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

Optimal. Leaf size=82 \[ -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}} \]

[Out]

-2*arctanh((2*f*x+e)*(-a*e+b*d)^(1/2)/e^(1/2)/(-4*a*f+b*e)^(1/2)/(f*x^2+e*x+d)^(1/2))*e^(1/2)/(-a*e+b*d)^(1/2)
/(-4*a*f+b*e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {982, 208} \[ -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e]*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(Sqrt[b*
d - a*e]*Sqrt[b*e - 4*a*f])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 982

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx &=-\left ((2 e) \operatorname {Subst}\left (\int \frac {1}{e (b e-4 a f)-(b d-a e) x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )\right )\\ &=-\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.39, size = 178, normalized size = 2.17 \[ \frac {\sqrt {e} \left (\tanh ^{-1}\left (\frac {-\sqrt {e} (e+2 f x) \sqrt {b e-4 a f}-\sqrt {b} \left (e^2-4 d f\right )}{4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b} \left (e^2-4 d f\right )-\sqrt {e} (e+2 f x) \sqrt {b e-4 a f}}{4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}}\right )\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[e]*(ArcTanh[(-(Sqrt[b]*(e^2 - 4*d*f)) - Sqrt[e]*Sqrt[b*e - 4*a*f]*(e + 2*f*x))/(4*Sqrt[b*d - a*e]*f*Sqrt
[d + x*(e + f*x)])] + ArcTanh[(Sqrt[b]*(e^2 - 4*d*f) - Sqrt[e]*Sqrt[b*e - 4*a*f]*(e + 2*f*x))/(4*Sqrt[b*d - a*
e]*f*Sqrt[d + x*(e + f*x)])]))/(Sqrt[b*d - a*e]*Sqrt[b*e - 4*a*f])

________________________________________________________________________________________

fricas [B]  time = 1.71, size = 1079, normalized size = 13.16 \[ \left [\frac {1}{2} \, \sqrt {\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}} \log \left (\frac {8 \, b^{2} d^{2} e^{4} - 8 \, a b d e^{5} + a^{2} e^{6} + 16 \, a^{2} d^{2} e^{2} f^{2} + {\left (b^{2} e^{4} f^{2} + 16 \, {\left (b^{2} d^{2} - 8 \, a b d e + 8 \, a^{2} e^{2}\right )} f^{4} + 8 \, {\left (3 \, b^{2} d e^{2} - 4 \, a b e^{3}\right )} f^{3}\right )} x^{4} + 2 \, {\left (b^{2} e^{5} f + 16 \, {\left (b^{2} d^{2} e - 8 \, a b d e^{2} + 8 \, a^{2} e^{3}\right )} f^{3} + 8 \, {\left (3 \, b^{2} d e^{3} - 4 \, a b e^{4}\right )} f^{2}\right )} x^{3} + {\left (b^{2} e^{6} - 32 \, {\left (3 \, a b d^{2} e - 4 \, a^{2} d e^{2}\right )} f^{3} + 16 \, {\left (3 \, b^{2} d^{2} e^{2} - 13 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} f^{2} + 2 \, {\left (16 \, b^{2} d e^{4} - 19 \, a b e^{5}\right )} f\right )} x^{2} - 8 \, {\left (4 \, a b d^{2} e^{3} - 3 \, a^{2} d e^{4}\right )} f + 2 \, {\left (4 \, b^{2} d e^{5} - 3 \, a b e^{6} - 16 \, {\left (3 \, a b d^{2} e^{2} - 4 \, a^{2} d e^{3}\right )} f^{2} + 8 \, {\left (2 \, b^{2} d^{2} e^{3} - 5 \, a b d e^{4} + 2 \, a^{2} e^{5}\right )} f\right )} x - 4 \, {\left (2 \, b^{3} d^{2} e^{4} - 3 \, a b^{2} d e^{5} + a^{2} b e^{6} - 2 \, {\left (16 \, {\left (a b^{2} d^{2} - 3 \, a^{2} b d e + 2 \, a^{3} e^{2}\right )} f^{4} - 4 \, {\left (b^{3} d^{2} e - 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} f^{3} - {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} f^{2}\right )} x^{3} + 16 \, {\left (a^{2} b d^{2} e^{2} - a^{3} d e^{3}\right )} f^{2} - 3 \, {\left (16 \, {\left (a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3}\right )} f^{3} - 4 \, {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} f^{2} - {\left (b^{3} d e^{4} - a b^{2} e^{5}\right )} f\right )} x^{2} - 4 \, {\left (3 \, a b^{2} d^{2} e^{3} - 4 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} f + {\left (b^{3} d e^{5} - a b^{2} e^{6} + 32 \, {\left (a^{2} b d^{2} e - a^{3} d e^{2}\right )} f^{3} - 40 \, {\left (a b^{2} d^{2} e^{2} - 2 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} f^{2} + 2 \, {\left (4 \, b^{3} d^{2} e^{3} - 11 \, a b^{2} d e^{4} + 7 \, a^{2} b e^{5}\right )} f\right )} x\right )} \sqrt {f x^{2} + e x + d} \sqrt {\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}}}{b^{2} f^{2} x^{4} + 2 \, b^{2} e f x^{3} + 2 \, a b e^{2} x + a^{2} e^{2} + {\left (b^{2} e^{2} + 2 \, a b e f\right )} x^{2}}\right ), -\sqrt {-\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}} \arctan \left (-\frac {{\left (2 \, b d e^{2} - a e^{3} - 4 \, a d e f + {\left (b e^{2} f + 4 \, {\left (b d - 2 \, a e\right )} f^{2}\right )} x^{2} + {\left (b e^{3} + 4 \, {\left (b d e - 2 \, a e^{2}\right )} f\right )} x\right )} \sqrt {f x^{2} + e x + d} \sqrt {-\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}}}{2 \, {\left (2 \, e f^{2} x^{3} + 3 \, e^{2} f x^{2} + d e^{2} + {\left (e^{3} + 2 \, d e f\right )} x\right )}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*log((8*b^2*d^2*e^4 - 8*a*b*d*e^5 + a^2*e^6 + 16*a^2*d^2
*e^2*f^2 + (b^2*e^4*f^2 + 16*(b^2*d^2 - 8*a*b*d*e + 8*a^2*e^2)*f^4 + 8*(3*b^2*d*e^2 - 4*a*b*e^3)*f^3)*x^4 + 2*
(b^2*e^5*f + 16*(b^2*d^2*e - 8*a*b*d*e^2 + 8*a^2*e^3)*f^3 + 8*(3*b^2*d*e^3 - 4*a*b*e^4)*f^2)*x^3 + (b^2*e^6 -
32*(3*a*b*d^2*e - 4*a^2*d*e^2)*f^3 + 16*(3*b^2*d^2*e^2 - 13*a*b*d*e^3 + 10*a^2*e^4)*f^2 + 2*(16*b^2*d*e^4 - 19
*a*b*e^5)*f)*x^2 - 8*(4*a*b*d^2*e^3 - 3*a^2*d*e^4)*f + 2*(4*b^2*d*e^5 - 3*a*b*e^6 - 16*(3*a*b*d^2*e^2 - 4*a^2*
d*e^3)*f^2 + 8*(2*b^2*d^2*e^3 - 5*a*b*d*e^4 + 2*a^2*e^5)*f)*x - 4*(2*b^3*d^2*e^4 - 3*a*b^2*d*e^5 + a^2*b*e^6 -
2*(16*(a*b^2*d^2 - 3*a^2*b*d*e + 2*a^3*e^2)*f^4 - 4*(b^3*d^2*e - 4*a*b^2*d*e^2 + 3*a^2*b*e^3)*f^3 - (b^3*d*e^
3 - a*b^2*e^4)*f^2)*x^3 + 16*(a^2*b*d^2*e^2 - a^3*d*e^3)*f^2 - 3*(16*(a*b^2*d^2*e - 3*a^2*b*d*e^2 + 2*a^3*e^3)
*f^3 - 4*(b^3*d^2*e^2 - 4*a*b^2*d*e^3 + 3*a^2*b*e^4)*f^2 - (b^3*d*e^4 - a*b^2*e^5)*f)*x^2 - 4*(3*a*b^2*d^2*e^3
- 4*a^2*b*d*e^4 + a^3*e^5)*f + (b^3*d*e^5 - a*b^2*e^6 + 32*(a^2*b*d^2*e - a^3*d*e^2)*f^3 - 40*(a*b^2*d^2*e^2
- 2*a^2*b*d*e^3 + a^3*e^4)*f^2 + 2*(4*b^3*d^2*e^3 - 11*a*b^2*d*e^4 + 7*a^2*b*e^5)*f)*x)*sqrt(f*x^2 + e*x + d)*
sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f)))/(b^2*f^2*x^4 + 2*b^2*e*f*x^3 + 2*a*b*e^2*x + a^2*e^2 + (b^2
*e^2 + 2*a*b*e*f)*x^2)), -sqrt(-e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*arctan(-1/2*(2*b*d*e^2 - a*e^3 -
4*a*d*e*f + (b*e^2*f + 4*(b*d - 2*a*e)*f^2)*x^2 + (b*e^3 + 4*(b*d*e - 2*a*e^2)*f)*x)*sqrt(f*x^2 + e*x + d)*sqr
t(-e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))/(2*e*f^2*x^3 + 3*e^2*f*x^2 + d*e^2 + (e^3 + 2*d*e*f)*x))]

