Integrand size = 25, antiderivative size = 300 \[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=-\frac {2 (2 c d-b e) f \text {arctanh}\left (\frac {b f+2 c f x+2 e \sqrt {-a+x (b+c x)}+2 \sqrt {c} \left (d+e x+f \sqrt {-a+x (b+c x)}\right )}{\sqrt {-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )}}\right )}{\left (e^2-c f^2\right ) \sqrt {-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )}}-\frac {\log \left (b+2 c x+2 \sqrt {c} \sqrt {-a+x (b+c x)}\right )}{e-\sqrt {c} f}+\frac {e \log \left (b d+a \left (e-\sqrt {c} f\right )+\left (2 \sqrt {c} d+b f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )+\left (e+\sqrt {c} f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )^2\right )}{e^2-c f^2} \] Output:
-2*(-b*e+2*c*d)*f*arctanh((b*f+2*c*f*x+2*e*(-a+x*(c*x+b))^(1/2)+2*c^(1/2)* (d+e*x+f*(-a+x*(c*x+b))^(1/2)))/(-4*b*d*e-4*a*e^2+b^2*f^2+4*c*(a*f^2+d^2)) ^(1/2))/(-c*f^2+e^2)/(-4*b*d*e-4*a*e^2+b^2*f^2+4*c*(a*f^2+d^2))^(1/2)-ln(b +2*c*x+2*c^(1/2)*(-a+x*(c*x+b))^(1/2))/(e-c^(1/2)*f)+e*ln(b*d+a*(e-c^(1/2) *f)+(2*c^(1/2)*d+b*f)*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2))+(e+c^(1/2)*f)*(c^(1/ 2)*x+(c*x^2+b*x-a)^(1/2))^2)/(-c*f^2+e^2)
Time = 1.73 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.96 \[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\frac {-\frac {2 (2 c d-b e) f \arctan \left (\frac {b f+2 c f x+2 e \sqrt {-a+x (b+c x)}-2 \sqrt {c} \left (d+e x+f \sqrt {-a+x (b+c x)}\right )}{\sqrt {4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )}}\right )}{\sqrt {4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )}}+\left (e-\sqrt {c} f\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {-a+x (b+c x)}\right )-e \log \left (b \left (d+e x-2 \sqrt {c} f x+f \sqrt {-a+x (b+c x)}\right )+2 \sqrt {c} \left (a f+\left (-d-e x+\sqrt {c} f x\right ) \left (-\sqrt {c} x+\sqrt {-a+x (b+c x)}\right )\right )\right )}{-e^2+c f^2} \] Input:
Integrate[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-1),x]
Output:
((-2*(2*c*d - b*e)*f*ArcTan[(b*f + 2*c*f*x + 2*e*Sqrt[-a + x*(b + c*x)] - 2*Sqrt[c]*(d + e*x + f*Sqrt[-a + x*(b + c*x)]))/Sqrt[4*b*d*e + 4*a*e^2 - b ^2*f^2 - 4*c*(d^2 + a*f^2)]])/Sqrt[4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*c*(d^2 + a*f^2)] + (e - Sqrt[c]*f)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[-a + x*(b + c*x )]] - e*Log[b*(d + e*x - 2*Sqrt[c]*f*x + f*Sqrt[-a + x*(b + c*x)]) + 2*Sqr t[c]*(a*f + (-d - e*x + Sqrt[c]*f*x)*(-(Sqrt[c]*x) + Sqrt[-a + x*(b + c*x) ]))])/(-e^2 + c*f^2)
Leaf count is larger than twice the leaf count of optimal. \(1084\) vs. \(2(300)=600\).
Time = 10.30 (sec) , antiderivative size = 1084, normalized size of antiderivative = 3.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7293, 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.
