Integrand size = 23, antiderivative size = 92 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=d x+\frac {e x^2}{2}+\frac {f (b+2 c x) \sqrt {-a+b x+c x^2}}{4 c}-\frac {\left (b^2+4 a c\right ) f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {-a+b x+c x^2}}\right )}{8 c^{3/2}} \] Output:
d*x+1/2*e*x^2+1/4*f*(2*c*x+b)*(c*x^2+b*x-a)^(1/2)/c-1/8*(4*a*c+b^2)*f*arct anh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x-a)^(1/2))/c^(3/2)
Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.35 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=d x+\frac {e x^2}{2}+\frac {f (b+2 c x) \sqrt {-a+x (b+c x)}}{4 c}+\frac {\left (-b^2-4 a c\right ) f \sqrt {-a+x (b+c x)} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {-a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt {-a+b x+c x^2}} \] Input:
Integrate[d + e*x + f*Sqrt[-a + b*x + c*x^2],x]
Output:
d*x + (e*x^2)/2 + (f*(b + 2*c*x)*Sqrt[-a + x*(b + c*x)])/(4*c) + ((-b^2 - 4*a*c)*f*Sqrt[-a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[-a + b *x + c*x^2])])/(8*c^(3/2)*Sqrt[-a + b*x + c*x^2])
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {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.
\(\displaystyle \int \left (f \sqrt {-a+b x+c x^2}+d+e x\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {f \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {-a+b x+c x^2}}\right )}{8 c^{3/2}}+\frac {f (b+2 c x) \sqrt {-a+b x+c x^2}}{4 c}+d x+\frac {e x^2}{2}\) |
Input:
Int[d + e*x + f*Sqrt[-a + b*x + c*x^2],x]
Output:
d*x + (e*x^2)/2 + (f*(b + 2*c*x)*Sqrt[-a + b*x + c*x^2])/(4*c) - ((b^2 + 4 *a*c)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[-a + b*x + c*x^2])])/(8*c^(3/2 ))
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88
method | result | size |
default | \(d x +\frac {e \,x^{2}}{2}+f \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x -a}}{4 c}+\frac {\left (-4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x -a}\right )}{8 c^{\frac {3}{2}}}\right )\) | \(81\) |
parts | \(d x +\frac {e \,x^{2}}{2}+f \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x -a}}{4 c}+\frac {\left (-4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x -a}\right )}{8 c^{\frac {3}{2}}}\right )\) | \(81\) |
Input:
int(d+e*x+f*(c*x^2+b*x-a)^(1/2),x,method=_RETURNVERBOSE)
Output:
d*x+1/2*e*x^2+f*(1/4*(2*c*x+b)/c*(c*x^2+b*x-a)^(1/2)+1/8*(-4*a*c-b^2)/c^(3 /2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x-a)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.41 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=\left [\frac {8 \, c^{2} e x^{2} + 16 \, c^{2} d x + {\left (b^{2} + 4 \, a c\right )} \sqrt {c} f \log \left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} - 4 \, \sqrt {c x^{2} + b x - a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, c^{2} f x + b c f\right )} \sqrt {c x^{2} + b x - a}}{16 \, c^{2}}, \frac {4 \, c^{2} e x^{2} + 8 \, c^{2} d x + {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} f \arctan \left (\frac {\sqrt {c x^{2} + b x - a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x - a c\right )}}\right ) + 2 \, {\left (2 \, c^{2} f x + b c f\right )} \sqrt {c x^{2} + b x - a}}{8 \, c^{2}}\right ] \] Input:
integrate(d+e*x+f*(c*x^2+b*x-a)^(1/2),x, algorithm="fricas")
Output:
[1/16*(8*c^2*e*x^2 + 16*c^2*d*x + (b^2 + 4*a*c)*sqrt(c)*f*log(8*c^2*x^2 + 8*b*c*x + b^2 - 4*sqrt(c*x^2 + b*x - a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*( 2*c^2*f*x + b*c*f)*sqrt(c*x^2 + b*x - a))/c^2, 1/8*(4*c^2*e*x^2 + 8*c^2*d* x + (b^2 + 4*a*c)*sqrt(-c)*f*arctan(1/2*sqrt(c*x^2 + b*x - a)*(2*c*x + b)* sqrt(-c)/(c^2*x^2 + b*c*x - a*c)) + 2*(2*c^2*f*x + b*c*f)*sqrt(c*x^2 + b*x - a))/c^2]
