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

Optimal. Leaf size=291 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac{f \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{a d x} \]

[Out]

-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d) +
(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^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (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^(3/2)*Sqrt[c*
d + b*Sqrt[d]*Sqrt[f] + a*f])

________________________________________________________________________________________

Rubi [A]  time = 0.65675, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {6725, 730, 724, 206, 984} \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac{f \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{a d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d) +
(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^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (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^(3/2)*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 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 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 984

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

Rubi steps

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

Mathematica [A]  time = 1.06665, size = 325, normalized size = 1.12 \[ \frac{\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{a^{3/2}}+\frac{f \tanh ^{-1}\left (\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \left (\sqrt{d}-\sqrt{f} x\right )+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{2 b \sqrt{d}}{a \sqrt{a+x (b+c x)}}-\frac{2 c \sqrt{d} x}{a \sqrt{a+x (b+c x)}}-\frac{2 \sqrt{d}}{x \sqrt{a+x (b+c x)}}}{2 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b*Sqrt[d])/(a*Sqrt[a + x*(b + c*x)]) - (2*Sqrt[d])/(x*Sqrt[a + x*(b + c*x)]) - (2*c*Sqrt[d]*x)/(a*Sqrt[a
+ x*(b + c*x)]) + (b*Sqrt[d]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/a^(3/2) + (f*ArcTanh[(b*S
qrt[d] + 2*a*Sqrt[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] + (f*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - 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])/(2*d^(3
/2))

________________________________________________________________________________________

Maple [A]  time = 0.265, size = 427, normalized size = 1.5 \begin{align*} -{\frac{f}{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{df}}}{\frac{1}{\sqrt{{\frac{1}{f} \left ( -b\sqrt{df}+af+cd \right ) }}}}}-{\frac{1}{adx}\sqrt{c{x}^{2}+bx+a}}+{\frac{b}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}+{\frac{f}{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{df}}}{\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^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)

[Out]

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

________________________________________________________________________________________

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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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^2), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- d x^{2} \sqrt{a + b x + c x^{2}} + f x^{4} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-d*x**2*sqrt(a + b*x + c*x**2) + f*x**4*sqrt(a + b*x + c*x**2)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError