\(\int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx\) [127]

Optimal result

Integrand size = 48, antiderivative size = 173 \[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\frac {2 \sqrt {-b c+a d} \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \] Output:

2*(a*d-b*c)^(1/2)*(b*x+a)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(- 
a*f+b*e))^(1/2)*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c 
)*f/d/(-a*f+b*e))^(1/2))/b/d^(1/2)/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c 
*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)
 

Mathematica [A] (verified)

Time = 11.35 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=-\frac {2 (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{b \sqrt {a-\frac {b c}{d}} \sqrt {(a+b x) (c+d x) (e+f x)}} \] Input:

Integrate[1/Sqrt[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d* 
f)*x^2 + b*d*f*x^3],x]
 

Output:

(-2*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/( 
f*(a + b*x))]*EllipticF[ArcSin[Sqrt[a - (b*c)/d]/Sqrt[a + b*x]], (b*d*e - 
a*d*f)/(b*c*f - a*d*f)])/(b*Sqrt[a - (b*c)/d]*Sqrt[(a + b*x)*(c + d*x)*(e 
+ f*x)])
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}} \, dx\)

\(\Big \downarrow \) 2481

\(\displaystyle \int \frac {1}{\sqrt {\frac {(-2 a d f+b c f+b d e) (-a d f-b c f+2 b d e) (a d f-2 b c f+b d e)}{27 b^2 d^2 f^2}+b d f \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )^3+\frac {\left (3 b d f (a c f+a d e+b c e)-(a d f+b c f+b d e)^2\right ) \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )}{3 b d f}}}d\left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )\)

Input:

Int[1/Sqrt[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 
 + b*d*f*x^3],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2481

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* 
d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 
*d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {2 \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\)	\(191\)
elliptic	\(\frac {2 \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\)	\(191\)

Input:

int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2 
),x,method=_RETURNVERBOSE)
 

Output:

2*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/ 
(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x 
+b*c*e*x+a*c*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/ 
f+a/b))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\frac {2 \, \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )}{b d f} \] Input:

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 
)^(1/2),x, algorithm="fricas")
 

Output:

2*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e 
*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^ 
3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b 
*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3) 
/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))/(b*d*f)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int \frac {1}{\sqrt {a c e + b d f x^{3} + x^{2} \left (a d f + b c f + b d e\right ) + x \left (a c f + a d e + b c e\right )}}\, dx \] Input:

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x* 
*3)**(1/2),x)
 

Output:

Integral(1/sqrt(a*c*e + b*d*f*x**3 + x**2*(a*d*f + b*c*f + b*d*e) + x*(a*c 
*f + a*d*e + b*c*e)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int { \frac {1}{\sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x}} \,d x } \] Input:

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 
)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/sqrt(b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e 
+ a*d*e + a*c*f)*x), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int { \frac {1}{\sqrt {b d f x^{3} + a c e + {\left (b d e + b c f + a d f\right )} x^{2} + {\left (b c e + a d e + a c f\right )} x}} \,d x } \] Input:

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 
)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/sqrt(b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e 
+ a*d*e + a*c*f)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int \frac {1}{\sqrt {b\,d\,f\,x^3+\left (a\,d\,f+b\,c\,f+b\,d\,e\right )\,x^2+\left (a\,c\,f+a\,d\,e+b\,c\,e\right )\,x+a\,c\,e}} \,d x \] Input:

int(1/(x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b 
*d*f*x^3)^(1/2),x)
 

Output:

int(1/(x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b 
*d*f*x^3)^(1/2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}d x \] Input:

int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2 
),x)
 

Output:

int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*e + a*c*f*x + a*d*e*x 
 + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3),x)