Integrand size = 25, antiderivative size = 126 \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a} d}-\frac {e \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d \sqrt {c d^2-b d e+a e^2}} \] Output:
-arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(1/2)/d-e*arctanh(1/ 2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2) )/d/(a*e^2-b*d*e+c*d^2)^(1/2)
Time = 0.71 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (-\frac {e \sqrt {-c d^2+e (b d-a e)} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{d} \] Input:
Integrate[1/(x*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
(2*(-((e*Sqrt[-(c*d^2) + e*(b*d - a*e)]*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt [a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a *e))) + ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]]/Sqrt[a]))/d
Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 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.
\(\displaystyle \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1289 |
\(\displaystyle \int \left (\frac {1}{d x \sqrt {a+b x+c x^2}}-\frac {e}{d (d+e x) \sqrt {a+b x+c x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{d \sqrt {a e^2-b d e+c d^2}}-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a} d}\) |
Input:
Int[1/(x*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(Sqrt[a]*d)) - (e *ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sq rt[a + b*x + c*x^2])])/(d*Sqrt[c*d^2 - b*d*e + a*e^2])
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}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Time = 1.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.54
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {\ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\) | \(194\) |
Input:
int(1/x/(e*x+d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/d/((a*e^2-b*d *e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2 *((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2 -b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (110) = 220\).
Time = 0.23 (sec) , antiderivative size = 1047, normalized size of antiderivative = 8.31 \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:
integrate(1/x/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/2*(sqrt(c*d^2 - b*d*e + a*e^2)*a*e*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^ 2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d ^2)) + (c*d^2 - b*d*e + a*e^2)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2))/(a*c*d^3 - a*b *d^2*e + a^2*d*e^2), -1/2*(2*sqrt(-c*d^2 + b*d*e - a*e^2)*a*e*arctan(-1/2* sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - (c*d^2 - b*d*e + a*e^2)*sqrt(a)*log( -(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a ) + 8*a^2)/x^2))/(a*c*d^3 - a*b*d^2*e + a^2*d*e^2), 1/2*(sqrt(c*d^2 - b*d* e + a*e^2)*a*e*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt (c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e ^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(c*d^2 - b*d*e + a*e^2)*sqrt(-a)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/( a*c*x^2 + a*b*x + a^2)))/(a*c*d^3 - a*b*d^2*e + a^2*d*e^2), -(sqrt(-c*d^2 + b*d*e - a*e^2)*a*e*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 ...
\[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{x \left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(1/x/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral(1/(x*(d + e*x)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} x} \,d x } \] Input:
integrate(1/x/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)*x), x)
Exception generated. \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{x\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/(x*(d + e*x)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/(x*(d + e*x)*(a + b*x + c*x^2)^(1/2)), x)
Time = 0.48 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.93 \[ \int \frac {1}{x (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a e -\sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a e +\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,e^{2}-\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b d e +\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) c \,d^{2}-\sqrt {a}\, \mathrm {log}\left (x \right ) a \,e^{2}+\sqrt {a}\, \mathrm {log}\left (x \right ) b d e -\sqrt {a}\, \mathrm {log}\left (x \right ) c \,d^{2}}{a d \left (a \,e^{2}-b d e +c \,d^{2}\right )} \] Input:
int(1/x/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)
Output:
(sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*e - sqrt(a*e**2 - b* d*e + c*d**2)*log(d + e*x)*a*e + sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x* *2) - 2*a - b*x)*a*e**2 - sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2 *a - b*x)*b*d*e + sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x )*c*d**2 - sqrt(a)*log(x)*a*e**2 + sqrt(a)*log(x)*b*d*e - sqrt(a)*log(x)*c *d**2)/(a*d*(a*e**2 - b*d*e + c*d**2))