Integrand size = 21, antiderivative size = 121 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\frac {(c d-b e) x}{c^2}+\frac {e x^2}{2 c}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:
(-b*e+c*d)*x/c^2+1/2*e*x^2/c-(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)*arctanh(( 2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(1/2)-1/2*(a*c*e-b^2*e+b*c*d )*ln(c*x^2+b*x+a)/c^3
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\frac {2 c (c d-b e) x+c^2 e x^2+\frac {2 \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (-b c d+b^2 e-a c e\right ) \log (a+x (b+c x))}{2 c^3} \] Input:
Integrate[(x^2*(d + e*x))/(a + b*x + c*x^2),x]
Output:
(2*c*(c*d - b*e)*x + c^2*e*x^2 + (2*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c *e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-(b*c*d) + b^2*e - a*c*e)*Log[a + x*(b + c*x)])/(2*c^3)
Time = 0.33 (sec) , antiderivative size = 121, 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 = {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.
\(\displaystyle \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (-\frac {x \left (a c e+b^2 (-e)+b c d\right )+a (c d-b e)}{c^2 \left (a+b x+c x^2\right )}+\frac {c d-b e}{c^2}+\frac {e x}{c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {x (c d-b e)}{c^2}+\frac {e x^2}{2 c}\) |
Input:
Int[(x^2*(d + e*x))/(a + b*x + c*x^2),x]
Output:
((c*d - b*e)*x)/c^2 + (e*x^2)/(2*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a* b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) - ( (b*c*d - b^2*e + a*c*e)*Log[a + b*x + c*x^2])/(2*c^3)
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]
Time = 1.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {-\frac {1}{2} c e \,x^{2}+b e x -c d x}{c^{2}}+\frac {\frac {\left (-a c e +e \,b^{2}-d b c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a b e -a c d -\frac {\left (-a c e +e \,b^{2}-d b c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2}}\) | \(127\) |
risch | \(\text {Expression too large to display}\) | \(2051\) |
Input:
int(x^2*(e*x+d)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/c^2*(-1/2*c*e*x^2+b*e*x-c*d*x)+1/c^2*(1/2*(-a*c*e+b^2*e-b*c*d)/c*ln(c*x ^2+b*x+a)+2*(a*b*e-a*c*d-1/2*(-a*c*e+b^2*e-b*c*d)*b/c)/(4*a*c-b^2)^(1/2)*a rctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 414, normalized size of antiderivative = 3.42 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{2} + \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - {\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x - {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{2} - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - {\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x - {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \] Input:
integrate(x^2*(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/2*((b^2*c^2 - 4*a*c^3)*e*x^2 + sqrt(b^2 - 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4* a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c - 4*a*b*c^2)*e)*x - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)* e)*log(c*x^2 + b*x + a))/(b^2*c^3 - 4*a*c^4), 1/2*((b^2*c^2 - 4*a*c^3)*e*x ^2 - 2*sqrt(-b^2 + 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*arctan (-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c - 4*a*b*c^2)*e)*x - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^2 + b*x + a))/(b^2*c^3 - 4*a*c^4)]
Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (119) = 238\).
Time = 1.12 (sec) , antiderivative size = 609, normalized size of antiderivative = 5.03 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=x \left (- \frac {b e}{c^{2}} + \frac {d}{c}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c e - a b^{2} e + a b c d + 4 a c^{3} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right ) - b^{2} c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c e - a b^{2} e + a b c d + 4 a c^{3} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right ) - b^{2} c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{2 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \frac {e x^{2}}{2 c} \] Input:
integrate(x**2*(e*x+d)/(c*x**2+b*x+a),x)
Output:
x*(-b*e/c**2 + d/c) + (-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3 *e + b**2*c*d)/(2*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(2*c**3) )*log(x + (2*a**2*c*e - a*b**2*e + a*b*c*d + 4*a*c**3*(-sqrt(-4*a*c + b**2 )*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(2*c**3*(4*a*c - b**2)) - ( a*c*e - b**2*e + b*c*d)/(2*c**3)) - b**2*c**2*(-sqrt(-4*a*c + b**2)*(3*a*b *c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(2*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(2*c**3)))/(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)) + (sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(2*c**3 *(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(2*c**3))*log(x + (2*a**2*c*e - a*b**2*e + a*b*c*d + 4*a*c**3*(sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2 *d - b**3*e + b**2*c*d)/(2*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d) /(2*c**3)) - b**2*c**2*(sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3 *e + b**2*c*d)/(2*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(2*c**3) ))/(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)) + e*x**2/(2*c)
Exception generated. \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\frac {c e x^{2} + 2 \, c d x - 2 \, b e x}{2 \, c^{2}} - \frac {{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} \] Input:
integrate(x^2*(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*(c*e*x^2 + 2*c*d*x - 2*b*e*x)/c^2 - 1/2*(b*c*d - b^2*e + a*c*e)*log(c* x^2 + b*x + a)/c^3 + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*arctan((2*c *x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.39 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=x\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,e\,a^2\,c^2-5\,e\,a\,b^2\,c+4\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}+\frac {e\,x^2}{2\,c}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (e\,b^3-d\,b^2\,c-3\,a\,e\,b\,c+2\,a\,d\,c^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \] Input:
int((x^2*(d + e*x))/(a + b*x + c*x^2),x)
Output:
x*(d/c - (b*e)/c^2) - (log(a + b*x + c*x^2)*(b^4*e + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e))/(2*(4*a*c^4 - b^2*c^3)) + (e*x^2)/(2*c) - ( atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(b^3*e + 2*a*c^2 *d - b^2*c*d - 3*a*b*c*e))/(c^3*(4*a*c - b^2)^(1/2))
Time = 0.24 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.65 \[ \int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx=\frac {6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c e -4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} d -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c d -4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} c^{2} e +5 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c e -4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b \,c^{2} d -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} e +\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3} c d -8 a b \,c^{2} e x +8 a \,c^{3} d x +4 a \,c^{3} e \,x^{2}+2 b^{3} c e x -2 b^{2} c^{2} d x -b^{2} c^{2} e \,x^{2}}{2 c^{3} \left (4 a c -b^{2}\right )} \] Input:
int(x^2*(e*x+d)/(c*x^2+b*x+a),x)
Output:
(6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*e - 4*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*d - 2*sqrt(4*a *c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e + 2*sqrt(4*a*c - b* *2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*d - 4*log(a + b*x + c*x**2 )*a**2*c**2*e + 5*log(a + b*x + c*x**2)*a*b**2*c*e - 4*log(a + b*x + c*x** 2)*a*b*c**2*d - log(a + b*x + c*x**2)*b**4*e + log(a + b*x + c*x**2)*b**3* c*d - 8*a*b*c**2*e*x + 8*a*c**3*d*x + 4*a*c**3*e*x**2 + 2*b**3*c*e*x - 2*b **2*c**2*d*x - b**2*c**2*e*x**2)/(2*c**3*(4*a*c - b**2))