Integrand size = 26, antiderivative size = 1382 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx =\text {Too large to display} \] Output:
-8*(-4*a*c+b^2)^(1/2)*(c^4*(8*a*c^2*d*f^2+e*(4*b*(-4*a*c+b^2)^(1/2)*d*e+4* a*(-4*a*c+b^2)^(1/2)*e^2-b^3*f^2-b^2*(-4*a*c+b^2)^(1/2)*f^2)+c*(4*(-4*a*c+ b^2)^(1/2)*d^2*e-2*b^2*d*f^2+4*a*(b+(-4*a*c+b^2)^(1/2))*e*f^2))-2*c^4*f*(b ^2*(b-(-4*a*c+b^2)^(1/2))*e^2+4*c^2*((-4*a*c+b^2)^(1/2)*d^2-2*a*d*e+2*a*(- 4*a*c+b^2)^(1/2)*f^2)+c*(8*a*(-4*a*c+b^2)^(1/2)*e^2+4*b*e*((-4*a*c+b^2)^(1 /2)*d-a*e)+2*b^2*(d*e-(-4*a*c+b^2)^(1/2)*f^2)))*(-c*x^2+b*x-a)^(1/2)/(b-(- 4*a*c+b^2)^(1/2)-2*c*x))/(2*c*d+(b-(-4*a*c+b^2)^(1/2))*e)^2/(4*b*d*e+4*a*e ^2-b^2*f^2+4*c*(a*f^2+d^2))/(c*(2*c*d+(b+(-4*a*c+b^2)^(1/2))*e)-4*c^2*(-4* a*c+b^2)^(1/2)*f*(-c*x^2+b*x-a)^(1/2)/(b-(-4*a*c+b^2)^(1/2)-2*c*x)-4*c^2*( 2*c*d+(b-(-4*a*c+b^2)^(1/2))*e)*(c*x^2-b*x+a)/(b-(-4*a*c+b^2)^(1/2)-2*c*x) ^2)^2+4*(-4*a*c+b^2)^(1/2)*(c^2*(3*b^3*(b-(-4*a*c+b^2)^(1/2))*e^2*f^2+16*c ^3*(2*a^2*f^4+a*d^2*f^2+2*d^4)+2*c*(16*a^2*e^4-2*b^3*d*e*f^2+b^4*f^4+2*a*b *e^2*(16*d*e+3*(-4*a*c+b^2)^(1/2)*f^2)+b^2*e*(16*d^2*e-3*(-4*a*c+b^2)^(1/2 )*d*f^2-14*a*e*f^2))+c^2*(16*b*d*e*(a*f^2+4*d^2)-4*b^2*f^2*(4*a*f^2+d^2)+8 *a*e*(8*d^2*e+3*(-4*a*c+b^2)^(1/2)*d*f^2+8*a*e*f^2)))/(2*c*d+(b-(-4*a*c+b^ 2)^(1/2))*e)^2+6*c^2*(b*e+2*c*d)*(b*(b-(-4*a*c+b^2)^(1/2))*e-2*c*((-4*a*c+ b^2)^(1/2)*d+2*a*e))*f*(-c*x^2+b*x-a)^(1/2)/(2*c*d+(b-(-4*a*c+b^2)^(1/2))* e)/(b-(-4*a*c+b^2)^(1/2)-2*c*x))/(4*b*d*e+4*a*e^2-b^2*f^2+4*c*(a*f^2+d^2)) ^2/(c*(2*c*d+(b+(-4*a*c+b^2)^(1/2))*e)-4*c^2*(-4*a*c+b^2)^(1/2)*f*(-c*x^2+ b*x-a)^(1/2)/(b-(-4*a*c+b^2)^(1/2)-2*c*x)-4*c^2*(2*c*d+(b-(-4*a*c+b^2)^...
Result contains complex when optimal does not.
