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

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

[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 = 0.07, 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 = {996, 214} \begin {gather*} -\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}} \end {gather*}

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 214

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

Rule 996

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) \text {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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.43, size = 162, normalized size = 1.98 \begin {gather*} e \text {RootSum}\left [-b d e^2+a e^3+b d^2 f+2 b d e \sqrt {f} \text {$\#$1}-4 a e^2 \sqrt {f} \text {$\#$1}+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-2 b e \sqrt {f} \text {$\#$1}^3+b f \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )}{b d \sqrt {f}-2 a e \sqrt {f}+b e \text {$\#$1}-b \sqrt {f} \text {$\#$1}^2}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

e*RootSum[-(b*d*e^2) + a*e^3 + b*d^2*f + 2*b*d*e*Sqrt[f]*#1 - 4*a*e^2*Sqrt[f]*#1 + b*e^2*#1^2 - 2*b*d*f*#1^2 +
4*a*e*f*#1^2 - 2*b*e*Sqrt[f]*#1^3 + b*f*#1^4 & , Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]/(b*d*Sqrt[f]
- 2*a*e*Sqrt[f] + b*e*#1 - b*Sqrt[f]*#1^2) & ]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(68)=136\).
time = 0.27, size = 491, normalized size = 5.99

method	result	size
default	\(e \left (\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{\sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{\sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}\right )\)	\(491\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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-%e*b>0)', see `assume?`
for more det

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (72) = 144\).
time = 2.43, size = 1062, normalized size = 12.95 \begin {gather*} \left [\frac {1}{2} \, \sqrt {-\frac {e}{4 \, a b d f + a b e^{2} - {\left (b^{2} d + 4 \, a^{2} f\right )} e}} \log \left (\frac {16 \, b^{2} d^{2} f^{4} x^{4} + 4 \, {\left (32 \, a b^{2} d^{2} f^{4} x^{3} + {\left (a b^{2} x - a^{2} b\right )} e^{6} + {\left (3 \, a b^{2} f x^{2} + 3 \, a b^{2} d + 4 \, a^{3} f - {\left (b^{3} d + 14 \, a^{2} b f\right )} x\right )} e^{5} + {\left (2 \, a b^{2} f^{2} x^{3} - 2 \, b^{3} d^{2} - 16 \, a^{2} b d f - 3 \, {\left (b^{3} d f + 12 \, a^{2} b f^{2}\right )} x^{2} + 2 \, {\left (11 \, a b^{2} d f + 20 \, a^{3} f^{2}\right )} x\right )} e^{4} + 2 \, {\left (6 \, a b^{2} d^{2} f + 8 \, a^{3} d f^{2} - {\left (b^{3} d f^{2} + 12 \, a^{2} b f^{3}\right )} x^{3} + 24 \, {\left (a b^{2} d f^{2} + 2 \, a^{3} f^{3}\right )} x^{2} - 4 \, {\left (b^{3} d^{2} f + 10 \, a^{2} b d f^{2}\right )} x\right )} e^{3} - 4 \, {\left (4 \, a^{2} b d^{2} f^{2} - 8 \, {\left (a b^{2} d f^{3} + 2 \, a^{3} f^{4}\right )} x^{3} + 3 \, {\left (b^{3} d^{2} f^{2} + 12 \, a^{2} b d f^{3}\right )} x^{2} - 2 \, {\left (5 \, a b^{2} d^{2} f^{2} + 4 \, a^{3} d f^{3}\right )} x\right )} e^{2} + 8 \, {\left (6 \, a b^{2} d^{2} f^{3} x^{2} - 4 \, a^{2} b d^{2} f^{3} x - {\left (b^{3} d^{2} f^{3} + 12 \, a^{2} b d f^{4}\right )} x^{3}\right )} e\right )} \sqrt {f x^{2} + x e + d} \sqrt {-\frac {e}{4 \, a b d f + a b e^{2} - {\left (b^{2} d + 4 \, a^{2} f\right )} e}} + {\left (b^{2} x^{2} - 6 \, a b x + a^{2}\right )} e^{6} + 2 \, {\left (b^{2} f x^{3} - 19 \, a b f x^{2} - 4 \, a b d + 4 \, {\left (b^{2} d + 4 \, a^{2} f\right )} x\right )} e^{5} + {\left (b^{2} f^{2} x^{4} - 64 \, a b f^{2} x^{3} - 80 \, a b d f x + 8 \, b^{2} d^{2} + 24 \, a^{2} d f + 32 \, {\left (b^{2} d f + 5 \, a^{2} f^{2}\right )} x^{2}\right )} e^{4} - 16 \, {\left (2 \, a b f^{3} x^{4} + 13 \, a b d f^{2} x^{2} + 2 \, a b d^{2} f - {\left (3 \, b^{2} d f^{2} + 16 \, a^{2} f^{3}\right )} x^{3} - 2 \, {\left (b^{2} d^{2} f + 4 \, a^{2} d f^{2}\right )} x\right )} e^{3} - 8 \, {\left (32 \, a b d f^{3} x^{3} + 12 \, a b d^{2} f^{2} x - 2 \, a^{2} d^{2} f^{2} - {\left (3 \, b^{2} d f^{3} + 16 \, a^{2} f^{4}\right )} x^{4} - 2 \, {\left (3 \, b^{2} d^{2} f^{2} + 8 \, a^{2} d f^{3}\right )} x^{2}\right )} e^{2} - 32 \, {\left (4 \, a b d f^{4} x^{4} - b^{2} d^{2} f^{3} x^{3} + 3 \, a b d^{2} f^{3} x^{2}\right )} e}{b^{2} f^{2} x^{4} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (b^{2} f x^{3} + a b f x^{2}\right )} e}\right ), \frac {\arctan \left (\frac {{\left (4 \, b d f^{2} x^{2} + {\left (b x - a\right )} e^{3} + {\left (b f x^{2} - 8 \, a f x + 2 \, b d\right )} e^{2} - 4 \, {\left (2 \, a f^{2} x^{2} - b d f x + a d f\right )} e\right )} \sqrt {f x^{2} + x e + d} e^{\frac {1}{2}}}{2 \, \sqrt {4 \, a b d f + a b e^{2} - {\left (b^{2} d + 4 \, a^{2} f\right )} e} {\left (x e^{3} + {\left (3 \, f x^{2} + d\right )} e^{2} + 2 \, {\left (f^{2} x^{3} + d f x\right )} e\right )}}\right ) e^{\frac {1}{2}}}{\sqrt {4 \, a b d f + a b e^{2} - {\left (b^{2} d + 4 \, a^{2} f\right )} e}}\right ] \end {gather*}

