\(\int \frac {(a+b x)^n (c+d x^3)}{x} \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 99 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )}{x} \, dx=\frac {a^2 d (a+b x)^{1+n}}{b^3 (1+n)}-\frac {2 a d (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )}{x} \, dx=\frac {(a+b x)^{1+n} \left (a d \left (2 a^2-2 a b (1+n) x+b^2 \left (2+3 n+n^2\right ) x^2\right )-b^3 c \left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a b^3 (1+n) (2+n) (3+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 2123

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (83) = 166\).

Time = 2.72 (sec) , antiderivative size = 675, normalized size of antiderivative = 6.82 \[ \int \frac {(a+b x)^n \left (c+d x^3\right )}{x} \, dx=d \left (\begin {cases} \frac {a^{n} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2}}{a b^{3} + b^{4} x} - \frac {2 a b x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} x^{2}}{a b^{3} + b^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} & \text {for}\: n = -1 \\\frac {2 a^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} & \text {otherwise} \end {cases}\right ) - \frac {b^{n + 1} c n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 1} c \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \] Input:

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

Output:

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)) - b**(n + 1)*c*n*(a/ 
b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2) 
) - b**(n + 1)*c*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n 
+ 1)/(a*gamma(n + 2))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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