Integrand size = 20, antiderivative size = 151 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:
e^2*(-b*e+3*c*d)*x/c^2+1/2*e^3*x^2/c-(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*(3* a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(1/2)+1/2 *e*(3*c^2*d^2+b^2*e^2-c*e*(a*e+3*b*d))*ln(c*x^2+b*x+a)/c^3
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\frac {2 c e^2 (3 c d-b e) x+c^2 e^3 x^2+\frac {2 (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (a+x (b+c x))}{2 c^3} \] Input:
Integrate[(d + e*x)^3/(a + b*x + c*x^2),x]
Output:
(2*c*e^2*(3*c*d - b*e)*x + c^2*e^3*x^2 + (2*(2*c*d - b*e)*(c^2*d^2 + b^2*e ^2 - c*e*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + x*(b + c*x) ])/(2*c^3)
Time = 0.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1143, 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.
\(\displaystyle \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle \int \left (\frac {e x \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+a b e^3-3 a c d e^2+c^2 d^3}{c^2 \left (a+b x+c x^2\right )}+\frac {e^2 (3 c d-b e)}{c^2}+\frac {e^3 x}{c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {e^3 x^2}{2 c}\) |
Input:
Int[(d + e*x)^3/(a + b*x + c*x^2),x]
Output:
(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^2)/(2*c) - ((2*c*d - b*e)*(c^2*d^2 + b^ 2*e^2 - c*e*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sq rt[b^2 - 4*a*c]) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + b* x + c*x^2])/(2*c^3)
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Time = 1.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.15
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{2} c e \,x^{2}+b e x -3 c d x \right )}{c^{2}}+\frac {\frac {\left (-a c \,e^{3}+b^{2} e^{3}-3 d \,e^{2} b c +3 d^{2} e \,c^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a \,e^{3} b -3 a d \,e^{2} c +c^{2} d^{3}-\frac {\left (-a c \,e^{3}+b^{2} e^{3}-3 d \,e^{2} b c +3 d^{2} e \,c^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2}}\) | \(174\) |
risch | \(\text {Expression too large to display}\) | \(4105\) |
Input:
int((e*x+d)^3/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-e^2/c^2*(-1/2*c*e*x^2+b*e*x-3*c*d*x)+1/c^2*(1/2*(-a*c*e^3+b^2*e^3-3*b*c*d *e^2+3*c^2*d^2*e)/c*ln(c*x^2+b*x+a)+2*(a*e^3*b-3*a*d*e^2*c+c^2*d^3-1/2*(-a *c*e^3+b^2*e^3-3*b*c*d*e^2+3*c^2*d^2*e)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c *x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.48 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{2} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x + {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{2} - 2 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x + {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/2*((b^2*c^2 - 4*a*c^3)*e^3*x^2 + (2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3)*sqrt(b^2 - 4*a*c)*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*(3*(b^2*c^2 - 4*a*c^3)*d*e^2 - (b^3*c - 4*a*b*c^2)*e^3)*x + (3*(b^2*c ^2 - 4*a*c^3)*d^2*e - 3*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 5*a*b^2*c + 4*a ^2*c^2)*e^3)*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^3*x^2 - 2*(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c* x + b)/(b^2 - 4*a*c)) + 2*(3*(b^2*c^2 - 4*a*c^3)*d*e^2 - (b^3*c - 4*a*b*c^ 2)*e^3)*x + (3*(b^2*c^2 - 4*a*c^3)*d^2*e - 3*(b^3*c - 4*a*b*c^2)*d*e^2 + ( b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^3)*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. 892 vs. \(2 (141) = 282\).
Time = 2.13 (sec) , antiderivative size = 892, normalized size of antiderivative = 5.91 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)**3/(c*x**2+b*x+a),x)
Output:
x*(-b*e**3/c**2 + 3*d*e**2/c) + (-e*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3* c**2*d**2)/(2*c**3) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2 *e**2 + b*c*d*e - c**2*d**2)/(2*c**3*(4*a*c - b**2)))*log(x + (2*a**2*c*e* *3 - a*b**2*e**3 + 3*a*b*c*d*e**2 + 4*a*c**3*(-e*(a*c*e**2 - b**2*e**2 + 3 *b*c*d*e - 3*c**2*d**2)/(2*c**3) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a* c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(2*c**3*(4*a*c - b**2))) - 6*a*c **2*d**2*e - b**2*c**2*(-e*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2 )/(2*c**3) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b *c*d*e - c**2*d**2)/(2*c**3*(4*a*c - b**2))) + b*c**2*d**3)/(3*a*b*c*e**3 - 6*a*c**2*d*e**2 - b**3*e**3 + 3*b**2*c*d*e**2 - 3*b*c**2*d**2*e + 2*c**3 *d**3)) + (-e*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/(2*c**3) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2 *d**2)/(2*c**3*(4*a*c - b**2)))*log(x + (2*a**2*c*e**3 - a*b**2*e**3 + 3*a *b*c*d*e**2 + 4*a*c**3*(-e*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2 )/(2*c**3) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b *c*d*e - c**2*d**2)/(2*c**3*(4*a*c - b**2))) - 6*a*c**2*d**2*e - b**2*c**2 *(-e*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/(2*c**3) + sqrt(-4*a *c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(2 *c**3*(4*a*c - b**2))) + b*c**2*d**3)/(3*a*b*c*e**3 - 6*a*c**2*d*e**2 - b* *3*e**3 + 3*b**2*c*d*e**2 - 3*b*c**2*d**2*e + 2*c**3*d**3)) + e**3*x**2...
