Integrand size = 21, antiderivative size = 108 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=-\frac {d \left (c d^2-b d e+a e^2\right ) x}{e^4}+\frac {\left (c d^2-b d e+a e^2\right ) x^2}{2 e^3}-\frac {(c d-b e) x^3}{3 e^2}+\frac {c x^4}{4 e}+\frac {d^2 \left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^5} \] Output:
-d*(a*e^2-b*d*e+c*d^2)*x/e^4+1/2*(a*e^2-b*d*e+c*d^2)*x^2/e^3-1/3*(-b*e+c*d )*x^3/e^2+1/4*c*x^4/e+d^2*(a*e^2-b*d*e+c*d^2)*ln(e*x+d)/e^5
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {e x \left (c \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 e \left (3 a e (-2 d+e x)+b \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )+12 \left (c d^4+d^2 e (-b d+a e)\right ) \log (d+e x)}{12 e^5} \] Input:
Integrate[(x^2*(a + b*x + c*x^2))/(d + e*x),x]
Output:
(e*x*(c*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*e*(3*a*e*(-2*d + e*x) + b*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + 12*(c*d^4 + d^2*e*(-(b*d) + a*e))*Log[d + e*x])/(12*e^5)
Time = 0.29 (sec) , antiderivative size = 108, 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.
\(\displaystyle \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {d^2 \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac {d \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {x \left (a e^2-b d e+c d^2\right )}{e^3}+\frac {x^2 (b e-c d)}{e^2}+\frac {c x^3}{e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d^2 \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac {d x \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {x^2 \left (a e^2-b d e+c d^2\right )}{2 e^3}-\frac {x^3 (c d-b e)}{3 e^2}+\frac {c x^4}{4 e}\) |
Input:
Int[(x^2*(a + b*x + c*x^2))/(d + e*x),x]
Output:
-((d*(c*d^2 - b*d*e + a*e^2)*x)/e^4) + ((c*d^2 - b*d*e + a*e^2)*x^2)/(2*e^ 3) - ((c*d - b*e)*x^3)/(3*e^2) + (c*x^4)/(4*e) + (d^2*(c*d^2 - b*d*e + a*e ^2)*Log[d + e*x])/e^5
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]
Time = 0.72 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.95
method | result | size |
norman | \(\frac {c \,x^{4}}{4 e}+\frac {\left (b e -c d \right ) x^{3}}{3 e^{2}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {d \left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{e^{4}}+\frac {d^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(103\) |
default | \(-\frac {-\frac {1}{4} c \,x^{4} e^{3}-\frac {1}{3} b \,e^{3} x^{3}+\frac {1}{3} c d \,x^{3} e^{2}-\frac {1}{2} a \,e^{3} x^{2}+\frac {1}{2} b d \,e^{2} x^{2}-\frac {1}{2} c \,d^{2} e \,x^{2}+d \,e^{2} a x -b \,d^{2} e x +d^{3} c x}{e^{4}}+\frac {d^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(115\) |
risch | \(\frac {c \,x^{4}}{4 e}+\frac {b \,x^{3}}{3 e}-\frac {c d \,x^{3}}{3 e^{2}}+\frac {a \,x^{2}}{2 e}-\frac {b d \,x^{2}}{2 e^{2}}+\frac {c \,d^{2} x^{2}}{2 e^{3}}-\frac {d a x}{e^{2}}+\frac {b \,d^{2} x}{e^{3}}-\frac {d^{3} c x}{e^{4}}+\frac {d^{2} \ln \left (e x +d \right ) a}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right ) b}{e^{4}}+\frac {d^{4} \ln \left (e x +d \right ) c}{e^{5}}\) | \(131\) |
parallelrisch | \(\frac {3 c \,x^{4} e^{4}+4 x^{3} b \,e^{4}-4 x^{3} c d \,e^{3}+6 x^{2} a \,e^{4}-6 x^{2} b d \,e^{3}+6 x^{2} c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) a \,d^{2} e^{2}-12 \ln \left (e x +d \right ) b \,d^{3} e +12 \ln \left (e x +d \right ) c \,d^{4}-12 x a d \,e^{3}+12 x b \,d^{2} e^{2}-12 x c \,d^{3} e}{12 e^{5}}\) | \(132\) |
Input:
int(x^2*(c*x^2+b*x+a)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/4*c*x^4/e+1/3/e^2*(b*e-c*d)*x^3+1/2*(a*e^2-b*d*e+c*d^2)*x^2/e^3-d*(a*e^2 -b*d*e+c*d^2)*x/e^4+d^2*(a*e^2-b*d*e+c*d^2)*ln(e*x+d)/e^5