________________________________________________________________________________________

giac [B]  time = 3.04, size = 847, normalized size = 10.33 \[ \frac {\sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} \log \left ({\left | -4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} b d f^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} a f^{2} e - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} b d f^{\frac {3}{2}} e + 4 \, b d^{2} f^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} b f e^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} a f^{\frac {3}{2}} e^{2} + 4 \, \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} f^{\frac {3}{2}} - 3 \, b d f e^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} b \sqrt {f} e^{3} + 4 \, \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} f e + 2 \, a f e^{3} + \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} \sqrt {f} e^{2} \right |}\right )}{4 \, a b d f - b^{2} d e - 4 \, a^{2} f e + a b e^{2}} - \frac {\sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} \log \left ({\left | -4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} b d f^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} a f^{2} e - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} b d f^{\frac {3}{2}} e + 4 \, b d^{2} f^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} b f e^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} a f^{\frac {3}{2}} e^{2} - 4 \, \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )}^{2} f^{\frac {3}{2}} - 3 \, b d f e^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} b \sqrt {f} e^{3} - 4 \, \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} f e + 2 \, a f e^{3} - \sqrt {-4 \, a b d f e + b^{2} d e^{2} + 4 \, a^{2} f e^{2} - a b e^{3}} \sqrt {f} e^{2} \right |}\right )}{4 \, a b d f - b^{2} d e - 4 \, a^{2} f e + a b e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2 - a*b*e^3)*log(abs(-4*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*b*d*f^
2 + 8*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*a*f^2*e - 4*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))*b*d*f^(3/2)*e + 4*
b*d^2*f^2 - (sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*b*f*e^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))*a*f^(3/2)*e^
2 + 4*sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2 - a*b*e^3)*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*f^(3/2) - 3
*b*d*f*e^2 - (sqrt(f)*x - sqrt(f*x^2 + x*e + d))*b*sqrt(f)*e^3 + 4*sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2
- a*b*e^3)*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))*f*e + 2*a*f*e^3 + sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2
- a*b*e^3)*sqrt(f)*e^2))/(4*a*b*d*f - b^2*d*e - 4*a^2*f*e + a*b*e^2) - sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f
*e^2 - a*b*e^3)*log(abs(-4*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + x*e + d
))^2*a*f^2*e - 4*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))*b*d*f^(3/2)*e + 4*b*d^2*f^2 - (sqrt(f)*x - sqrt(f*x^2 + x
*e + d))^2*b*f*e^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))*a*f^(3/2)*e^2 - 4*sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4
*a^2*f*e^2 - a*b*e^3)*(sqrt(f)*x - sqrt(f*x^2 + x*e + d))^2*f^(3/2) - 3*b*d*f*e^2 - (sqrt(f)*x - sqrt(f*x^2 +
x*e + d))*b*sqrt(f)*e^3 - 4*sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2 - a*b*e^3)*(sqrt(f)*x - sqrt(f*x^2 + x
*e + d))*f*e + 2*a*f*e^3 - sqrt(-4*a*b*d*f*e + b^2*d*e^2 + 4*a^2*f*e^2 - a*b*e^3)*sqrt(f)*e^2))/(4*a*b*d*f - b
^2*d*e - 4*a^2*f*e + a*b*e^2)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 491, normalized size = 5.99 \[ -\frac {e \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-\left (4 a f -b e \right ) b e}\, \left (x -\frac {-b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )^{2} f +\frac {\sqrt {-\left (4 a f -b e \right ) b e}\, \left (x -\frac {-b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}}\right )}{\sqrt {-\left (4 a f -b e \right ) b e}\, \sqrt {-\frac {a e -b d}{b}}}+\frac {e \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-\left (4 a f -b e \right ) b e}\, \left (x +\frac {b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )^{2} f -\frac {\sqrt {-\left (4 a f -b e \right ) b e}\, \left (x +\frac {b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {b e +\sqrt {-\left (4 a f -b e \right ) b e}}{2 b f}}\right )}{\sqrt {-\left (4 a f -b e \right ) b e}\, \sqrt {-\frac {a e -b d}{b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+
(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e
*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*
a*f-b*e))^(1/2))/b/f))+e/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)-(-b*e*(4*a*f-b*e))
^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(
1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x+
1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*f-b*e>0)', see `assume?` f
or more details)Is 4*a*f-b*e    positive, negative or zero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {f\,x^2+e\,x+d}\,\left (a+b\,x+\frac {b\,f\,x^2}{e}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e \int \frac {1}{a e \sqrt {d + e x + f x^{2}} + b e x \sqrt {d + e x + f x^{2}} + b f x^{2} \sqrt {d + e x + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________