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

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

_______________________________________________________________________________________

Rubi [A]  time = 1.36445, 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 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac{f \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{a d x} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 131.872, size = 262, normalized size = 0.9 \[ - \frac{f \operatorname{atanh}{\left (\frac{- 2 a \sqrt{f} - b \sqrt{d} + x \left (- b \sqrt{f} - 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f + b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 d^{\frac{3}{2}} \sqrt{a f + b \sqrt{d} \sqrt{f} + c d}} - \frac{f \operatorname{atanh}{\left (\frac{2 a \sqrt{f} - b \sqrt{d} + x \left (b \sqrt{f} - 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f - b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 d^{\frac{3}{2}} \sqrt{a f - b \sqrt{d} \sqrt{f} + c d}} - \frac{\sqrt{a + b x + c x^{2}}}{a d x} + \frac{b \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 a^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)

[Out]

-f*atanh((-2*a*sqrt(f) - b*sqrt(d) + x*(-b*sqrt(f) - 2*c*sqrt(d)))/(2*sqrt(a + b
*x + c*x**2)*sqrt(a*f + b*sqrt(d)*sqrt(f) + c*d)))/(2*d**(3/2)*sqrt(a*f + b*sqrt
(d)*sqrt(f) + c*d)) - f*atanh((2*a*sqrt(f) - b*sqrt(d) + x*(b*sqrt(f) - 2*c*sqrt
(d)))/(2*sqrt(a + b*x + c*x**2)*sqrt(a*f - b*sqrt(d)*sqrt(f) + c*d)))/(2*d**(3/2
)*sqrt(a*f - b*sqrt(d)*sqrt(f) + c*d)) - sqrt(a + b*x + c*x**2)/(a*d*x) + b*atan
h((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(2*a**(3/2)*d)

_______________________________________________________________________________________

Mathematica [A]  time = 2.27305, size = 432, normalized size = 1.48 \[ \frac{\frac{b \sqrt{d} \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{a^{3/2}}-\frac{b \sqrt{d} \log (x)}{a^{3/2}}-\frac{f \log \left (\sqrt{d} \sqrt{f}-f x\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{f \log \left (\sqrt{d} \sqrt{f}+f x\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{f \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{2 b \sqrt{d}}{a \sqrt{a+x (b+c x)}}-\frac{2 c \sqrt{d} x}{a \sqrt{a+x (b+c x)}}-\frac{2 \sqrt{d}}{x \sqrt{a+x (b+c x)}}}{2 d^{3/2}} \]

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]*Log[x])/a^(3/2) - (f*
Log[Sqrt[d]*Sqrt[f] - f*x])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] + (f*Log[Sqrt[d]
*Sqrt[f] + f*x])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + (b*Sqrt[d]*Log[2*a + b*x
+ 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/a^(3/2) - (f*Log[Sqrt[d]*(-(b*Sqrt[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]*S
qrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + (f*Log[Sqrt[d]*(b*
(Sqrt[d] + Sqrt[f]*x) + 2*(a*Sqrt[f] + c*Sqrt[d]*x + 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 = 0.024, size = 427, normalized size = 1.5 \[ -{\frac{1}{adx}\sqrt{c{x}^{2}+bx+a}}+{\frac{b}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}+{\frac{f}{2\,d}\ln \left ({1 \left ( 2\,{\frac{b\sqrt{df}+fa+cd}{f}}+{\frac{1}{f} \left ( 2\,c\sqrt{df}+bf \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{b\sqrt{df}+fa+cd}{f}}}\sqrt{ \left ( x-{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{2\,c\sqrt{df}+bf}{f} \left ( x-{\frac{\sqrt{df}}{f}} \right ) }+{\frac{b\sqrt{df}+fa+cd}{f}}} \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{df}}}{\frac{1}{\sqrt{{\frac{1}{f} \left ( b\sqrt{df}+fa+cd \right ) }}}}}-{\frac{f}{2\,d}\ln \left ({1 \left ( 2\,{\frac{-b\sqrt{df}+fa+cd}{f}}+{\frac{1}{f} \left ( -2\,c\sqrt{df}+bf \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{-b\sqrt{df}+fa+cd}{f}}}\sqrt{ \left ( x+{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{-2\,c\sqrt{df}+bf}{f} \left ( x+{\frac{\sqrt{df}}{f}} \right ) }+{\frac{-b\sqrt{df}+fa+cd}{f}}} \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{df}}}{\frac{1}{\sqrt{{\frac{1}{f} \left ( -b\sqrt{df}+fa+cd \right ) }}}}} \]

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]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(c*x^2 + b*x + a)*(f*x^2 - d)*x^2),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

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/XCAS [A]  time = 0.65815, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x