x)^n}{x}dx}{c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right )}{c x}}{2 c}-\frac {a^2 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^2 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\frac {\int \frac {\left (6 b^2 x c^3+a d n \left (12 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right ) (c+d x)^n}{x}dx}{c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right )}{c x}}{2 c}-\frac {a^2 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^2 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 90

\(\displaystyle -\frac {-\frac {\frac {a d n \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right ) \int \frac {(c+d x)^n}{x}dx+\frac {6 b^2 c^3 (c+d x)^{n+1}}{d (n+1)}}{c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right )}{c x}}{2 c}-\frac {a^2 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^2 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {-\frac {a^2 d (2-n) (c+d x)^{n+1}}{2 c x^2}-\frac {\frac {\frac {6 b^2 c^3 (c+d x)^{n+1}}{d (n+1)}-\frac {a d n (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}}{c}-\frac {a (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+12 b c^2\right )}{c x}}{2 c}}{3 c}-\frac {a^2 (c+d x)^{n+1}}{3 c x^3}\)

Input:

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

Output:

-1/3*(a^2*(c + d*x)^(1 + n))/(c*x^3) - (-1/2*(a^2*d*(2 - n)*(c + d*x)^(1 + 
 n))/(c*x^2) - (-((a*(12*b*c^2 + a*d^2*(2 - 3*n + n^2))*(c + d*x)^(1 + n)) 
/(c*x)) + ((6*b^2*c^3*(c + d*x)^(1 + n))/(d*(1 + n)) - (a*d*n*(12*b*c^2 + 
a*d^2*(2 - 3*n + n^2))*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n 
, 1 + (d*x)/c])/(c*(1 + n)))/c)/(2*c))/(3*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 90

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 11.60 (sec) , antiderivative size = 3016, normalized size of antiderivative = 22.68 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

(d*x + c)^(n + 1)*b^2/(d*(n + 1)) + integrate((2*a*b*x^2 + a^2)*(d*x + c)^ 
n/x^4, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^4} \, dx=\frac {-2 \left (d x +c \right )^{n} a^{2} c^{2} d n -2 \left (d x +c \right )^{n} a^{2} c^{2} d -\left (d x +c \right )^{n} a^{2} c \,d^{2} n^{2} x -\left (d x +c \right )^{n} a^{2} c \,d^{2} n x -\left (d x +c \right )^{n} a^{2} d^{3} n^{3} x^{2}+\left (d x +c \right )^{n} a^{2} d^{3} n^{2} x^{2}+2 \left (d x +c \right )^{n} a^{2} d^{3} n \,x^{2}-12 \left (d x +c \right )^{n} a b \,c^{2} d n \,x^{2}-12 \left (d x +c \right )^{n} a b \,c^{2} d \,x^{2}+6 \left (d x +c \right )^{n} b^{2} c^{3} x^{3}+6 \left (d x +c \right )^{n} b^{2} c^{2} d \,x^{4}+\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{4} x^{3}-2 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{3} x^{3}-\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n^{2} x^{3}+2 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a^{2} d^{4} n \,x^{3}+12 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{2} n^{2} x^{3}+12 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a b \,c^{2} d^{2} n \,x^{3}}{6 c^{2} d \,x^{3} \left (n +1\right )} \] Input:

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

Output:

( - 2*(c + d*x)**n*a**2*c**2*d*n - 2*(c + d*x)**n*a**2*c**2*d - (c + d*x)* 
*n*a**2*c*d**2*n**2*x - (c + d*x)**n*a**2*c*d**2*n*x - (c + d*x)**n*a**2*d 
**3*n**3*x**2 + (c + d*x)**n*a**2*d**3*n**2*x**2 + 2*(c + d*x)**n*a**2*d** 
3*n*x**2 - 12*(c + d*x)**n*a*b*c**2*d*n*x**2 - 12*(c + d*x)**n*a*b*c**2*d* 
x**2 + 6*(c + d*x)**n*b**2*c**3*x**3 + 6*(c + d*x)**n*b**2*c**2*d*x**4 + i 
nt((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**4*n**4*x**3 - 2*int((c + d*x)**n 
/(c*x + d*x**2),x)*a**2*d**4*n**3*x**3 - int((c + d*x)**n/(c*x + d*x**2),x 
)*a**2*d**4*n**2*x**3 + 2*int((c + d*x)**n/(c*x + d*x**2),x)*a**2*d**4*n*x 
**3 + 12*int((c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**2*n**2*x**3 + 12*i 
nt((c + d*x)**n/(c*x + d*x**2),x)*a*b*c**2*d**2*n*x**3)/(6*c**2*d*x**3*(n 
+ 1))