3.10.35 \(\int \frac {(a+b x)^n (c+d x)^3}{x^3} \, dx\) [935]

3.10.35.1 Optimal result

Integrand size = 18, antiderivative size = 176 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^3} \, dx=\frac {d (a+b x)^{1+n} (c+d x)^2}{b (1+n) x^2}-\frac {c (a+b x)^{1+n} \left (a c (2 a d+b c (1+n))+\left (4 a^2 d^2+6 a b c d (1+n)-b^2 c^2 \left (1-n^2\right )\right ) x\right )}{2 a^2 b (1+n) x^2}-\frac {c \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right ) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{2 a^3 (1+n)} \]

output

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

3.10.35.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^3} \, dx=\frac {(a+b x)^{1+n} \left (a \left (-b^2 c^3 \left (-1+n^2\right ) x+2 a^2 d^3 x^2-a b c^2 (1+n) (c+6 d x)\right )-b c \left (6 a^2 d^2+6 a b c d n+b^2 c^2 (-1+n) n\right ) x^2 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{2 a^3 b (1+n) x^2} \]

input

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

output

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

3.10.35.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {111, 27, 162, 75}

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.

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 75

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

input

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

output

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

3.10.35.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

3.10.35.4 Maple [F]

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

input

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

output

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

3.10.35.5 Fricas [F]

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

input

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

3.10.35.6 Sympy [A] (verification not implemented)

Time = 4.55 (sec) , antiderivative size = 1085, normalized size of antiderivative = 6.16 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^3} \, dx=\text {Too large to display} \]

input

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

output

a**2*b**(n + 3)*c**3*n**3*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1) 
*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b** 
2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3)*c**3*n*(a/b + x)**(n + 1)*l 
erchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b 
*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3) 
*c**3*n*(a/b + x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x 
*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3)*c 
**3*(a/b + x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gam 
ma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a*b*b**(n + 3)*c**3 
*n**3*x*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2* 
a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gam 
ma(n + 2)) - a*b*b**(n + 3)*c**3*n**2*x*(a/b + x)**(n + 1)*gamma(n + 1)/(- 
2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*g 
amma(n + 2)) - 2*a*b*b**(n + 3)*c**3*n*x*(a/b + x)**(n + 1)*lerchphi(1 + b 
*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 
2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a*b*b**(n + 3)*c**3*x*(a/b + 
 x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) 
+ 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - b**2*b**(n + 3)*c**3*n**3*(a/b 
+ x)**2*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2* 
a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*...
 

3.10.35.7 Maxima [F]

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

input

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

output

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

3.10.35.8 Giac [F]

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

input

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

output

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

3.10.35.9 Mupad [F(-1)]

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

input

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

output

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