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.46 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=d x + \frac {e x^{2}}{2} + f \left (\begin {cases} \left (- \frac {a}{2} - \frac {b^{2}}{8 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {- a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a + \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (\frac {b}{4 c} + \frac {x}{2}\right ) \sqrt {- a + b x + c x^{2}} & \text {for}\: c \neq 0 \\\frac {2 \left (- a + b x\right )^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\x \sqrt {- a} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(d+e*x+f*(c*x**2+b*x-a)**(1/2),x)
Output:
d*x + e*x**2/2 + f*Piecewise(((-a/2 - b**2/(8*c))*Piecewise((log(b + 2*sqr t(c)*sqrt(-a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a + b**2/(4*c), 0)), ((b /(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) + (b/(4*c) + x/2)*sqrt(-a + b*x + c*x**2), Ne(c, 0)), (2*(-a + b*x)**(3/2)/(3*b), Ne(b , 0)), (x*sqrt(-a), True))
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=\frac {1}{2} \, e x^{2} + \frac {1}{8} \, {\left (4 \, \sqrt {c x^{2} + b x - a} x - \frac {b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x - a} \sqrt {c}\right )}{c^{\frac {3}{2}}} - \frac {4 \, a \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x - a} \sqrt {c}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {c x^{2} + b x - a} b}{c}\right )} f + d x \] Input:
integrate(d+e*x+f*(c*x^2+b*x-a)^(1/2),x, algorithm="maxima")
Output:
1/2*e*x^2 + 1/8*(4*sqrt(c*x^2 + b*x - a)*x - b^2*log(2*c*x + b + 2*sqrt(c* x^2 + b*x - a)*sqrt(c))/c^(3/2) - 4*a*log(2*c*x + b + 2*sqrt(c*x^2 + b*x - a)*sqrt(c))/sqrt(c) + 2*sqrt(c*x^2 + b*x - a)*b/c)*f + d*x
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=\frac {1}{2} \, e x^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {c x^{2} + b x - a} {\left (2 \, x + \frac {b}{c}\right )} + \frac {{\left (b^{2} + 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x - a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}}\right )} f + d x \] Input:
integrate(d+e*x+f*(c*x^2+b*x-a)^(1/2),x, algorithm="giac")
Output:
1/2*e*x^2 + 1/8*(2*sqrt(c*x^2 + b*x - a)*(2*x + b/c) + (b^2 + 4*a*c)*log(a bs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x - a))*sqrt(c) + b))/c^(3/2))*f + d*x
Time = 22.96 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=d\,x+\frac {e\,x^2}{2}+f\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x-a}-\frac {f\,\ln \left (\sqrt {c\,x^2+b\,x-a}+\frac {\frac {b}{2}+c\,x}{\sqrt {c}}\right )\,\left (\frac {b^2}{4}+a\,c\right )}{2\,c^{3/2}} \] Input:
int(d + f*(b*x - a + c*x^2)^(1/2) + e*x,x)
Output:
d*x + (e*x^2)/2 + f*(x/2 + b/(4*c))*(b*x - a + c*x^2)^(1/2) - (f*log((b*x - a + c*x^2)^(1/2) + (b/2 + c*x)/c^(1/2))*(a*c + b^2/4))/(2*c^(3/2))
Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.60 \[ \int \left (d+e x+f \sqrt {-a+b x+c x^2}\right ) \, dx=\frac {2 \sqrt {c \,x^{2}+b x -a}\, b c f +4 \sqrt {c \,x^{2}+b x -a}\, c^{2} f x -4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x -a}+b +2 c x}{\sqrt {4 a c +b^{2}}}\right ) a c f -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x -a}+b +2 c x}{\sqrt {4 a c +b^{2}}}\right ) b^{2} f +8 c^{2} d x +4 c^{2} e \,x^{2}}{8 c^{2}} \] Input:
int(d+e*x+f*(c*x^2+b*x-a)^(1/2),x)
Output:
(2*sqrt( - a + b*x + c*x**2)*b*c*f + 4*sqrt( - a + b*x + c*x**2)*c**2*f*x - 4*sqrt(c)*log((2*sqrt(c)*sqrt( - a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a *c + b**2))*a*c*f - sqrt(c)*log((2*sqrt(c)*sqrt( - a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c + b**2))*b**2*f + 8*c**2*d*x + 4*c**2*e*x**2)/(8*c**2)