Integrand size = 17, antiderivative size = 187 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\frac {c^3 \left (a+\frac {b}{x}\right )^{1+m} x^2}{b (1-m)}+\frac {d^2 (12 a c-b d (3-m)) \left (a+\frac {b}{x}\right )^{1+m} x^3}{12 a^2}+\frac {d^3 \left (a+\frac {b}{x}\right )^{1+m} x^4}{4 a}-\frac {b \left (24 a^3 c^3-b d \left (36 a^2 c^2-b d (12 a c-b d (3-m)) (2-m)\right ) (1-m)\right ) \left (a+\frac {b}{x}\right )^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,1+\frac {b}{a x}\right )}{12 a^5 \left (1-m^2\right )} \] Output:
c^3*(a+b/x)^(1+m)*x^2/b/(1-m)+1/12*d^2*(12*a*c-b*d*(3-m))*(a+b/x)^(1+m)*x^ 3/a^2+1/4*d^3*(a+b/x)^(1+m)*x^4/a-1/12*b*(24*a^3*c^3-b*d*(36*a^2*c^2-b*d*( 12*a*c-b*d*(3-m))*(2-m))*(1-m))*(a+b/x)^(1+m)*hypergeom([3, 1+m],[2+m],1+b /a/x)/a^5/(-m^2+1)
Time = 0.19 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\frac {\left (a+\frac {b}{x}\right )^m (b+a x) \left (-a^3 (1+m) x^2 \left (12 a^2 c^3-b^2 d^3 \left (3-4 m+m^2\right ) x-3 a b d^2 (-1+m) x (4 c+d x)\right )+b^2 \left (24 a^3 c^3+36 a^2 b c^2 d (-1+m)+12 a b^2 c d^2 \left (2-3 m+m^2\right )+b^3 d^3 \left (-6+11 m-6 m^2+m^3\right )\right ) \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,1+\frac {b}{a x}\right )\right )}{12 a^5 b (-1+m) (1+m) x} \] Input:
Integrate[(a + b/x)^m*(c + d*x)^3,x]
Output:
((a + b/x)^m*(b + a*x)*(-(a^3*(1 + m)*x^2*(12*a^2*c^3 - b^2*d^3*(3 - 4*m + m^2)*x - 3*a*b*d^2*(-1 + m)*x*(4*c + d*x))) + b^2*(24*a^3*c^3 + 36*a^2*b* c^2*d*(-1 + m) + 12*a*b^2*c*d^2*(2 - 3*m + m^2) + b^3*d^3*(-6 + 11*m - 6*m ^2 + m^3))*Hypergeometric2F1[3, 1 + m, 2 + m, 1 + b/(a*x)]))/(12*a^5*b*(-1 + m)*(1 + m)*x)
Time = 0.65 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {941, 948, 111, 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.
| \(\displaystyle \int (c+d x)^3 \left (a+\frac {b}{x}\right )^m \, dx\) |
| \(\Big \downarrow \) 941 |
| \(\displaystyle \int x^3 \left (\frac {c}{x}+d\right )^3 \left (a+\frac {b}{x}\right )^mdx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\int \left (a+\frac {b}{x}\right )^m \left (\frac {c}{x}+d\right )^3 x^5d\frac {1}{x}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\int \left (a+\frac {b}{x}\right )^m \left (\frac {c}{x}+d\right ) \left (d (4 a c-b d (1-m))+\frac {c (2 a c+b d (m+1))}{x}\right ) x^5d\frac {1}{x}}{b (1-m)}+\frac {c x^4 \left (\frac {c}{x}+d\right )^2 \left (a+\frac {b}{x}\right )^{m+1}}{b (1-m)}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {\frac {\left (24 a^3 c^3-36 a^2 b c^2 d (1-m)+12 a b^2 c d^2 \left (m^2-3 m+2\right )-b^3 d^3 \left (-m^3+6 m^2-11 m+6\right )\right ) \int \left (a+\frac {b}{x}\right )^m x^3d\frac {1}{x}}{12 a^2}-\frac {d x^4 \left (a+\frac {b}{x}\right )^{m+1} \left (\frac {24 a^2 c^2-12 a b c d (1-m)+b^2 d^2 \left (m^2-4 m+3\right )}{x}+3 a d (4 a c-b d (1-m))\right )}{12 a^2}}{b (1-m)}+\frac {c x^4 \left (\frac {c}{x}+d\right )^2 \left (a+\frac {b}{x}\right )^{m+1}}{b (1-m)}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {-\frac {d x^4 \left (a+\frac {b}{x}\right )^{m+1} \left (\frac {24 a^2 c^2-12 a b c d (1-m)+b^2 d^2 \left (m^2-4 m+3\right )}{x}+3 a d (4 a c-b d (1-m))\right )}{12 a^2}-\frac {b^2 \left (a+\frac {b}{x}\right )^{m+1} \left (24 a^3 c^3-36 a^2 b c^2 d (1-m)+12 a b^2 c d^2 \left (m^2-3 m+2\right )-b^3 d^3 \left (-m^3+6 m^2-11 m+6\right )\right ) \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,\frac {b}{a x}+1\right )}{12 a^5 (m+1)}}{b (1-m)}+\frac {c x^4 \left (\frac {c}{x}+d\right )^2 \left (a+\frac {b}{x}\right )^{m+1}}{b (1-m)}\) |