Time = 72.55 (sec) , antiderivative size = 3558, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x + f*Sqrt[-a + b*x - c*x^2])^(-3),x]
Output:
(2*(-4*c^2*d^4*e*f^2 - 5*b*c*d^3*e^2*f^2 - b^2*d^2*e^3*f^2 - 2*a*c*d^2*e^3 *f^2 + a*b*d*e^4*f^2 + 2*a^2*e^5*f^2 - b*c^2*d^3*f^4 - 6*a*c^2*d^2*e*f^4 - 3*a*b*c*d*e^2*f^4 - a*b^2*e^3*f^4 + 2*a^2*c*e^3*f^4 - 6*c^2*d^3*e^2*f^2*x - 9*b*c*d^2*e^3*f^2*x - 3*b^2*d*e^4*f^2*x - 6*a*c*d*e^4*f^2*x - 3*a*b*e^5 *f^2*x + 2*c^3*d^3*f^4*x + 3*b*c^2*d^2*e*f^4*x + 3*b^2*c*d*e^2*f^4*x - 6*a *c^2*d*e^2*f^4*x + b^3*e^3*f^4*x - 3*a*b*c*e^3*f^4*x))/((e^2 + c*f^2)^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) + (-8*c^2*d^4*e^3 - 16*b*c*d^3*e^4 - 8* b^2*d^2*e^5 - 16*a*c*d^2*e^5 - 16*a*b*d*e^6 - 8*a^2*e^7 + 24*c^3*d^4*e*f^2 + 48*b*c^2*d^3*e^2*f^2 + 34*b^2*c*d^2*e^3*f^2 + 8*a*c^2*d^2*e^3*f^2 + 7*b ^3*d*e^4*f^2 + 20*a*b*c*d*e^4*f^2 + 4*a*b^2*e^5*f^2 + 8*a^2*c*e^5*f^2 - 6* b^2*c^2*d^2*e*f^4 + 24*a*c^3*d^2*e*f^4 - 12*b^3*c*d*e^2*f^4 + 48*a*b*c^2*d *e^2*f^4 - 2*b^4*e^3*f^4 - 2*a*b^2*c*e^3*f^4 + 40*a^2*c^2*e^3*f^4 - 3*b^3* c^2*d*f^6 + 12*a*b*c^3*d*f^6 - 6*a*b^2*c^2*e*f^6 + 24*a^2*c^3*e*f^6 + 6*b^ 2*c*d*e^4*f^2*x - 24*a*c^2*d*e^4*f^2*x + 3*b^3*e^5*f^2*x - 12*a*b*c*e^5*f^ 2*x + 12*b^2*c^2*d*e^2*f^4*x - 48*a*c^3*d*e^2*f^4*x + 6*b^3*c*e^3*f^4*x - 24*a*b*c^2*e^3*f^4*x + 6*b^2*c^3*d*f^6*x - 24*a*c^4*d*f^6*x + 3*b^3*c^2*e* f^6*x - 12*a*b*c^3*e*f^6*x)/((e^2 + c*f^2)^2*(4*c*d^2 + 4*b*d*e + 4*a*e^2 - b^2*f^2 + 4*a*c*f^2)^2*(d^2 + a*f^2 + 2*d*e*x - b*f^2*x + e^2*x^2 + c*f^ 2*x^2)) + Sqrt[-a + b*x - c*x^2]*((2*(2*c*d^3*e*f + 2*b*d^2*e^2*f + 2*a...
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.
\(\displaystyle \int \frac {1}{\left (f \sqrt {-a+b x-c x^2}+d+e x\right )^3} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 f^2 (d+e x) \left (-a+b x-c x^2\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}-\frac {3 f \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^2}+\frac {d+e x}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^2}+\frac {4 b f^3 x \sqrt {-a+b x-c x^2}}{\left (-a f^2-x \left (2 d e-b f^2\right )-x^2 \left (c f^2+e^2\right )-d^2\right )^3}+\frac {4 c f^3 x^2 \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}+\frac {4 a f^3 \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {4 f^2 (d+e x) \left (-a+b x-c x^2\right )}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}-\frac {3 f \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^2}+\frac {d+e x}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^2}+\frac {4 b f^3 x \sqrt {-a+b x-c x^2}}{\left (-a f^2-x \left (2 d e-b f^2\right )-x^2 \left (c f^2+e^2\right )-d^2\right )^3}+\frac {4 c f^3 x^2 \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}+\frac {4 a f^3 \sqrt {-a+b x-c x^2}}{\left (a f^2+x \left (2 d e-b f^2\right )+x^2 \left (c f^2+e^2\right )+d^2\right )^3}\right )dx\) |
Input:
Int[(d + e*x + f*Sqrt[-a + b*x - c*x^2])^(-3),x]
Output:
$Aborted
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 255.15 (sec) , antiderivative size = 5961416, normalized size of antiderivative = 4313.62
Input:
int(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(d+e*x+f*(-c*x**2+b*x-a)**(1/2))**3,x)
Output:
Timed out
\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\int { \frac {1}{{\left (e x + \sqrt {-c x^{2} + b x - a} f + d\right )}^{3}} \,d x } \] Input:
integrate(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,x, algorithm="maxima")
Output:
integrate((e*x + sqrt(-c*x^2 + b*x - a)*f + d)^(-3), x)
Exception generated. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,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. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\int \frac {1}{{\left (d+f\,\sqrt {-c\,x^2+b\,x-a}+e\,x\right )}^3} \,d x \] Input:
int(1/(d + f*(b*x - a - c*x^2)^(1/2) + e*x)^3,x)
Output:
int(1/(d + f*(b*x - a - c*x^2)^(1/2) + e*x)^3, x)
\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^3} \, dx=\int \frac {1}{\left (d +e x +f \sqrt {-c \,x^{2}+b x -a}\right )^{3}}d x \] Input:
int(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,x)
Output:
int(1/(d+e*x+f*(-c*x^2+b*x-a)^(1/2))^3,x)