Exception generated. \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^3/(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 = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\frac {c e^{3} x^{2} + 6 \, c d e^{2} x - 2 \, b e^{3} x}{2 \, c^{2}} + \frac {{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*(c*e^3*x^2 + 6*c*d*e^2*x - 2*b*e^3*x)/c^2 + 1/2*(3*c^2*d^2*e - 3*b*c*d *e^2 + b^2*e^3 - a*c*e^3)*log(c*x^2 + b*x + a)/c^3 + (2*c^3*d^3 - 3*b*c^2* d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*arctan((2*c *x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\frac {e^3\,x^2}{2\,c}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,a^2\,c^2\,e^3-5\,a\,b^2\,c\,e^3+12\,a\,b\,c^2\,d\,e^2-12\,a\,c^3\,d^2\,e+b^4\,e^3-3\,b^3\,c\,d\,e^2+3\,b^2\,c^2\,d^2\,e\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}-x\,\left (\frac {b\,e^3}{c^2}-\frac {3\,d\,e^2}{c}\right )-\frac {\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-b\,c\,d\,e+c^2\,d^2-3\,a\,c\,e^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \] Input:
int((d + e*x)^3/(a + b*x + c*x^2),x)
Output:
(e^3*x^2)/(2*c) - (log(a + b*x + c*x^2)*(b^4*e^3 + 4*a^2*c^2*e^3 + 3*b^2*c ^2*d^2*e - 5*a*b^2*c*e^3 - 12*a*c^3*d^2*e - 3*b^3*c*d*e^2 + 12*a*b*c^2*d*e ^2))/(2*(4*a*c^4 - b^2*c^3)) - x*((b*e^3)/c^2 - (3*d*e^2)/c) - (atan((b + 2*c*x)/(4*a*c - b^2)^(1/2))*(b*e - 2*c*d)*(b^2*e^2 + c^2*d^2 - 3*a*c*e^2 - b*c*d*e))/(c^3*(4*a*c - b^2)^(1/2))
Time = 0.20 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.21 \[ \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx=\frac {6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c \,e^{3}-12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} d \,e^{2}-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e^{3}+6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c d \,e^{2}-6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} d^{2} e +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) c^{3} d^{3}-4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} c^{2} e^{3}+5 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c \,e^{3}-12 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b \,c^{2} d \,e^{2}+12 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,c^{3} d^{2} e -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} e^{3}+3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3} c d \,e^{2}-3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{2} c^{2} d^{2} e -8 a b \,c^{2} e^{3} x +24 a \,c^{3} d \,e^{2} x +4 a \,c^{3} e^{3} x^{2}+2 b^{3} c \,e^{3} x -6 b^{2} c^{2} d \,e^{2} x -b^{2} c^{2} e^{3} x^{2}}{2 c^{3} \left (4 a c -b^{2}\right )} \] Input:
int((e*x+d)^3/(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**3 - 12 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*d*e**2 - 2 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e**3 + 6*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*d*e**2 - 6*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d**2*e + 4*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**3*d**3 - 4*log(a + b*x + c*x**2)*a**2*c**2*e**3 + 5*log(a + b*x + c*x**2)*a*b**2*c*e**3 - 12 *log(a + b*x + c*x**2)*a*b*c**2*d*e**2 + 12*log(a + b*x + c*x**2)*a*c**3*d **2*e - log(a + b*x + c*x**2)*b**4*e**3 + 3*log(a + b*x + c*x**2)*b**3*c*d *e**2 - 3*log(a + b*x + c*x**2)*b**2*c**2*d**2*e - 8*a*b*c**2*e**3*x + 24* a*c**3*d*e**2*x + 4*a*c**3*e**3*x**2 + 2*b**3*c*e**3*x - 6*b**2*c**2*d*e** 2*x - b**2*c**2*e**3*x**2)/(2*c**3*(4*a*c - b**2))