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/(4*a*b*d*f + a*b*e^2 - (b^2*d + 4*a^2*f)*e))*log((16*b^2*d^2*f^4*x^4 + 4*(32*a*b^2*d^2*f^4*x^3 +
(a*b^2*x - a^2*b)*e^6 + (3*a*b^2*f*x^2 + 3*a*b^2*d + 4*a^3*f - (b^3*d + 14*a^2*b*f)*x)*e^5 + (2*a*b^2*f^2*x^3
- 2*b^3*d^2 - 16*a^2*b*d*f - 3*(b^3*d*f + 12*a^2*b*f^2)*x^2 + 2*(11*a*b^2*d*f + 20*a^3*f^2)*x)*e^4 + 2*(6*a*b^
2*d^2*f + 8*a^3*d*f^2 - (b^3*d*f^2 + 12*a^2*b*f^3)*x^3 + 24*(a*b^2*d*f^2 + 2*a^3*f^3)*x^2 - 4*(b^3*d^2*f + 10*
a^2*b*d*f^2)*x)*e^3 - 4*(4*a^2*b*d^2*f^2 - 8*(a*b^2*d*f^3 + 2*a^3*f^4)*x^3 + 3*(b^3*d^2*f^2 + 12*a^2*b*d*f^3)*
x^2 - 2*(5*a*b^2*d^2*f^2 + 4*a^3*d*f^3)*x)*e^2 + 8*(6*a*b^2*d^2*f^3*x^2 - 4*a^2*b*d^2*f^3*x - (b^3*d^2*f^3 + 1
2*a^2*b*d*f^4)*x^3)*e)*sqrt(f*x^2 + x*e + d)*sqrt(-e/(4*a*b*d*f + a*b*e^2 - (b^2*d + 4*a^2*f)*e)) + (b^2*x^2 -
6*a*b*x + a^2)*e^6 + 2*(b^2*f*x^3 - 19*a*b*f*x^2 - 4*a*b*d + 4*(b^2*d + 4*a^2*f)*x)*e^5 + (b^2*f^2*x^4 - 64*a
*b*f^2*x^3 - 80*a*b*d*f*x + 8*b^2*d^2 + 24*a^2*d*f + 32*(b^2*d*f + 5*a^2*f^2)*x^2)*e^4 - 16*(2*a*b*f^3*x^4 + 1
3*a*b*d*f^2*x^2 + 2*a*b*d^2*f - (3*b^2*d*f^2 + 16*a^2*f^3)*x^3 - 2*(b^2*d^2*f + 4*a^2*d*f^2)*x)*e^3 - 8*(32*a*
b*d*f^3*x^3 + 12*a*b*d^2*f^2*x - 2*a^2*d^2*f^2 - (3*b^2*d*f^3 + 16*a^2*f^4)*x^4 - 2*(3*b^2*d^2*f^2 + 8*a^2*d*f
^3)*x^2)*e^2 - 32*(4*a*b*d*f^4*x^4 - b^2*d^2*f^3*x^3 + 3*a*b*d^2*f^3*x^2)*e)/(b^2*f^2*x^4 + (b^2*x^2 + 2*a*b*x
+ a^2)*e^2 + 2*(b^2*f*x^3 + a*b*f*x^2)*e)), arctan(1/2*(4*b*d*f^2*x^2 + (b*x - a)*e^3 + (b*f*x^2 - 8*a*f*x +
2*b*d)*e^2 - 4*(2*a*f^2*x^2 - b*d*f*x + a*d*f)*e)*sqrt(f*x^2 + x*e + d)*e^(1/2)/(sqrt(4*a*b*d*f + a*b*e^2 - (b
^2*d + 4*a^2*f)*e)*(x*e^3 + (3*f*x^2 + d)*e^2 + 2*(f^2*x^3 + d*f*x)*e)))*e^(1/2)/sqrt(4*a*b*d*f + a*b*e^2 - (b
^2*d + 4*a^2*f)*e)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (72) = 144\).
time = 6.49, size = 847, normalized size = 10.33 \begin {gather*} \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}} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________