\(\int \frac {(c+d x)^n (a+b x^2)}{x^5} \, dx\) [193]

Optimal result

Integrand size = 18, antiderivative size = 111 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^5} \, dx=-\frac {a (c+d x)^{1+n}}{4 c x^4}-\frac {b (c+d x)^{1+n}}{d (2-n) x^3}-\frac {d^2 \left (12 b c^2+a d^2 \left (6-5 n+n^2\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {d x}{c}\right )}{4 c^5 \left (2+n-n^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^5} \, dx=-\frac {d^2 (c+d x)^{1+n} \left (b c^2 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \operatorname {Hypergeometric2F1}\left (5,1+n,2+n,1+\frac {d x}{c}\right )\right )}{c^5 (1+n)} \] Input:

Integrate[((c + d*x)^n*(a + b*x^2))/x^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) (c+d x)^n}{x^5} \, dx\)

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 75

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

Input:

Int[((c + d*x)^n*(a + b*x^2))/x^5,x]
 

Output:

-1/4*(a*(c + d*x)^(1 + n))/(c*x^4) - (-1/3*(a*d*(3 - n)*(c + d*x)^(1 + n)) 
/(c*x^3) + (d^2*(12*b*c^2 + a*d^2*(2 - n)*(3 - n))*(c + d*x)^(1 + n)*Hyper 
geometric2F1[3, 1 + n, 2 + n, 1 + (d*x)/c])/(3*c^4*(1 + n)))/(4*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]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )}{x^{5}}d x\]

Input:

int((d*x+c)^n*(b*x^2+a)/x^5,x)
 

Output:

int((d*x+c)^n*(b*x^2+a)/x^5,x)
 

Fricas [F]

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

integrate((d*x+c)^n*(b*x^2+a)/x^5,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)*(d*x + c)^n/x^5, x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6387 vs. \(2 (92) = 184\).

Time = 85.81 (sec) , antiderivative size = 6387, normalized size of antiderivative = 57.54 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^5} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)**n*(b*x**2+a)/x**5,x)
 

Output:

3*a*c**4*d**(n + 5)*n**5*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)* 
gamma(n + 1)/(-72*c**9*gamma(n + 2) - 96*c**8*d*x*gamma(n + 2) + 144*c**7* 
d**2*(c/d + x)**2*gamma(n + 2) - 96*c**6*d**3*(c/d + x)**3*gamma(n + 2) + 
24*c**5*d**4*(c/d + x)**4*gamma(n + 2)) - 15*a*c**4*d**(n + 5)*n**4*(c/d + 
 x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-72*c**9*gamma(n 
+ 2) - 96*c**8*d*x*gamma(n + 2) + 144*c**7*d**2*(c/d + x)**2*gamma(n + 2) 
- 96*c**6*d**3*(c/d + x)**3*gamma(n + 2) + 24*c**5*d**4*(c/d + x)**4*gamma 
(n + 2)) - 2*a*c**4*d**(n + 5)*n**4*(c/d + x)**(n + 1)*gamma(n + 1)/(-72*c 
**9*gamma(n + 2) - 96*c**8*d*x*gamma(n + 2) + 144*c**7*d**2*(c/d + x)**2*g 
amma(n + 2) - 96*c**6*d**3*(c/d + x)**3*gamma(n + 2) + 24*c**5*d**4*(c/d + 
 x)**4*gamma(n + 2)) + 15*a*c**4*d**(n + 5)*n**3*(c/d + x)**(n + 1)*lerchp 
hi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-72*c**9*gamma(n + 2) - 96*c**8*d*x* 
gamma(n + 2) + 144*c**7*d**2*(c/d + x)**2*gamma(n + 2) - 96*c**6*d**3*(c/d 
 + x)**3*gamma(n + 2) + 24*c**5*d**4*(c/d + x)**4*gamma(n + 2)) + 11*a*c** 
4*d**(n + 5)*n**3*(c/d + x)**(n + 1)*gamma(n + 1)/(-72*c**9*gamma(n + 2) - 
 96*c**8*d*x*gamma(n + 2) + 144*c**7*d**2*(c/d + x)**2*gamma(n + 2) - 96*c 
**6*d**3*(c/d + x)**3*gamma(n + 2) + 24*c**5*d**4*(c/d + x)**4*gamma(n + 2 
)) + 15*a*c**4*d**(n + 5)*n**2*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n 
 + 1)*gamma(n + 1)/(-72*c**9*gamma(n + 2) - 96*c**8*d*x*gamma(n + 2) + 144 
*c**7*d**2*(c/d + x)**2*gamma(n + 2) - 96*c**6*d**3*(c/d + x)**3*gamma(...
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((d*x+c)^n*(b*x^2+a)/x^5,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)*(d*x + c)^n/x^5, x)
 

