3.10.33 \(\int \frac {(a+b x)^n (c+d x)^3}{x} \, dx\) [933]

3.10.33.1 Optimal result

Integrand size = 18, antiderivative size = 131 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c^3 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \]

output

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

3.10.33.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=(a+b x)^{1+n} \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)}{b^3 (2+n)}+\frac {d^3 (a+b x)^2}{b^3 (3+n)}-\frac {c^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b x}{a}\right )}{a+a n}\right ) \]

input

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

output

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

3.10.33.3 Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 131, 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 = {99, 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.

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.10.33.3.1 Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

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

3.10.33.4 Maple [F]

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

input

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

output

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

3.10.33.5 Fricas [F]

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

input

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

output

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

3.10.33.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (116) = 232\).

Time = 3.38 (sec) , antiderivative size = 923, normalized size of antiderivative = 7.05 \[ \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx=3 c^{2} d \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + 3 c d^{2} \left (\begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases}\right ) + d^{3} \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^{3} 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^{3} \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+c)**3/x,x)
 

output

3*c**2*d*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 
 1), Ne(n, -1)), (log(a + b*x), True))/b, True)) + 3*c*d**2*Piecewise((a** 
n*x**2/2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x) + a/(a*b**2 + b**3* 
x) + b*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(n, -2)), (-a*log(a/b + x)/b**2 
 + x/b, Eq(n, -1)), (-a**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + 
a*b*n*x*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*n*x**2*(a + b* 
x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*x**2*(a + b*x)**n/(b**2*n**2 
+ 3*b**2*n + 2*b**2), True)) + d**3*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...
 

3.10.33.7 Maxima [F]

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

input

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

output

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

3.10.33.8 Giac [F]

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

input

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

output

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

3.10.33.9 Mupad [F(-1)]

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

input

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

output

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