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*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])

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Rubi [A]  time = 0.110623, antiderivative size = 82, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.388, 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])

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Maple [B]  time = 0.388, size = 491, normalized size = 6. \begin{align*} -{e\ln \left ({ \left ( -2\,{\frac{ae-bd}{b}}+{\frac{1}{b}\sqrt{-be \left ( 4\,af-be \right ) } \left ( x-{\frac{1}{2\,bf} \left ( -be+\sqrt{-be \left ( 4\,af-be \right ) } \right ) } \right ) }+2\,\sqrt{-{\frac{ae-bd}{b}}}\sqrt{ \left ( x-1/2\,{\frac{-be+\sqrt{-be \left ( 4\,af-be \right ) }}{bf}} \right ) ^{2}f+{\frac{\sqrt{-be \left ( 4\,af-be \right ) }}{b} \left ( x-1/2\,{\frac{-be+\sqrt{-be \left ( 4\,af-be \right ) }}{bf}} \right ) }-{\frac{ae-bd}{b}}} \right ) \left ( x-{\frac{1}{2\,bf} \left ( -be+\sqrt{-be \left ( 4\,af-be \right ) } \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{-be \left ( 4\,af-be \right ) }}}{\frac{1}{\sqrt{-{\frac{ae-bd}{b}}}}}}+{e\ln \left ({ \left ( -2\,{\frac{ae-bd}{b}}-{\frac{1}{b}\sqrt{-be \left ( 4\,af-be \right ) } \left ( x+{\frac{1}{2\,bf} \left ( be+\sqrt{-be \left ( 4\,af-be \right ) } \right ) } \right ) }+2\,\sqrt{-{\frac{ae-bd}{b}}}\sqrt{ \left ( x+1/2\,{\frac{be+\sqrt{-be \left ( 4\,af-be \right ) }}{bf}} \right ) ^{2}f-{\frac{\sqrt{-be \left ( 4\,af-be \right ) }}{b} \left ( x+1/2\,{\frac{be+\sqrt{-be \left ( 4\,af-be \right ) }}{bf}} \right ) }-{\frac{ae-bd}{b}}} \right ) \left ( x+{\frac{1}{2\,bf} \left ( be+\sqrt{-be \left ( 4\,af-be \right ) } \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{-be \left ( 4\,af-be \right ) }}}{\frac{1}{\sqrt{-{\frac{ae-bd}{b}}}}}} \end{align*}

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

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

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

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Fricas [B]  time = 2.13968, size = 2226, normalized size = 27.15 \begin{align*} \text{result too large to display} \end{align*}

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))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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{align*}

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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out