3.2.85 \(\int \frac {(a+b x)^n (c+d x^3)^3}{x} \, dx\) [185]

3.2.85.1 Optimal result

Integrand size = 20, antiderivative size = 358 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx=\frac {a^2 d \left (3 b^6 c^2-3 a^3 b^3 c d+a^6 d^2\right ) (a+b x)^{1+n}}{b^9 (1+n)}-\frac {a d \left (6 b^6 c^2-15 a^3 b^3 c d+8 a^6 d^2\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac {d \left (3 b^6 c^2-30 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{3+n}}{b^9 (3+n)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{4+n}}{b^9 (4+n)}-\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{5+n}}{b^9 (5+n)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac {28 a^2 d^3 (a+b x)^{7+n}}{b^9 (7+n)}-\frac {8 a d^3 (a+b x)^{8+n}}{b^9 (8+n)}+\frac {d^3 (a+b x)^{9+n}}{b^9 (9+n)}-\frac {c^3 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \]

output

a^2*d*(a^6*d^2-3*a^3*b^3*c*d+3*b^6*c^2)*(b*x+a)^(1+n)/b^9/(1+n)-a*d*(8*a^6 
*d^2-15*a^3*b^3*c*d+6*b^6*c^2)*(b*x+a)^(2+n)/b^9/(2+n)+d*(28*a^6*d^2-30*a^ 
3*b^3*c*d+3*b^6*c^2)*(b*x+a)^(3+n)/b^9/(3+n)+2*a^2*d^2*(-28*a^3*d+15*b^3*c 
)*(b*x+a)^(4+n)/b^9/(4+n)-5*a*d^2*(-14*a^3*d+3*b^3*c)*(b*x+a)^(5+n)/b^9/(5 
+n)+d^2*(-56*a^3*d+3*b^3*c)*(b*x+a)^(6+n)/b^9/(6+n)+28*a^2*d^3*(b*x+a)^(7+ 
n)/b^9/(7+n)-8*a*d^3*(b*x+a)^(8+n)/b^9/(8+n)+d^3*(b*x+a)^(9+n)/b^9/(9+n)-c 
^3*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/a)/a/(1+n)
 

3.2.85.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx=(a+b x)^{1+n} \left (\frac {a^2 d \left (3 b^6 c^2-3 a^3 b^3 c d+a^6 d^2\right )}{b^9 (1+n)}-\frac {a d \left (6 b^6 c^2-15 a^3 b^3 c d+8 a^6 d^2\right ) (a+b x)}{b^9 (2+n)}+\frac {d \left (3 b^6 c^2-30 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^2}{b^9 (3+n)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^3}{b^9 (4+n)}+\frac {5 a d^2 \left (-3 b^3 c+14 a^3 d\right ) (a+b x)^4}{b^9 (5+n)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^5}{b^9 (6+n)}+\frac {28 a^2 d^3 (a+b x)^6}{b^9 (7+n)}-\frac {8 a d^3 (a+b x)^7}{b^9 (8+n)}+\frac {d^3 (a+b x)^8}{b^9 (9+n)}-\frac {c^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b x}{a}\right )}{a+a n}\right ) \]

input

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

output

(a + b*x)^(1 + n)*((a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2))/(b^9*(1 + 
 n)) - (a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x))/(b^9*(2 + 
n)) + (d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^2)/(b^9*(3 + 
n)) + (2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^3)/(b^9*(4 + n)) + (5*a*d 
^2*(-3*b^3*c + 14*a^3*d)*(a + b*x)^4)/(b^9*(5 + n)) + (d^2*(3*b^3*c - 56*a 
^3*d)*(a + b*x)^5)/(b^9*(6 + n)) + (28*a^2*d^3*(a + b*x)^6)/(b^9*(7 + n)) 
- (8*a*d^3*(a + b*x)^7)/(b^9*(8 + n)) + (d^3*(a + b*x)^8)/(b^9*(9 + n)) - 
(c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))
 

3.2.85.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 358, 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.

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

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {5 a d^2 \left (14 a^3 d-3 b^3 c\right ) (a+b x)^{n+4}}{b^8}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+5}}{b^8}+\frac {28 a^2 d^3 (a+b x)^{n+6}}{b^8}-\frac {a d \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right ) (a+b x)^{n+1}}{b^8}+\frac {d \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+2}}{b^8}-\frac {2 a^2 d^2 \left (28 a^3 d-15 b^3 c\right ) (a+b x)^{n+3}}{b^8}+\frac {a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^n}{b^8}-\frac {8 a d^3 (a+b x)^{n+7}}{b^8}+\frac {d^3 (a+b x)^{n+8}}{b^8}+\frac {c^3 (a+b x)^n}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac {28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac {a d \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac {d \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac {a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+1}}{b^9 (n+1)}-\frac {8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac {d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac {c^3 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a (n+1)}\)

input

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

output

(a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2)*(a + b*x)^(1 + n))/(b^9*(1 + 
n)) - (a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x)^(2 + n))/(b^ 
9*(2 + n)) + (d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n 
))/(b^9*(3 + n)) + (2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^(4 + n))/(b^ 
9*(4 + n)) - (5*a*d^2*(3*b^3*c - 14*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n) 
) + (d^2*(3*b^3*c - 56*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d 
^3*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 
+ n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 + n)) - (c^3*(a + b*x)^(1 + n)*Hyp 
ergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(1 + n))
 

3.2.85.3.1 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])
 

3.2.85.4 Maple [F]

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

input

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

output

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

3.2.85.5 Fricas [F]

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

input

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

output

integral((d^3*x^9 + 3*c*d^2*x^6 + 3*c^2*d*x^3 + c^3)*(b*x + a)^n/x, x)
 

3.2.85.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12492 vs. \(2 (338) = 676\).

Time = 13.33 (sec) , antiderivative size = 17189, normalized size of antiderivative = 48.01 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx=\text {Too large to display} \]

input

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

output

3*c**2*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)) + 3*c*d**2*Pi 
ecewise((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*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 ...
 

3.2.85.7 Maxima [F]

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

input

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

output

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

3.2.85.8 Giac [F]

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

input

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

output

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

3.2.85.9 Mupad [F(-1)]

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

input

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

output

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