\(\displaystyle \int \frac {1}{f \sqrt {-a+b x+c x^2}+d+e x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {f \sqrt {-a+b x+c x^2}}{-a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2}+\frac {d+e x}{a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(2 c d-b e) f \text {arctanh}\left (\frac {-b f^2+2 d e+2 \left (e^2-c f^2\right ) x}{f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}\right )}{\left (e^2-c f^2\right ) \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}-\frac {\sqrt {c} f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x-a}}\right )}{e^2-c f^2}-\frac {\sqrt {-2 a e^4-2 b d e^3+2 c d^2 e^2+b^2 f^2 e^2+2 a c f^2 e^2-2 b c d f^2 e-(2 c d-b e) f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )} e+2 c^2 d^2 f^2} \text {arctanh}\left (\frac {4 a e^2+2 b d e-b^2 f^2-4 a c f^2+2 \left (-b e^2+2 c d e-c f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}\right ) x-b f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}{2 \sqrt {2} \sqrt {-2 a e^4-2 b d e^3+2 c d^2 e^2+b^2 f^2 e^2+2 a c f^2 e^2-2 b c d f^2 e-(2 c d-b e) f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )} e+2 c^2 d^2 f^2} \sqrt {c x^2+b x-a}}\right )}{\sqrt {2} \left (e^2-c f^2\right ) \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}+\frac {\sqrt {-2 a e^4-2 b d e^3+2 c d^2 e^2+b^2 f^2 e^2+2 a c f^2 e^2-2 b c d f^2 e+(2 c d-b e) f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )} e+2 c^2 d^2 f^2} \text {arctanh}\left (\frac {4 a e^2+2 b d e-b^2 f^2-4 a c f^2+2 \left (-b e^2+2 c d e+c f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}\right ) x+b f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}{2 \sqrt {2} \sqrt {-2 a e^4-2 b d e^3+2 c d^2 e^2+b^2 f^2 e^2+2 a c f^2 e^2-2 b c d f^2 e+(2 c d-b e) f \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )} e+2 c^2 d^2 f^2} \sqrt {c x^2+b x-a}}\right )}{\sqrt {2} \left (e^2-c f^2\right ) \sqrt {-4 a e^2-4 b d e+b^2 f^2+4 c \left (d^2+a f^2\right )}}+\frac {e \log \left (d^2+a f^2+\left (e^2-c f^2\right ) x^2+\left (2 d e-b f^2\right ) x\right )}{2 \left (e^2-c f^2\right )}\) |
Input:
Int[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-1),x]
Output:
((2*c*d - b*e)*f*ArcTanh[(2*d*e - b*f^2 + 2*(e^2 - c*f^2)*x)/(f*Sqrt[-4*b* d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)])])/((e^2 - c*f^2)*Sqrt[-4*b*d *e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)]) - (Sqrt[c]*f*ArcTanh[(b + 2*c *x)/(2*Sqrt[c]*Sqrt[-a + b*x + c*x^2])])/(e^2 - c*f^2) - (Sqrt[2*c*d^2*e^2 - 2*b*d*e^3 - 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2*f^2 + 2*a *c*e^2*f^2 - e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^ 2 + a*f^2)]]*ArcTanh[(2*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2 - b*f*Sqrt[- 4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)] + 2*(2*c*d*e - b*e^2 - c* f*Sqrt[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)])*x)/(2*Sqrt[2]*Sq rt[2*c*d^2*e^2 - 2*b*d*e^3 - 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2 *e^2*f^2 + 2*a*c*e^2*f^2 - e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e - 4*a*e^2 + b^2 *f^2 + 4*c*(d^2 + a*f^2)]]*Sqrt[-a + b*x + c*x^2])])/(Sqrt[2]*(e^2 - c*f^2 )*Sqrt[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)]) + (Sqrt[2*c*d^2* e^2 - 2*b*d*e^3 - 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2*f^2 + 2*a*c*e^2*f^2 + e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c* (d^2 + a*f^2)]]*ArcTanh[(2*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2 + b*f*Sqr t[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)] + 2*(2*c*d*e - b*e^2 + c*f*Sqrt[-4*b*d*e - 4*a*e^2 + b^2*f^2 + 4*c*(d^2 + a*f^2)])*x)/(2*Sqrt[2] *Sqrt[2*c*d^2*e^2 - 2*b*d*e^3 - 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2*f^2 + 2*a*c*e^2*f^2 + e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e - 4*a*e^2...
