Integrand size = 25, antiderivative size = 385 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\frac {2 \left (2 b \left (c d^2-a e^2\right )-4 a c^{3/2} d f-b^2 d \left (2 e+\sqrt {c} f\right )+\left (4 c^{3/2} d^2-4 \sqrt {c} e (b d+a e)-b^2 e f-4 a c e f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )\right )}{\left (e+\sqrt {c} f\right ) \left (4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )\right ) \left (b d+a \left (e-\sqrt {c} f\right )+\left (2 \sqrt {c} d+b f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )+\left (e+\sqrt {c} f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )^2\right )}-\frac {4 \left (b^2+4 a c\right ) f \text {arctanh}\left (\frac {b f+2 c f x+2 e \sqrt {-a+x (b+c x)}+2 \sqrt {c} \left (d+e x+f \sqrt {-a+x (b+c x)}\right )}{\sqrt {-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )}}\right )}{\left (-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )\right )^{3/2}} \] Output:
2*(2*b*(-a*e^2+c*d^2)-4*a*c^(3/2)*d*f-b^2*d*(2*e+c^(1/2)*f)+(4*c^(3/2)*d^2 -4*c^(1/2)*e*(a*e+b*d)-b^2*e*f-4*a*c*e*f)*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2))) /(e+c^(1/2)*f)/(4*b*d*e+4*a*e^2-b^2*f^2-4*c*(a*f^2+d^2))/(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)-4*(4*a*c+b^2)*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))/(-4*b*d*e-4*a*e^2+b^2*f^2+4*c*(a*f^2+d^ 2))^(3/2)
Result contains complex when optimal does not.
Time = 50.69 (sec) , antiderivative size = 2401, normalized size of antiderivative = 6.24 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-2),x]
Output:
(-2*(-2*c*d^3*e + 2*b*d^2*e^2 + 2*a*d*e^3 - b*c*d^2*f^2 - 4*a*c*d*e*f^2 + a*b*e^2*f^2 - 2*c*d^2*e^2*x + 2*b*d*e^3*x + 2*a*e^4*x - 2*c^2*d^2*f^2*x + 2*b*c*d*e*f^2*x - b^2*e^2*f^2*x - 2*a*c*e^2*f^2*x))/((e^2 - c*f^2)*(-4*c*d ^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2)*(d^2 + a*f^2 + 2*d*e*x - b*f ^2*x + e^2*x^2 - c*f^2*x^2)) + (2*(b*d*f + 2*a*e*f + 2*c*d*f*x - b*e*f*x)* Sqrt[-a + b*x + c*x^2])/((-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f ^2)*(d^2 + a*f^2 + 2*d*e*x - b*f^2*x + e^2*x^2 - c*f^2*x^2)) + (2*(b^2 + 4 *a*c)*f*ArcTan[(2*d*e - b*f^2 + 2*e^2*x - 2*c*f^2*x)/(f*Sqrt[-4*c*d^2 + 4* b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2])])/(-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2)^(3/2) + ((b^2 + 4*a*c)*f*((-2*I)*c*d*f + I*b*e*f + e* Sqrt[-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2])*Log[-((Sqrt[2]*( e^2 - c*f^2)*(-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2)^(3/2)*(( -2*I)*b*d*e - (4*I)*a*e^2 + I*b^2*f^2 + (4*I)*a*c*f^2 + b*f*Sqrt[-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2] - (4*I)*c*d*e*x + (2*I)*b*e^2*x + 2*c*f*Sqrt[-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2]*x))/((b^ 2 + 4*a*c)*f*(-2*c*d*f + b*e*f - I*e*Sqrt[-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b ^2*f^2 - 4*a*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 + (2*I)*c*d*e*f*Sqrt[-4*c* d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2] - I*b*e^2*f*Sqrt[-4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 - 4*a*c*f^2]]*(2*d*e - b*f^2 + I*f*Sqrt[-4...
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.
\(\displaystyle \int \frac {1}{\left (f \sqrt {-a+b x+c x^2}+d+e x\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 f^2 \left (-a+b x+c x^2\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}-\frac {2 d f \sqrt {-a+b x+c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}-\frac {2 e f x \sqrt {-a+b x+c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {1}{a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 d \left (e x-f \sqrt {x (b+c x)-a}\right )+x \left (-2 e f \sqrt {x (b+c x)-a}+b f^2+c f^2 x+e^2 x\right )+d^2 \left (1-\frac {a f^2}{d^2}\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {d^2-a f^2}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {2 d \left (e x-f \sqrt {-a+b x+c x^2}\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {x \left (-2 e f \sqrt {-a+b x+c x^2}+b f^2+e^2 x \left (\frac {c f^2}{e^2}+1\right )\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {d^2-a f^2}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {2 d \left (e x-f \sqrt {-a+b x+c x^2}\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {x \left (-2 e f \sqrt {-a+b x+c x^2}+b f^2+e^2 x \left (\frac {c f^2}{e^2}+1\right )\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}\right )dx\) |
Input:
Int[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-2),x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.25 (sec) , antiderivative size = 508167, normalized size of antiderivative = 1319.91
Input:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 2877 vs. \(2 (346) = 692\).
Time = 20.29 (sec) , antiderivative size = 6919, normalized size of antiderivative = 17.97 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{\left (d + e x + f \sqrt {- a + b x + c x^{2}}\right )^{2}}\, dx \] Input:
integrate(1/(d+e*x+f*(c*x**2+b*x-a)**(1/2))**2,x)
Output:
Integral((d + e*x + f*sqrt(-a + b*x + c*x**2))**(-2), x)
\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\int { \frac {1}{{\left (e x + \sqrt {c x^{2} + b x - a} f + d\right )}^{2}} \,d x } \] Input:
integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^2,x, algorithm="maxima")
Output:
integrate((e*x + sqrt(c*x^2 + b*x - a)*f + d)^(-2), x)
Exception generated. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^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. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{{\left (d+f\,\sqrt {c\,x^2+b\,x-a}+e\,x\right )}^2} \,d x \] Input:
int(1/(d + f*(b*x - a + c*x^2)^(1/2) + e*x)^2,x)
Output:
int(1/(d + f*(b*x - a + c*x^2)^(1/2) + e*x)^2, x)
\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{\left (d +e x +f \sqrt {c \,x^{2}+b x -a}\right )^{2}}d x \] Input:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^2,x)
Output:
int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^2,x)