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {3 \, c e^{4} x^{4} - 4 \, {\left (c d e^{3} - b e^{4}\right )} x^{3} + 6 \, {\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} x^{2} - 12 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x + 12 \, {\left (c d^{4} - b d^{3} e + a d^{2} e^{2}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \] Input:
integrate(x^2*(c*x^2+b*x+a)/(e*x+d),x, algorithm="fricas")
Output:
1/12*(3*c*e^4*x^4 - 4*(c*d*e^3 - b*e^4)*x^3 + 6*(c*d^2*e^2 - b*d*e^3 + a*e ^4)*x^2 - 12*(c*d^3*e - b*d^2*e^2 + a*d*e^3)*x + 12*(c*d^4 - b*d^3*e + a*d ^2*e^2)*log(e*x + d))/e^5
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {c x^{4}}{4 e} + \frac {d^{2} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + x^{3} \left (\frac {b}{3 e} - \frac {c d}{3 e^{2}}\right ) + x^{2} \left (\frac {a}{2 e} - \frac {b d}{2 e^{2}} + \frac {c d^{2}}{2 e^{3}}\right ) + x \left (- \frac {a d}{e^{2}} + \frac {b d^{2}}{e^{3}} - \frac {c d^{3}}{e^{4}}\right ) \] Input:
integrate(x**2*(c*x**2+b*x+a)/(e*x+d),x)
Output:
c*x**4/(4*e) + d**2*(a*e**2 - b*d*e + c*d**2)*log(d + e*x)/e**5 + x**3*(b/ (3*e) - c*d/(3*e**2)) + x**2*(a/(2*e) - b*d/(2*e**2) + c*d**2/(2*e**3)) + x*(-a*d/e**2 + b*d**2/e**3 - c*d**3/e**4)
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {3 \, c e^{3} x^{4} - 4 \, {\left (c d e^{2} - b e^{3}\right )} x^{3} + 6 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x^{2} - 12 \, {\left (c d^{3} - b d^{2} e + a d e^{2}\right )} x}{12 \, e^{4}} + \frac {{\left (c d^{4} - b d^{3} e + a d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \] Input:
integrate(x^2*(c*x^2+b*x+a)/(e*x+d),x, algorithm="maxima")
Output:
1/12*(3*c*e^3*x^4 - 4*(c*d*e^2 - b*e^3)*x^3 + 6*(c*d^2*e - b*d*e^2 + a*e^3 )*x^2 - 12*(c*d^3 - b*d^2*e + a*d*e^2)*x)/e^4 + (c*d^4 - b*d^3*e + a*d^2*e ^2)*log(e*x + d)/e^5
Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {3 \, c e^{3} x^{4} - 4 \, c d e^{2} x^{3} + 4 \, b e^{3} x^{3} + 6 \, c d^{2} e x^{2} - 6 \, b d e^{2} x^{2} + 6 \, a e^{3} x^{2} - 12 \, c d^{3} x + 12 \, b d^{2} e x - 12 \, a d e^{2} x}{12 \, e^{4}} + \frac {{\left (c d^{4} - b d^{3} e + a d^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} \] Input:
integrate(x^2*(c*x^2+b*x+a)/(e*x+d),x, algorithm="giac")
Output:
1/12*(3*c*e^3*x^4 - 4*c*d*e^2*x^3 + 4*b*e^3*x^3 + 6*c*d^2*e*x^2 - 6*b*d*e^ 2*x^2 + 6*a*e^3*x^2 - 12*c*d^3*x + 12*b*d^2*e*x - 12*a*d*e^2*x)/e^4 + (c*d ^4 - b*d^3*e + a*d^2*e^2)*log(abs(e*x + d))/e^5
Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=x^3\,\left (\frac {b}{3\,e}-\frac {c\,d}{3\,e^2}\right )+x^2\,\left (\frac {a}{2\,e}-\frac {d\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )}{2\,e}\right )+\frac {c\,x^4}{4\,e}+\frac {\ln \left (d+e\,x\right )\,\left (c\,d^4-b\,d^3\,e+a\,d^2\,e^2\right )}{e^5}-\frac {d\,x\,\left (\frac {a}{e}-\frac {d\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )}{e}\right )}{e} \] Input:
int((x^2*(a + b*x + c*x^2))/(d + e*x),x)
Output:
x^3*(b/(3*e) - (c*d)/(3*e^2)) + x^2*(a/(2*e) - (d*(b/e - (c*d)/e^2))/(2*e) ) + (c*x^4)/(4*e) + (log(d + e*x)*(c*d^4 + a*d^2*e^2 - b*d^3*e))/e^5 - (d* x*(a/e - (d*(b/e - (c*d)/e^2))/e))/e
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {12 \,\mathrm {log}\left (e x +d \right ) a \,d^{2} e^{2}-12 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} e +12 \,\mathrm {log}\left (e x +d \right ) c \,d^{4}-12 a d \,e^{3} x +6 a \,e^{4} x^{2}+12 b \,d^{2} e^{2} x -6 b d \,e^{3} x^{2}+4 b \,e^{4} x^{3}-12 c \,d^{3} e x +6 c \,d^{2} e^{2} x^{2}-4 c d \,e^{3} x^{3}+3 c \,e^{4} x^{4}}{12 e^{5}} \] Input:
int(x^2*(c*x^2+b*x+a)/(e*x+d),x)
Output:
(12*log(d + e*x)*a*d**2*e**2 - 12*log(d + e*x)*b*d**3*e + 12*log(d + e*x)* c*d**4 - 12*a*d*e**3*x + 6*a*e**4*x**2 + 12*b*d**2*e**2*x - 6*b*d*e**3*x** 2 + 4*b*e**4*x**3 - 12*c*d**3*e*x + 6*c*d**2*e**2*x**2 - 4*c*d*e**3*x**3 + 3*c*e**4*x**4)/(12*e**5)