3.683 \(\int \frac{\sqrt{c+d x^2}}{x^3 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=113 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2}-\frac{\sqrt{c+d x^2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.361708, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2}-\frac{\sqrt{c+d x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 46.9083, size = 97, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{2}}}{2 a x^{2}} - \frac{\sqrt{b} \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2}} - \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(1/2)/x**3/(b*x**2+a),x)

[Out]

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

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Mathematica [C]  time = 1.64559, size = 281, normalized size = 2.49 \[ -\frac{\sqrt{b} \sqrt{b c-a d} \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\sqrt{b} \left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{3/2}}\right )+\sqrt{b} \sqrt{b c-a d} \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\sqrt{b} \left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{3/2}}\right )+\frac{(a d-2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}+\frac{\log (x) (2 b c-a d)}{\sqrt{c}}+\frac{a \sqrt{c+d x^2}}{x^2}}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)),x]

[Out]

-((a*Sqrt[c + d*x^2])/x^2 + ((2*b*c - a*d)*Log[x])/Sqrt[c] + ((-2*b*c + a*d)*Log
[c + Sqrt[c]*Sqrt[c + d*x^2]])/Sqrt[c] + Sqrt[b]*Sqrt[b*c - a*d]*Log[(2*a^2*(Sqr
t[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b]*(b*c - a*d)^
(3/2)*(I*Sqrt[a] + Sqrt[b]*x))] + Sqrt[b]*Sqrt[b*c - a*d]*Log[(2*a^2*(Sqrt[b]*c
+ I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b]*(b*c - a*d)^(3/2)*(
(-I)*Sqrt[a] + Sqrt[b]*x))])/(2*a^2)

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Maple [B]  time = 0.018, size = 1054, normalized size = 9.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(1/2)/x^3/(b*x^2+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^3), x)

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Fricas [A]  time = 0.340166, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b^{2} c - a b d} \sqrt{c} x^{2} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) -{\left (2 \, b c - a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} \sqrt{c} x^{2}}, \frac{2 \,{\left (2 \, b c - a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) + \sqrt{b^{2} c - a b d} \sqrt{-c} x^{2} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} x^{2}}, -\frac{2 \, \sqrt{-b^{2} c + a b d} \sqrt{c} x^{2} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}\right ) +{\left (2 \, b c - a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} \sqrt{c} x^{2}}, -\frac{\sqrt{-b^{2} c + a b d} \sqrt{-c} x^{2} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}\right ) -{\left (2 \, b c - a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) + \sqrt{d x^{2} + c} a \sqrt{-c}}{2 \, a^{2} \sqrt{-c} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="fricas")

[Out]

[1/4*(sqrt(b^2*c - a*b*d)*sqrt(c)*x^2*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d +
a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c
- a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (2*b*c - a*d)*x^2*log(-
((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2) - 2*sqrt(d*x^2 + c)*a*sqrt(c)
)/(a^2*sqrt(c)*x^2), 1/4*(2*(2*b*c - a*d)*x^2*arctan(sqrt(-c)/sqrt(d*x^2 + c)) +
sqrt(b^2*c - a*b*d)*sqrt(-c)*x^2*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2
*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c - a*
b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*sqrt(d*x^2 + c)*a*sqrt(-c
))/(a^2*sqrt(-c)*x^2), -1/4*(2*sqrt(-b^2*c + a*b*d)*sqrt(c)*x^2*arctan(1/2*(b*d*
x^2 + 2*b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c))) + (2*b*c - a*d)*x^2*l
og(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2) + 2*sqrt(d*x^2 + c)*a*sqr
t(c))/(a^2*sqrt(c)*x^2), -1/2*(sqrt(-b^2*c + a*b*d)*sqrt(-c)*x^2*arctan(1/2*(b*d
*x^2 + 2*b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c))) - (2*b*c - a*d)*x^2*
arctan(sqrt(-c)/sqrt(d*x^2 + c)) + sqrt(d*x^2 + c)*a*sqrt(-c))/(a^2*sqrt(-c)*x^2
)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{x^{3} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**(1/2)/x**3/(b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.25255, size = 163, normalized size = 1.44 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c - a b d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{2} + c}}{a d^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="giac")

[Out]

1/2*d^2*(2*(b^2*c - a*b*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(
-b^2*c + a*b*d)*a^2*d^2) - (2*b*c - a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*s
qrt(-c)*d^2) - sqrt(d*x^2 + c)/(a*d^2*x^2))