\(\int \frac {(a+b x)^n (c+d x^3)^2}{x} \, dx\) [34]

Optimal result

Integrand size = 20, antiderivative size = 209 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^2}{x} \, dx=\frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)}-\frac {c^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \] Output:

a^2*d*(-a^3*d+2*b^3*c)*(b*x+a)^(1+n)/b^6/(1+n)-a*d*(-5*a^3*d+4*b^3*c)*(b*x 
+a)^(2+n)/b^6/(2+n)+2*d*(-5*a^3*d+b^3*c)*(b*x+a)^(3+n)/b^6/(3+n)+10*a^2*d^ 
2*(b*x+a)^(4+n)/b^6/(4+n)-5*a*d^2*(b*x+a)^(5+n)/b^6/(5+n)+d^2*(b*x+a)^(6+n 
)/b^6/(6+n)-c^2*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/a)/a/(1+n)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^2}{x} \, dx=(a+b x)^{1+n} \left (\frac {a^2 d \left (2 b^3 c-a^3 d\right )}{b^6 (1+n)}+\frac {a d \left (-4 b^3 c+5 a^3 d\right ) (a+b x)}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^2}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^3}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^4}{b^6 (5+n)}+\frac {d^2 (a+b x)^5}{b^6 (6+n)}-\frac {c^2 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b x}{a}\right )}{a+a n}\right ) \] Input:

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

Output:

(a + b*x)^(1 + n)*((a^2*d*(2*b^3*c - a^3*d))/(b^6*(1 + n)) + (a*d*(-4*b^3* 
c + 5*a^3*d)*(a + b*x))/(b^6*(2 + n)) + (2*d*(b^3*c - 5*a^3*d)*(a + b*x)^2 
)/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^3)/(b^6*(4 + n)) - (5*a*d^2*(a + b 
*x)^4)/(b^6*(5 + n)) + (d^2*(a + b*x)^5)/(b^6*(6 + n)) - (c^2*Hypergeometr 
ic2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.

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

\(\Big \downarrow \) 2123

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{n+2}}{b^6 (n+2)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {10 a^2 d^2 (a+b x)^{n+4}}{b^6 (n+4)}+\frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^6 (n+1)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)}-\frac {c^2 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a (n+1)}\)

Input:

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

Output:

(a^2*d*(2*b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*d*(4*b^3*c 
- 5*a^3*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) + (2*d*(b^3*c - 5*a^3*d)*(a + 
b*x)^(3 + n))/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^(4 + n))/(b^6*(4 + n)) 
 - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^ 
6*(6 + n)) - (c^2*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + 
 (b*x)/a])/(a*(1 + n))
 

Defintions of rubi rules used

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4007 vs. \(2 (187) = 374\).

Time = 4.54 (sec) , antiderivative size = 4690, normalized size of antiderivative = 22.44 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^2}{x} \, dx=\text {Too large to display} \] Input:

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

Output:

2*c*d*Piecewise((a**n*x**3/3, Eq(b, 0)), (2*a**2*log(a/b + x)/(2*a**2*b**3 
 + 4*a*b**4*x + 2*b**5*x**2) + 3*a**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x 
**2) + 4*a*b*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a 
*b*x/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*x**2*log(a/b + x)/( 
2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2), Eq(n, -3)), (-2*a**2*log(a/b + x) 
/(a*b**3 + b**4*x) - 2*a**2/(a*b**3 + b**4*x) - 2*a*b*x*log(a/b + x)/(a*b* 
*3 + b**4*x) + b**2*x**2/(a*b**3 + b**4*x), Eq(n, -2)), (a**2*log(a/b + x) 
/b**3 - a*x/b**2 + x**2/(2*b), Eq(n, -1)), (2*a**3*(a + b*x)**n/(b**3*n**3 
 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*n*x*(a + b*x)**n/(b**3*n** 
3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n**2*x**2*(a + b*x)**n/(b** 
3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n*x**2*(a + b*x)**n/(b 
**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + b**3*n**2*x**3*(a + b*x)**n 
/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*n*x**3*(a + b*x)* 
*n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*x**3*(a + b*x)* 
*n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3), True)) + d**2*Piecewise 
((a**n*x**6/6, Eq(b, 0)), (60*a**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b 
**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b* 
*11*x**5) + 137*a**5/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 
+ 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a**4*b*x*lo 
g(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 120*(a + b*x)**n*a**6*d**2*n + 120*(a + b*x)**n*a**5*b*d**2*n**2*x - 6 
0*(a + b*x)**n*a**4*b**2*d**2*n**3*x**2 - 60*(a + b*x)**n*a**4*b**2*d**2*n 
**2*x**2 + 4*(a + b*x)**n*a**3*b**3*c*d*n**4 + 60*(a + b*x)**n*a**3*b**3*c 
*d*n**3 + 296*(a + b*x)**n*a**3*b**3*c*d*n**2 + 480*(a + b*x)**n*a**3*b**3 
*c*d*n + 20*(a + b*x)**n*a**3*b**3*d**2*n**4*x**3 + 60*(a + b*x)**n*a**3*b 
**3*d**2*n**3*x**3 + 40*(a + b*x)**n*a**3*b**3*d**2*n**2*x**3 - 4*(a + b*x 
)**n*a**2*b**4*c*d*n**5*x - 60*(a + b*x)**n*a**2*b**4*c*d*n**4*x - 296*(a 
+ b*x)**n*a**2*b**4*c*d*n**3*x - 480*(a + b*x)**n*a**2*b**4*c*d*n**2*x - 5 
*(a + b*x)**n*a**2*b**4*d**2*n**5*x**4 - 30*(a + b*x)**n*a**2*b**4*d**2*n* 
*4*x**4 - 55*(a + b*x)**n*a**2*b**4*d**2*n**3*x**4 - 30*(a + b*x)**n*a**2* 
b**4*d**2*n**2*x**4 + 2*(a + b*x)**n*a*b**5*c*d*n**6*x**2 + 32*(a + b*x)** 
n*a*b**5*c*d*n**5*x**2 + 178*(a + b*x)**n*a*b**5*c*d*n**4*x**2 + 388*(a + 
b*x)**n*a*b**5*c*d*n**3*x**2 + 240*(a + b*x)**n*a*b**5*c*d*n**2*x**2 + (a 
+ b*x)**n*a*b**5*d**2*n**6*x**5 + 10*(a + b*x)**n*a*b**5*d**2*n**5*x**5 + 
35*(a + b*x)**n*a*b**5*d**2*n**4*x**5 + 50*(a + b*x)**n*a*b**5*d**2*n**3*x 
**5 + 24*(a + b*x)**n*a*b**5*d**2*n**2*x**5 + (a + b*x)**n*b**6*c**2*n**6 
+ 21*(a + b*x)**n*b**6*c**2*n**5 + 175*(a + b*x)**n*b**6*c**2*n**4 + 735*( 
a + b*x)**n*b**6*c**2*n**3 + 1624*(a + b*x)**n*b**6*c**2*n**2 + 1764*(a + 
b*x)**n*b**6*c**2*n + 720*(a + b*x)**n*b**6*c**2 + 2*(a + b*x)**n*b**6*c*d 
*n**6*x**3 + 36*(a + b*x)**n*b**6*c*d*n**5*x**3 + 242*(a + b*x)**n*b**6...