\(\int \frac {(d+e x)^m (a+b x+c x^2)}{x} \, dx\) [121]

Optimal result

Integrand size = 21, antiderivative size = 83 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x} \, dx=-\frac {(c d-b e) (d+e x)^{1+m}}{e^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 (2+m)}-\frac {a (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )}{d (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x} \, dx=-\frac {(d+e x)^{1+m} \left (-b d e (2+m)+c d (d-e (1+m) x)+a e^2 (2+m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )\right )}{d e^2 (1+m) (2+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1195, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 3.06 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.71 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x} \, dx=- \frac {a e^{m + 1} m \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d \Gamma \left (m + 2\right )} - \frac {a e^{m + 1} \left (\frac {d}{e} + x\right )^{m + 1} \Phi \left (1 + \frac {e x}{d}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{d \Gamma \left (m + 2\right )} + b \left (\begin {cases} d^{m} x & \text {for}\: e = 0 \\\frac {\begin {cases} \frac {\left (d + e x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (d + e x \right )} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {d^{m} x^{2}}{2} & \text {for}\: e = 0 \\\frac {d \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {d}{d e^{2} + e^{3} x} + \frac {e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -2 \\- \frac {d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {x}{e} & \text {for}\: m = -1 \\- \frac {d^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {d e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-a*e**(m + 1)*m*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 
 1)/(d*gamma(m + 2)) - a*e**(m + 1)*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 
 1, m + 1)*gamma(m + 1)/(d*gamma(m + 2)) + b*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)) + c*Piecewise((d**m*x**2/2, Eq(e, 0)), (d*log(d/e + x)/(d*e**2 
+ e**3*x) + d/(d*e**2 + e**3*x) + e*x*log(d/e + x)/(d*e**2 + e**3*x), Eq(m 
, -2)), (-d*log(d/e + x)/e**2 + x/e, Eq(m, -1)), (-d**2*(d + e*x)**m/(e**2 
*m**2 + 3*e**2*m + 2*e**2) + d*e*m*x*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 
2*e**2) + e**2*m*x**2*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2) + e**2* 
x**2*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2), True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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