\(\int \frac {x^2 (d+e x)^m}{a+b x+c x^2} \, dx\) [126]

Optimal result

Integrand size = 23, antiderivative size = 237 \[ \int \frac {x^2 (d+e x)^m}{a+b x+c x^2} \, dx=\frac {(d+e x)^{1+m}}{c e (1+m)}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)} \] Output:

(e*x+d)^(1+m)/c/e/(1+m)+(b-(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*(e*x+d)^(1+m)* 
hypergeom([1, 1+m],[2+m],2*c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e))/c/( 
2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(1+m)+(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))* 
(e*x+d)^(1+m)*hypergeom([1, 1+m],[2+m],2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^ 
(1/2))*e))/c/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(1+m)
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (d+e x)^m}{a+b x+c x^2} \, dx=-\frac {2 (d+e x)^{1+m} \left (2 \sqrt {b^2-4 a c} \left (c d^2+e (-b d+a e)\right )+e \left (-b^2 d+2 a c d+b \sqrt {b^2-4 a c} d+a b e-a \sqrt {b^2-4 a c} e\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )+e \left (b^2 d-2 a c d+b \sqrt {b^2-4 a c} d-a b e-a \sqrt {b^2-4 a c} e\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )\right )}{\sqrt {b^2-4 a c} e \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)} \] Input:

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

Output:

(-2*(d + e*x)^(1 + m)*(2*Sqrt[b^2 - 4*a*c]*(c*d^2 + e*(-(b*d) + a*e)) + e* 
(-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]* 
e)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[ 
b^2 - 4*a*c])*e)] + e*(b^2*d - 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - a*b*e - a 
*Sqrt[b^2 - 4*a*c]*e)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/( 
2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]))/(Sqrt[b^2 - 4*a*c]*e*(2*c*d + (-b + 
Sqrt[b^2 - 4*a*c])*e)*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + m))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1200, 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.

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

\(\Big \downarrow \) 1200

\(\displaystyle \int \left (\frac {\left (\frac {b^2-2 a c}{c \sqrt {b^2-4 a c}}-\frac {b}{c}\right ) (d+e x)^m}{-\sqrt {b^2-4 a c}+b+2 c x}+\frac {\left (-\frac {b^2-2 a c}{c \sqrt {b^2-4 a c}}-\frac {b}{c}\right ) (d+e x)^m}{\sqrt {b^2-4 a c}+b+2 c x}+\frac {(d+e x)^m}{c}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{c (m+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}+\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c (m+1) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}+\frac {(d+e x)^{m+1}}{c e (m+1)}\)

Input:

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

Output:

(d + e*x)^(1 + m)/(c*e*(1 + m)) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*( 
d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d 
 - (b - Sqrt[b^2 - 4*a*c])*e)])/(c*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(1 
+ m)) + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + m)*Hypergeom 
etric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c]) 
*e)])/(c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(1 + m))
 

Defintions of rubi rules used

rule 1200

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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