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

Optimal. Leaf size=267 \[ -\frac{\sqrt{f} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\sqrt{f} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d} \]

[Out]

-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(Sqrt[a]*d)) - (Sqrt[f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt
[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c
*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (Sqrt[f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*S
qrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])

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Rubi [A]  time = 0.663489, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6725, 724, 206, 1033} \[ -\frac{\sqrt{f} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\sqrt{f} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(Sqrt[a]*d)) - (Sqrt[f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt
[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c
*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (Sqrt[f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*S
qrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1033

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)
/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[-(a*c)]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx &=\int \left (\frac{1}{d x \sqrt{a+b x+c x^2}}-\frac{f x}{d \sqrt{a+b x+c x^2} \left (-d+f x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{d}-\frac{f \int \frac{x}{\sqrt{a+b x+c x^2} \left (-d+f x^2\right )} \, dx}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{d}-\frac{f \int \frac{1}{\left (-\sqrt{d} \sqrt{f}+f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d}-\frac{f \int \frac{1}{\left (\sqrt{d} \sqrt{f}+f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d}\\ &=-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d}+\frac{f \operatorname{Subst}\left (\int \frac{1}{4 c d f-4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{-b \sqrt{d} \sqrt{f}+2 a f-\left (2 c \sqrt{d} \sqrt{f}-b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{d}+\frac{f \operatorname{Subst}\left (\int \frac{1}{4 c d f+4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{b \sqrt{d} \sqrt{f}+2 a f-\left (-2 c \sqrt{d} \sqrt{f}-b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{d}\\ &=-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{b \sqrt{d}-2 a \sqrt{f}+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x}{2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 d \sqrt{c d-b \sqrt{d} \sqrt{f}+a f}}+\frac{\sqrt{f} \tanh ^{-1}\left (\frac{b \sqrt{d}+2 a \sqrt{f}+\left (2 c \sqrt{d}+b \sqrt{f}\right ) x}{2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 d \sqrt{c d+b \sqrt{d} \sqrt{f}+a f}}\\ \end{align*}

Mathematica [A]  time = 0.472324, size = 252, normalized size = 0.94 \[ \frac{\sqrt{f} \left (\frac{\tanh ^{-1}\left (\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )-\frac{2 \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{\sqrt{a}}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + Sqrt[f]*(-(ArcTanh[(b*Sqrt[d] - 2*a*Sqr
t[f] + 2*c*Sqrt[d]*x - b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d -
b*Sqrt[d]*Sqrt[f] + a*f]) + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sq
rt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(2*d)

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Maple [A]  time = 0.28, size = 391, normalized size = 1.5 \begin{align*} -{\frac{1}{d}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{1}{2\,d}\ln \left ({ \left ( 2\,{\frac{-b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( -2\,c\sqrt{df}+bf \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{-b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x+{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{-2\,c\sqrt{df}+bf}{f} \left ( x+{\frac{\sqrt{df}}{f}} \right ) }+{\frac{-b\sqrt{df}+af+cd}{f}}} \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{1}{f} \left ( -b\sqrt{df}+af+cd \right ) }}}}}+{\frac{1}{2\,d}\ln \left ({ \left ( 2\,{\frac{b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( 2\,c\sqrt{df}+bf \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x-{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{2\,c\sqrt{df}+bf}{f} \left ( x-{\frac{\sqrt{df}}{f}} \right ) }+{\frac{b\sqrt{df}+af+cd}{f}}} \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{1}{f} \left ( b\sqrt{df}+af+cd \right ) }}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (f x^{2} - d\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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/x/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- d x \sqrt{a + b x + c x^{2}} + f x^{3} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2