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

Optimal. Leaf size=286 \[ \frac{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 1.54478, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 176.368, size = 257, normalized size = 0.9 \[ - \frac{\sqrt{a + b x + c x^{2}}}{d x} + \frac{\sqrt{a f - b \sqrt{d} \sqrt{f} + c d} \operatorname{atanh}{\left (\frac{- 2 a \sqrt{f} + b \sqrt{d} + x \left (- b \sqrt{f} + 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f - b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 d^{\frac{3}{2}}} - \frac{\sqrt{a f + b \sqrt{d} \sqrt{f} + c d} \operatorname{atanh}{\left (\frac{- 2 a \sqrt{f} - b \sqrt{d} + x \left (- b \sqrt{f} - 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f + b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 d^{\frac{3}{2}}} - \frac{b \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.39548, size = 379, normalized size = 1.33 \[ \frac{-\log \left (\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+\log \left (\sqrt{d} \sqrt{f}+f x\right ) \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}-\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )+\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )-\frac{2 \sqrt{d} \sqrt{a+x (b+c x)}}{x}-\frac{b \sqrt{d} \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}+\frac{b \sqrt{d} \log (x)}{\sqrt{a}}}{2 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 1819, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d/a/x*(c*x^2+b*x+a)^(3/2)+1/d*b/a*(c*x^2+b*x+a)^(1/2)-1/2/d*b/a^(1/2)*ln((2*a
+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/d*c/a*(c*x^2+b*x+a)^(1/2)*x+1/d*c^(1/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/2*f/d/(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)-1/2/d*ln((1/2*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(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))*c^(1/2)-1/4*f/d/(d*f)^(1/2)*ln((1/2*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^
(1/2)/f))/c^(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))/c^(1/2)*b+1/2/d/((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))*b+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))*a+1/2/(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))*c
+1/2*f/d/(d*f)^(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)-1/2/d*ln((1/2/f*(-2*c*(d*f)^(1/2)+b
*f)+c*(x+(d*f)^(1/2)/f))/c^(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))*c^(1/2)+1/4*f/d/(d*f)^
(1/2)*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(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))/c^(1/2)*b+1/2/d/(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))*b-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))*a-1/2/(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))*c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 25.1813, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(sqrt(a)*d*x*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log((2*b*c*x + 2*s
qrt(c*x^2 + b*x + a)*b*d*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3) + b^2 + (b*
d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - sqrt(a)*d*x*sqrt((d^3*sqrt(b^2*f/d^5) + c
*d + a*f)/d^3)*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt((d^3*sqrt(b^2*f/d
^5) + c*d + a*f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) + sqrt(a)*
d*x*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3)*log((2*b*c*x + 2*sqrt(c*x^2 + b
*x + a)*b*d*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3) + b^2 - (b*d^2*x + 2*a*
d^2)*sqrt(b^2*f/d^5))/x) - sqrt(a)*d*x*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d
^3)*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d
- a*f)/d^3) + b^2 - (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) + b*x*log((4*(a*b*x
+ 2*a^2)*sqrt(c*x^2 + b*x + a) - (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*sqrt(a))/
x^2) - 4*sqrt(c*x^2 + b*x + a)*sqrt(a))/(sqrt(a)*d*x), 1/4*(sqrt(-a)*d*x*sqrt((d
^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*d*
sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2
*f/d^5))/x) - sqrt(-a)*d*x*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log((2*b*
c*x - 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3) +
b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) + sqrt(-a)*d*x*sqrt(-(d^3*sqrt(b^2
*f/d^5) - c*d - a*f)/d^3)*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt(-(d^3*
sqrt(b^2*f/d^5) - c*d - a*f)/d^3) + b^2 - (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x
) - sqrt(-a)*d*x*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3)*log((2*b*c*x - 2*s
qrt(c*x^2 + b*x + a)*b*d*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3) + b^2 - (b
*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - 2*b*x*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(s
qrt(c*x^2 + b*x + a)*a)) - 4*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*d*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError