\(\int \frac {(c+d x)^n (a+b x^2)^3}{x^4} \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 193 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^4} \, dx=\frac {b^2 \left (b c^2+3 a d^2\right ) (c+d x)^{1+n}}{d^3 (1+n)}-\frac {a^3 (c+d x)^{1+n}}{3 c x^3}-\frac {3 a^2 b (c+d x)^{1+n}}{d (1-n) x^2}-\frac {2 b^3 c (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b^3 (c+d x)^{3+n}}{d^3 (3+n)}+\frac {a^2 d \left (18 b c^2+a d^2 \left (2-3 n+n^2\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )}{3 c^4 \left (1-n^2\right )} \] Output:

b^2*(3*a*d^2+b*c^2)*(d*x+c)^(1+n)/d^3/(1+n)-1/3*a^3*(d*x+c)^(1+n)/c/x^3-3* 
a^2*b*(d*x+c)^(1+n)/d/(1-n)/x^2-2*b^3*c*(d*x+c)^(2+n)/d^3/(2+n)+b^3*(d*x+c 
)^(3+n)/d^3/(3+n)+1/3*a^2*d*(18*b*c^2+a*d^2*(n^2-3*n+2))*(d*x+c)^(1+n)*hyp 
ergeom([3, 1+n],[2+n],1+d*x/c)/c^4/(-n^2+1)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^4} \, dx=\frac {(c+d x)^{1+n} \left (b^2 c^4 \left (3 a d^2 \left (6+5 n+n^2\right )+b \left (2 c^2-2 c d (1+n) x+d^2 \left (2+3 n+n^2\right ) x^2\right )\right )+3 a^2 b c^2 d^4 \left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )+a^3 d^6 \left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {d x}{c}\right )\right )}{c^4 d^3 (1+n) (2+n) (3+n)} \] Input:

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

Output:

((c + d*x)^(1 + n)*(b^2*c^4*(3*a*d^2*(6 + 5*n + n^2) + b*(2*c^2 - 2*c*d*(1 
 + n)*x + d^2*(2 + 3*n + n^2)*x^2)) + 3*a^2*b*c^2*d^4*(6 + 5*n + n^2)*Hype 
rgeometric2F1[2, 1 + n, 2 + n, 1 + (d*x)/c] + a^3*d^6*(6 + 5*n + n^2)*Hype 
rgeometric2F1[4, 1 + n, 2 + n, 1 + (d*x)/c]))/(c^4*d^3*(1 + n)*(2 + n)*(3 
+ n))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {520, 2124, 2124, 25, 2123, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-3 b^3 c x^5-9 a b^2 c x^3-9 a^2 b c x+a^3 d (2-n)\right )}{x^3}dx}{3 c}-\frac {a^3 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (6 b^3 c^2 x^4+18 a b^2 c^2 x^2+a^2 \left (18 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right )}{x^2}dx}{2 c}-\frac {a^3 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^3 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {-\frac {\int -\frac {(c+d x)^n \left (6 b^3 x^3 c^3+18 a b^2 x c^3+a^2 d n \left (18 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right )}{x}dx}{c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+18 b c^2\right )}{c x}}{2 c}-\frac {a^3 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^3 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\frac {\int \frac {(c+d x)^n \left (6 b^3 x^3 c^3+18 a b^2 x c^3+a^2 d n \left (18 b c^2+a d^2 \left (n^2-3 n+2\right )\right )\right )}{x}dx}{c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+18 b c^2\right )}{c x}}{2 c}-\frac {a^3 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^3 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 2123

\(\displaystyle -\frac {-\frac {\frac {\int \left (\frac {6 b^2 c^3 \left (b c^2+3 a d^2\right ) (c+d x)^n}{d^2}+\frac {\left (a^3 n^3 d^3-3 a^3 n^2 d^3+2 a^3 n d^3+18 a^2 b c^2 n d\right ) (c+d x)^n}{x}-\frac {12 b^3 c^4 (c+d x)^{n+1}}{d^2}+\frac {6 b^3 c^3 (c+d x)^{n+2}}{d^2}\right )dx}{c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+18 b c^2\right )}{c x}}{2 c}-\frac {a^3 d (2-n) (c+d x)^{n+1}}{2 c x^2}}{3 c}-\frac {a^3 (c+d x)^{n+1}}{3 c x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 (c+d x)^{n+1}}{3 c x^3}-\frac {-\frac {a^3 d (2-n) (c+d x)^{n+1}}{2 c x^2}-\frac {\frac {-\frac {a^2 d n (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+18 b c^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}+\frac {6 b^2 c^3 \left (3 a d^2+b c^2\right ) (c+d x)^{n+1}}{d^3 (n+1)}-\frac {12 b^3 c^4 (c+d x)^{n+2}}{d^3 (n+2)}+\frac {6 b^3 c^3 (c+d x)^{n+3}}{d^3 (n+3)}}{c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-3 n+2\right )+18 b c^2\right )}{c x}}{2 c}}{3 c}\)