Input:
Int[(a + b/x)^m*(c + d*x)^3,x]
Output:
(c*(a + b/x)^(1 + m)*(d + c/x)^2*x^4)/(b*(1 - m)) + (-1/12*(d*(a + b/x)^(1 + m)*(3*a*d*(4*a*c - b*d*(1 - m)) + (24*a^2*c^2 - 12*a*b*c*d*(1 - m) + b^ 2*d^2*(3 - 4*m + m^2))/x)*x^4)/a^2 - (b^2*(24*a^3*c^3 - 36*a^2*b*c^2*d*(1 - m) + 12*a*b^2*c*d^2*(2 - 3*m + m^2) - b^3*d^3*(6 - 11*m + 6*m^2 - m^3))* (a + b/x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, 1 + b/(a*x)])/(12*a^5 *(1 + m)))/(b*(1 - m))
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])
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]
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]))
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
\[\int \left (a +\frac {b}{x}\right )^{m} \left (d x +c \right )^{3}d x\]
Input:
int((a+b/x)^m*(d*x+c)^3,x)
Output:
int((a+b/x)^m*(d*x+c)^3,x)
\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:
integrate((a+b/x)^m*(d*x+c)^3,x, algorithm="fricas")
Output:
integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*((a*x + b)/x)^m, x)
Result contains complex when optimal does not.
Time = 155.51 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\frac {b^{m} c^{3} x^{1 - m} \Gamma \left (1 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 1 - m \\ 2 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac {3 b^{m} c^{2} d x^{2 - m} \Gamma \left (2 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 2 - m \\ 3 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} + \frac {3 b^{m} c d^{2} x^{3 - m} \Gamma \left (3 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 3 - m \\ 4 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (4 - m\right )} + \frac {b^{m} d^{3} x^{4 - m} \Gamma \left (4 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 4 - m \\ 5 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (5 - m\right )} \] Input:
integrate((a+b/x)**m*(d*x+c)**3,x)
Output:
b**m*c**3*x**(1 - m)*gamma(1 - m)*hyper((-m, 1 - m), (2 - m,), a*x*exp_pol ar(I*pi)/b)/gamma(2 - m) + 3*b**m*c**2*d*x**(2 - m)*gamma(2 - m)*hyper((-m , 2 - m), (3 - m,), a*x*exp_polar(I*pi)/b)/gamma(3 - m) + 3*b**m*c*d**2*x* *(3 - m)*gamma(3 - m)*hyper((-m, 3 - m), (4 - m,), a*x*exp_polar(I*pi)/b)/ gamma(4 - m) + b**m*d**3*x**(4 - m)*gamma(4 - m)*hyper((-m, 4 - m), (5 - m ,), a*x*exp_polar(I*pi)/b)/gamma(5 - m)
\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:
integrate((a+b/x)^m*(d*x+c)^3,x, algorithm="maxima")
Output:
integrate((d*x + c)^3*(a + b/x)^m, x)
\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:
integrate((a+b/x)^m*(d*x+c)^3,x, algorithm="giac")
Output:
integrate((d*x + c)^3*(a + b/x)^m, x)