Leaf count of result is larger than twice the leaf count of optimal. \(4828\) vs. \(2(270)=540\).
Time = 0.13 (sec) , antiderivative size = 4829, normalized size of antiderivative = 16.10
Input:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2)),x,method=_RETURNVERBOSE)
Output:
f*(-2*(c*f^2-e^2)/(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)/ (2*c*f^2-2*e^2)*(1/2*(4*(x+(b*f^2-2*d*e+(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4* b*d*e+4*c*d^2))^(1/2))/(2*c*f^2-2*e^2))^2*c-4*(b*e^2-2*c*d*e+(f^2*(4*a*c*f ^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)*(x+(b*f^2-2*d*e+ (f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2))/(2*c*f^2-2*e^2))+ 2*(2*a*c*e^2*f^2+b^2*e^2*f^2-2*b*c*d*e*f^2+2*c^2*d^2*f^2-2*e^4*a-2*d*e^3*b +2*d^2*e^2*c+(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*b*e^2 -2*(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c*d*e)/(c*f^2-e ^2)^2)^(1/2)-1/2*(b*e^2-2*c*d*e+(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4* c*d^2))^(1/2)*c)/(c*f^2-e^2)*ln((-1/2*(b*e^2-2*c*d*e+(f^2*(4*a*c*f^2+b^2*f ^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)+c*(x+(b*f^2-2*d*e+(f^2*( 4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2))/(2*c*f^2-2*e^2)))/c^(1/ 2)+((x+(b*f^2-2*d*e+(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2 ))/(2*c*f^2-2*e^2))^2*c-(b*e^2-2*c*d*e+(f^2*(4*a*c*f^2+b^2*f^2-4*a*e^2-4*b *d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)*(x+(b*f^2-2*d*e+(f^2*(4*a*c*f^2+b^2*f^ 2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2))/(2*c*f^2-2*e^2))+1/2*(2*a*c*e^2*f^2+b^2 *e^2*f^2-2*b*c*d*e*f^2+2*c^2*d^2*f^2-2*e^4*a-2*d*e^3*b+2*d^2*e^2*c+(f^2*(4 *a*c*f^2+b^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*b*e^2-2*(f^2*(4*a*c*f^2+b ^2*f^2-4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c*d*e)/(c*f^2-e^2)^2)^(1/2))/c^(1/2 )-1/2*(2*a*c*e^2*f^2+b^2*e^2*f^2-2*b*c*d*e*f^2+2*c^2*d^2*f^2-2*e^4*a-2*...
Timed out. \[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2)),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\int \frac {1}{d + e x + f \sqrt {- a + b x + c x^{2}}}\, dx \] Input:
integrate(1/(d+e*x+f*(c*x**2+b*x-a)**(1/2)),x)
Output:
Integral(1/(d + e*x + f*sqrt(-a + b*x + c*x**2)), x)
\[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\int { \frac {1}{e x + \sqrt {c x^{2} + b x - a} f + d} \,d x } \] Input:
integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/(e*x + sqrt(c*x^2 + b*x - a)*f + d), x)
Exception generated. \[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(d+e*x+f*(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. \[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\int \frac {1}{d+f\,\sqrt {c\,x^2+b\,x-a}+e\,x} \,d x \] Input:
int(1/(d + f*(b*x - a + c*x^2)^(1/2) + e*x),x)
Output:
int(1/(d + f*(b*x - a + c*x^2)^(1/2) + e*x), x)
\[ \int \frac {1}{d+e x+f \sqrt {-a+b x+c x^2}} \, dx=\int \frac {1}{d +e x +f \sqrt {c \,x^{2}+b x -a}}d x \] Input:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2)),x)
Output:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2)),x)