\(\int \frac {(d+e x)^m (a+b x+c x^2)}{x^2} \, dx\) [122]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\frac {c (d+e x)^{1+m}}{e (1+m)}-\frac {a (d+e x)^{1+m}}{d x}-\frac {(b d+a e m) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )}{d^2 (1+m)} \] Output:

c*(e*x+d)^(1+m)/e/(1+m)-a*(e*x+d)^(1+m)/d/x-(a*e*m+b*d)*(e*x+d)^(1+m)*hype 
rgeom([1, 1+m],[2+m],1+e*x/d)/d^2/(1+m)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\frac {(d+e x)^{1+m} \left (c d^2-b d e \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )+a e^2 \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,1+\frac {e x}{d}\right )\right )}{d^2 e (1+m)} \] Input:

Integrate[((d + e*x)^m*(a + b*x + c*x^2))/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1193, 25, 90, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right ) (d+e x)^m}{x^2} \, dx\)

\(\Big \downarrow \) 1193

\(\displaystyle -\frac {\int -\frac {(b d+c x d+a e m) (d+e x)^m}{x}dx}{d}-\frac {a (d+e x)^{m+1}}{d x}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(b d+c x d+a e m) (d+e x)^m}{x}dx}{d}-\frac {a (d+e x)^{m+1}}{d x}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(a e m+b d) \int \frac {(d+e x)^m}{x}dx+\frac {c d (d+e x)^{m+1}}{e (m+1)}}{d}-\frac {a (d+e x)^{m+1}}{d x}\)

\(\Big \downarrow \) 75

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

Input:

Int[((d + e*x)^m*(a + b*x + c*x^2))/x^2,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 90

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

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

Maple [F]

\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{x^{2}}d x\]

Input:

int((e*x+d)^m*(c*x^2+b*x+a)/x^2,x)
 

Output:

int((e*x+d)^m*(c*x^2+b*x+a)/x^2,x)
 

Fricas [F]

\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{2}} \,d x } \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)/x^2,x, algorithm="fricas")
 

Output:

integral((c*x^2 + b*x + a)*(e*x + d)^m/x^2, x)
 

Sympy [A] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.09 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=- \frac {a e^{m + 2} m \left (\frac {d}{e} + x\right )^{m + 1} \Gamma \left (m + 1\right )}{d e x \Gamma \left (m + 2\right )} - \frac {a e^{m + 2} \left (\frac {d}{e} + x\right )^{m + 1} \Gamma \left (m + 1\right )}{d e x \Gamma \left (m + 2\right )} - \frac {a e^{m + 2} m^{2} \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d^{2} \Gamma \left (m + 2\right )} - \frac {a e^{m + 2} m \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d^{2} \Gamma \left (m + 2\right )} - \frac {b e^{m + 1} m \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d \Gamma \left (m + 2\right )} - \frac {b e^{m + 1} \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d \Gamma \left (m + 2\right )} + c \left (\begin {cases} d^{m} x & \text {for}\: e = 0 \\\frac {\begin {cases} \frac {\left (d + e x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (d + e x \right )} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((e*x+d)**m*(c*x**2+b*x+a)/x**2,x)
 

Output:

-a*e**(m + 2)*m*(d/e + x)**(m + 1)*gamma(m + 1)/(d*e*x*gamma(m + 2)) - a*e 
**(m + 2)*(d/e + x)**(m + 1)*gamma(m + 1)/(d*e*x*gamma(m + 2)) - a*e**(m + 
 2)*m**2*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(d* 
*2*gamma(m + 2)) - a*e**(m + 2)*m*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1 
, m + 1)*gamma(m + 1)/(d**2*gamma(m + 2)) - b*e**(m + 1)*m*(d/e + x)**(m + 
 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(d*gamma(m + 2)) - b*e**(m 
+ 1)*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(d*gamm 
a(m + 2)) + c*Piecewise((d**m*x, Eq(e, 0)), (Piecewise(((d + e*x)**(m + 1) 
/(m + 1), Ne(m, -1)), (log(d + e*x), True))/e, True))
 

Maxima [F]

\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{2}} \,d x } \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)/x^2,x, algorithm="maxima")
 

Output:

(e*x + d)^(m + 1)*c/(e*(m + 1)) + integrate((b*x + a)*(e*x + d)^m/x^2, x)
 

Giac [F]

\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{2}} \,d x } \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)/x^2,x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)*(e*x + d)^m/x^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{x^2} \,d x \] Input:

int(((d + e*x)^m*(a + b*x + c*x^2))/x^2,x)
 

Output:

int(((d + e*x)^m*(a + b*x + c*x^2))/x^2, x)
 

Reduce [F]

\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^2} \, dx=\frac {-\left (e x +d \right )^{m} a e \,m^{2}-\left (e x +d \right )^{m} a e m +\left (e x +d \right )^{m} b e m x +\left (e x +d \right )^{m} b e x +\left (e x +d \right )^{m} c d m x +\left (e x +d \right )^{m} c e m \,x^{2}+\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) a \,e^{2} m^{3} x +\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) a \,e^{2} m^{2} x +\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) b d e \,m^{2} x +\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) b d e m x}{e m x \left (m +1\right )} \] Input:

int((e*x+d)^m*(c*x^2+b*x+a)/x^2,x)
 

Output:

( - (d + e*x)**m*a*e*m**2 - (d + e*x)**m*a*e*m + (d + e*x)**m*b*e*m*x + (d 
 + e*x)**m*b*e*x + (d + e*x)**m*c*d*m*x + (d + e*x)**m*c*e*m*x**2 + int((d 
 + e*x)**m/(d*x + e*x**2),x)*a*e**2*m**3*x + int((d + e*x)**m/(d*x + e*x** 
2),x)*a*e**2*m**2*x + int((d + e*x)**m/(d*x + e*x**2),x)*b*d*e*m**2*x + in 
t((d + e*x)**m/(d*x + e*x**2),x)*b*d*e*m*x)/(e*m*x*(m + 1))