Timed out. \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\int {\left (a+\frac {b}{x}\right )}^m\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int((a + b/x)^m*(c + d*x)^3,x)
Output:
int((a + b/x)^m*(c + d*x)^3, x)
\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^3 \, dx=\frac {24 \left (a x +b \right )^{m} a^{3} c^{3} x +36 \left (a x +b \right )^{m} a^{3} c^{2} d \,x^{2}+24 \left (a x +b \right )^{m} a^{3} c \,d^{2} x^{3}+6 \left (a x +b \right )^{m} a^{3} d^{3} x^{4}+36 \left (a x +b \right )^{m} a^{2} b \,c^{2} d m x +12 \left (a x +b \right )^{m} a^{2} b c \,d^{2} m \,x^{2}+2 \left (a x +b \right )^{m} a^{2} b \,d^{3} m \,x^{3}+12 \left (a x +b \right )^{m} a \,b^{2} c \,d^{2} m^{2} x -24 \left (a x +b \right )^{m} a \,b^{2} c \,d^{2} m x +\left (a x +b \right )^{m} a \,b^{2} d^{3} m^{2} x^{2}-3 \left (a x +b \right )^{m} a \,b^{2} d^{3} m \,x^{2}+\left (a x +b \right )^{m} b^{3} d^{3} m^{3} x -5 \left (a x +b \right )^{m} b^{3} d^{3} m^{2} x +6 \left (a x +b \right )^{m} b^{3} d^{3} m x +24 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a^{3} b \,c^{3} m +36 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a^{2} b^{2} c^{2} d \,m^{2}-36 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a^{2} b^{2} c^{2} d m +12 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a \,b^{3} c \,d^{2} m^{3}-36 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a \,b^{3} c \,d^{2} m^{2}+24 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a \,b^{3} c \,d^{2} m +x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{4} d^{3} m^{4}-6 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{4} d^{3} m^{3}+11 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{4} d^{3} m^{2}-6 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{4} d^{3} m}{24 x^{m} a^{3}} \] Input:
int((a+b/x)^m*(d*x+c)^3,x)
Output:
(24*(a*x + b)**m*a**3*c**3*x + 36*(a*x + b)**m*a**3*c**2*d*x**2 + 24*(a*x + b)**m*a**3*c*d**2*x**3 + 6*(a*x + b)**m*a**3*d**3*x**4 + 36*(a*x + b)**m *a**2*b*c**2*d*m*x + 12*(a*x + b)**m*a**2*b*c*d**2*m*x**2 + 2*(a*x + b)**m *a**2*b*d**3*m*x**3 + 12*(a*x + b)**m*a*b**2*c*d**2*m**2*x - 24*(a*x + b)* *m*a*b**2*c*d**2*m*x + (a*x + b)**m*a*b**2*d**3*m**2*x**2 - 3*(a*x + b)**m *a*b**2*d**3*m*x**2 + (a*x + b)**m*b**3*d**3*m**3*x - 5*(a*x + b)**m*b**3* d**3*m**2*x + 6*(a*x + b)**m*b**3*d**3*m*x + 24*x**m*int((a*x + b)**m/(x** m*a*x + x**m*b),x)*a**3*b*c**3*m + 36*x**m*int((a*x + b)**m/(x**m*a*x + x* *m*b),x)*a**2*b**2*c**2*d*m**2 - 36*x**m*int((a*x + b)**m/(x**m*a*x + x**m *b),x)*a**2*b**2*c**2*d*m + 12*x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x )*a*b**3*c*d**2*m**3 - 36*x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*a*b **3*c*d**2*m**2 + 24*x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*a*b**3*c *d**2*m + x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*b**4*d**3*m**4 - 6* x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*b**4*d**3*m**3 + 11*x**m*int( (a*x + b)**m/(x**m*a*x + x**m*b),x)*b**4*d**3*m**2 - 6*x**m*int((a*x + b)* *m/(x**m*a*x + x**m*b),x)*b**4*d**3*m)/(24*x**m*a**3)