Input:

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

Output:

-1/3*(a^3*(c + d*x)^(1 + n))/(c*x^3) - (-1/2*(a^3*d*(2 - n)*(c + d*x)^(1 + 
 n))/(c*x^2) - (-((a^2*(18*b*c^2 + a*d^2*(2 - 3*n + n^2))*(c + d*x)^(1 + n 
))/(c*x)) + ((6*b^2*c^3*(b*c^2 + 3*a*d^2)*(c + d*x)^(1 + n))/(d^3*(1 + n)) 
 - (12*b^3*c^4*(c + d*x)^(2 + n))/(d^3*(2 + n)) + (6*b^3*c^3*(c + d*x)^(3 
+ n))/(d^3*(3 + n)) - (a^2*d*n*(18*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 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 2009

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

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

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]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{3}}{x^{4}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((d*x+c)^n*(b*x^2+a)^3/x^4,x, algorithm="fricas")
 

Output:

integral((b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*(d*x + c)^n/x^4, x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (168) = 336\).

Time = 12.30 (sec) , antiderivative size = 3628, normalized size of antiderivative = 18.80 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*a**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*c**2*d...
 

Maxima [F]

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

integrate((d*x+c)^n*(b*x^2+a)^3/x^4,x, algorithm="maxima")
 

Output:

3*(d*x + c)^(n + 1)*a*b^2/(d*(n + 1)) + integrate((b^3*x^6 + 3*a^2*b*x^2 + 
 a^3)*(d*x + c)^n/x^4, x)
 

Giac [F]

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

integrate((d*x+c)^n*(b*x^2+a)^3/x^4,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*(c + d*x)**n*a**3*c**2*d**3*n**3 - 12*(c + d*x)**n*a**3*c**2*d**3*n* 
*2 - 22*(c + d*x)**n*a**3*c**2*d**3*n - 12*(c + d*x)**n*a**3*c**2*d**3 - ( 
c + d*x)**n*a**3*c*d**4*n**4*x - 6*(c + d*x)**n*a**3*c*d**4*n**3*x - 11*(c 
 + d*x)**n*a**3*c*d**4*n**2*x - 6*(c + d*x)**n*a**3*c*d**4*n*x - (c + d*x) 
**n*a**3*d**5*n**5*x**2 - 4*(c + d*x)**n*a**3*d**5*n**4*x**2 + (c + d*x)** 
n*a**3*d**5*n**3*x**2 + 16*(c + d*x)**n*a**3*d**5*n**2*x**2 + 12*(c + d*x) 
**n*a**3*d**5*n*x**2 - 18*(c + d*x)**n*a**2*b*c**2*d**3*n**3*x**2 - 108*(c 
 + d*x)**n*a**2*b*c**2*d**3*n**2*x**2 - 198*(c + d*x)**n*a**2*b*c**2*d**3* 
n*x**2 - 108*(c + d*x)**n*a**2*b*c**2*d**3*x**2 + 18*(c + d*x)**n*a*b**2*c 
**3*d**2*n**2*x**3 + 90*(c + d*x)**n*a*b**2*c**3*d**2*n*x**3 + 108*(c + d* 
x)**n*a*b**2*c**3*d**2*x**3 + 18*(c + d*x)**n*a*b**2*c**2*d**3*n**2*x**4 + 
 90*(c + d*x)**n*a*b**2*c**2*d**3*n*x**4 + 108*(c + d*x)**n*a*b**2*c**2*d* 
*3*x**4 + 12*(c + d*x)**n*b**3*c**5*x**3 - 12*(c + d*x)**n*b**3*c**4*d*n*x 
**4 + 6*(c + d*x)**n*b**3*c**3*d**2*n**2*x**5 + 6*(c + d*x)**n*b**3*c**3*d 
**2*n*x**5 + 6*(c + d*x)**n*b**3*c**2*d**3*n**2*x**6 + 18*(c + d*x)**n*b** 
3*c**2*d**3*n*x**6 + 12*(c + d*x)**n*b**3*c**2*d**3*x**6 + int((c + d*x)** 
n/(c*x + d*x**2),x)*a**3*d**6*n**6*x**3 + 3*int((c + d*x)**n/(c*x + d*x**2 
),x)*a**3*d**6*n**5*x**3 - 5*int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6* 
n**4*x**3 - 15*int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**3*x**3 + 4* 
int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**2*x**3 + 12*int((c + d*...