\(\int \frac {x^m (c+d x)^2}{a+b x} \, dx\) [530]

Optimal result

Integrand size = 18, antiderivative size = 99 \[ \int \frac {x^m (c+d x)^2}{a+b x} \, dx=\frac {c d x^{1+m}}{b (1+m)}+\frac {d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d^2 x^{2+m}}{b (2+m)}+\frac {(b c-a d)^2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a b^2 (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78 \[ \int \frac {x^m (c+d x)^2}{a+b x} \, dx=\frac {x^{1+m} \left (a d (2 b c (2+m)-a d (2+m)+b d (1+m) x)+(b c-a d)^2 (2+m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )\right )}{a b^2 (1+m) (2+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 99, 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.

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.56 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.14 \[ \int \frac {x^m (c+d x)^2}{a+b x} \, dx=\frac {c^{2} m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {c^{2} x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {2 c d m x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {4 c d x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {d^{2} m x^{m + 3} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac {3 d^{2} x^{m + 3} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} \] Input:

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

Output:

c**2*m*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/( 
a*gamma(m + 2)) + c**2*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1 
)*gamma(m + 1)/(a*gamma(m + 2)) + 2*c*d*m*x**(m + 2)*lerchphi(b*x*exp_pola 
r(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a*gamma(m + 3)) + 4*c*d*x**(m + 2)*lerc 
hphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a*gamma(m + 3)) + d**2 
*m*x**(m + 3)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 3)*gamma(m + 3)/(a*ga 
mma(m + 4)) + 3*d**2*x**(m + 3)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 3)* 
gamma(m + 3)/(a*gamma(m + 4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^m (c+d x)^2}{a+b x} \, dx=\frac {x^{m} a^{2} d^{2} m^{2}+3 x^{m} a^{2} d^{2} m +2 x^{m} a^{2} d^{2}-2 x^{m} a b c d \,m^{2}-6 x^{m} a b c d m -4 x^{m} a b c d -x^{m} a b \,d^{2} m^{2} x -2 x^{m} a b \,d^{2} m x +x^{m} b^{2} c^{2} m^{2}+3 x^{m} b^{2} c^{2} m +2 x^{m} b^{2} c^{2}+2 x^{m} b^{2} c d \,m^{2} x +4 x^{m} b^{2} c d m x +x^{m} b^{2} d^{2} m^{2} x^{2}+x^{m} b^{2} d^{2} m \,x^{2}-\left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{3} d^{2} m^{3}-3 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{3} d^{2} m^{2}-2 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{3} d^{2} m +2 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{2} b c d \,m^{3}+6 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{2} b c d \,m^{2}+4 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a^{2} b c d m -\left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a \,b^{2} c^{2} m^{3}-3 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a \,b^{2} c^{2} m^{2}-2 \left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a \,b^{2} c^{2} m}{b^{3} m \left (m^{2}+3 m +2\right )} \] Input:

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

Output:

(x**m*a**2*d**2*m**2 + 3*x**m*a**2*d**2*m + 2*x**m*a**2*d**2 - 2*x**m*a*b* 
c*d*m**2 - 6*x**m*a*b*c*d*m - 4*x**m*a*b*c*d - x**m*a*b*d**2*m**2*x - 2*x* 
*m*a*b*d**2*m*x + x**m*b**2*c**2*m**2 + 3*x**m*b**2*c**2*m + 2*x**m*b**2*c 
**2 + 2*x**m*b**2*c*d*m**2*x + 4*x**m*b**2*c*d*m*x + x**m*b**2*d**2*m**2*x 
**2 + x**m*b**2*d**2*m*x**2 - int(x**m/(a*x + b*x**2),x)*a**3*d**2*m**3 - 
3*int(x**m/(a*x + b*x**2),x)*a**3*d**2*m**2 - 2*int(x**m/(a*x + b*x**2),x) 
*a**3*d**2*m + 2*int(x**m/(a*x + b*x**2),x)*a**2*b*c*d*m**3 + 6*int(x**m/( 
a*x + b*x**2),x)*a**2*b*c*d*m**2 + 4*int(x**m/(a*x + b*x**2),x)*a**2*b*c*d 
*m - int(x**m/(a*x + b*x**2),x)*a*b**2*c**2*m**3 - 3*int(x**m/(a*x + b*x** 
2),x)*a*b**2*c**2*m**2 - 2*int(x**m/(a*x + b*x**2),x)*a*b**2*c**2*m)/(b**3 
*m*(m**2 + 3*m + 2))