Integrand size = 23, antiderivative size = 1477 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^3} \, dx =\text {Too large to display} \] Output:
(b^4*d*e*f^2-b^3*(d+a^(1/2)*f)*(2*d*e^2+a^(1/2)*e^2*f+c*d*f^2)-4*b*(2*a^(1 /2)*c^2*d^3*f+a^(3/2)*c*d*f*(-c*f^2+3*e^2)+a*c*d^2*(-c*f^2+4*e^2)+a^2*(-c* e^2*f^2+2*e^4))+8*a*c*e*(a*e^2*(d+a^(1/2)*f)+c*(d^3-a^(3/2)*f^3))+2*b^2*(a *e^3*(5*d+a^(1/2)*f)+c*e*(d^3+6*a^(1/2)*d^2*f-2*a*d*f^2+a^(3/2)*f^3))+(4*a ^(1/2)*b*d*(-b*e+c*d)*(-b*f^2+2*d*e)-16*a^(5/2)*(-c*e^2*f^2+e^4)+a^2*(-4*b *c*e*f^3-8*c^2*d*f^3+8*b*e^3*f)-b^2*d*f*(2*c*d^2-b*(-b*f^2+3*d*e))+4*a^(3/ 2)*(4*c^2*d^2*f^2+b*e^2*(-b*f^2+6*d*e)-4*c*d*e*(b*f^2+d*e))-a*f*(8*c^2*d^3 +b^2*e*(-b*f^2+12*d*e)-6*b*c*d*(b*f^2+2*d*e)))*(a^(1/2)-(c*x^2+b*x+a)^(1/2 ))/x)/(d-a^(1/2)*f)^3/(4*b*d*e-4*a*e^2-b^2*f^2-4*c*(-a*f^2+d^2))/(b*e-c*(d +a^(1/2)*f)+(2*a^(1/2)*e-b*f)*(a^(1/2)-(c*x^2+b*x+a)^(1/2))/x+(d-a^(1/2)*f )*(a^(1/2)-(c*x^2+b*x+a)^(1/2))^2/x^2)^2+((b^5*d*f^4-b^4*e*f^2*(5*d^2+6*a^ (1/2)*d*f-a*f^2)-16*a*e*(2*a^2*e^4+4*a*c*e^2*(-a*f^2+d^2)+c^2*(d-a^(1/2)*f )^2*(2*d^2+7*a^(1/2)*d*f+2*a*f^2))-2*b^3*(3*a^(3/2)*e^2*f^3-a*d*f^2*(-7*c* f^2+18*e^2)+3*a^(1/2)*d^2*f*(-2*c*f^2+e^2)-d^3*(c*f^2+8*e^2))+8*b*(10*a^2* d*e^4+c^2*d*(d-a^(1/2)*f)^2*(2*d^2+4*a^(1/2)*d*f+5*a*f^2)+3*a*c*e^2*(4*d^3 +a^(1/2)*d^2*f-6*a*d*f^2+a^(3/2)*f^3))-4*b^2*(4*a*e^3*(a*f^2+4*d^2)+c*e*(8 *d^4-3*a^(1/2)*d^3*f+5*a*d^2*f^2-9*a^(3/2)*d*f^3-a^2*f^4)))/(d-a^(1/2)*f)^ 2+2*(3*b^2*d^2*(-b*e+2*c*d)*f-12*a^2*c*(-b*e+2*c*d)*f^3-3*a*(-b*e+2*c*d)*f *(-b^2*f^2+4*c*d^2)+16*a^(5/2)*(-c*f^2+e^2)^2+8*a^(3/2)*(2*c^2*d^2*f^2-b*e ^2*(-b*f^2+4*d*e)+c*(-b^2*f^4+b*d*e*f^2+4*d^2*e^2))+a^(1/2)*(16*c^2*d^4...
Leaf count is larger than twice the leaf count of optimal. \(3355\) vs. \(2(1477)=2954\).
Time = 21.96 (sec) , antiderivative size = 3355, normalized size of antiderivative = 2.27 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^3} \, dx=\text {Result too large to show} \] 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 |
\(\displaystyle \int \left (\frac {4 f^2 (d+e x) \left (a+b x+c x^2\right )}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^3}-\frac {3 f \sqrt {a+b x+c x^2}}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {d+e x}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {4 c f^3 x^2 \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}+\frac {4 a f^3 \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}+\frac {4 b f^3 x \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {4 f^2 (d+e x) \left (a+b x+c x^2\right )}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^3}-\frac {3 f \sqrt {a+b x+c x^2}}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {d+e x}{\left (-a f^2+x \left (2 d e-b f^2\right )+x^2 \left (e^2-c f^2\right )+d^2\right )^2}+\frac {4 c f^3 x^2 \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}+\frac {4 a f^3 \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}+\frac {4 b f^3 x \sqrt {a+b x+c x^2}}{\left (a f^2-x \left (2 d e-b f^2\right )-x^2 \left (e^2-c f^2\right )-d^2\right )^3}\right )dx\) |
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 = 256.43 (sec) , antiderivative size = 6017742, normalized size of antiderivative = 4074.30
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
\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^3} \, dx=\int \frac {1}{\left (d + e x + f \sqrt {a + b x + c x^{2}}\right )^{3}}\, dx \] Input:
integrate(1/(d+e*x+f*(c*x**2+b*x+a)**(1/2))**3,x)
Output:
Integral((d + e*x + f*sqrt(a + b*x + c*x**2))**(-3), x)
\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^3} \, dx=\int { \frac {1}{{\left (e x + \sqrt {c x^{2} + b x + a} f + d\right )}^{3}} \,d x } \] 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. \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^3} \, dx=\int \frac {1}{{\left (d+e\,x+f\,\sqrt {c\,x^2+b\,x+a}\right )}^3} \,d x \] Input:
int(1/(d + e*x + f*(a + b*x + c*x^2)^(1/2))^3,x)
Output:
int(1/(d + e*x + f*(a + b*x + c*x^2)^(1/2))^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)