Giac [F]

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

integrate((d*x+c)^n*(b*x^2+a)/x^5,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)*(d*x + c)^n/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^5} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (c+d\,x\right )}^n}{x^5} \,d x \] Input:

int(((a + b*x^2)*(c + d*x)^n)/x^5,x)
 

Output:

int(((a + b*x^2)*(c + d*x)^n)/x^5, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^5} \, dx=\frac {-6 \left (d x +c \right )^{n} a \,c^{3}-2 \left (d x +c \right )^{n} a \,c^{2} d n x -\left (d x +c \right )^{n} a c \,d^{2} n^{2} x^{2}+3 \left (d x +c \right )^{n} a c \,d^{2} n \,x^{2}-\left (d x +c \right )^{n} a \,d^{3} n^{3} x^{3}+5 \left (d x +c \right )^{n} a \,d^{3} n^{2} x^{3}-6 \left (d x +c \right )^{n} a \,d^{3} n \,x^{3}-12 \left (d x +c \right )^{n} b \,c^{3} x^{2}-12 \left (d x +c \right )^{n} b \,c^{2} d n \,x^{3}+\left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a \,d^{4} n^{4} x^{4}-6 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a \,d^{4} n^{3} x^{4}+11 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a \,d^{4} n^{2} x^{4}-6 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) a \,d^{4} n \,x^{4}+12 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) b \,c^{2} d^{2} n^{2} x^{4}-12 \left (\int \frac {\left (d x +c \right )^{n}}{d \,x^{2}+c x}d x \right ) b \,c^{2} d^{2} n \,x^{4}}{24 c^{3} x^{4}} \] Input:

int((d*x+c)^n*(b*x^2+a)/x^5,x)
 

Output:

( - 6*(c + d*x)**n*a*c**3 - 2*(c + d*x)**n*a*c**2*d*n*x - (c + d*x)**n*a*c 
*d**2*n**2*x**2 + 3*(c + d*x)**n*a*c*d**2*n*x**2 - (c + d*x)**n*a*d**3*n** 
3*x**3 + 5*(c + d*x)**n*a*d**3*n**2*x**3 - 6*(c + d*x)**n*a*d**3*n*x**3 - 
12*(c + d*x)**n*b*c**3*x**2 - 12*(c + d*x)**n*b*c**2*d*n*x**3 + int((c + d 
*x)**n/(c*x + d*x**2),x)*a*d**4*n**4*x**4 - 6*int((c + d*x)**n/(c*x + d*x* 
*2),x)*a*d**4*n**3*x**4 + 11*int((c + d*x)**n/(c*x + d*x**2),x)*a*d**4*n** 
2*x**4 - 6*int((c + d*x)**n/(c*x + d*x**2),x)*a*d**4*n*x**4 + 12*int((c + 
d*x)**n/(c*x + d*x**2),x)*b*c**2*d**2*n**2*x**4 - 12*int((c + d*x)**n/(c*x 
 + d*x**2),x)*b*c**2*d**2*n*x**4)/(24*c**3*x**4)