3.736 \(\int \frac{\sqrt{c+d x^2}}{x \left (a+b x^2\right )^2} \, dx\)
Optimal. Leaf size=119 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x^2}}{2 a \left (a+b x^2\right )} \]
[Out]
Sqrt[c + d*x^2]/(2*a*(a + b*x^2)) - (Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a
^2 + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^2*S
qrt[b]*Sqrt[b*c - a*d])
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Rubi [A] time = 0.341224, antiderivative size = 119, 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
\[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x^2}}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]
[Out]
Sqrt[c + d*x^2]/(2*a*(a + b*x^2)) - (Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a
^2 + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^2*S
qrt[b]*Sqrt[b*c - a*d])
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Rubi in Sympy [A] time = 39.6491, size = 99, normalized size = 0.83 \[ \frac{\sqrt{c + d x^{2}}}{2 a \left (a + b x^{2}\right )} - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2}} + \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \sqrt{b} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(1/2)/x/(b*x**2+a)**2,x)
[Out]
sqrt(c + d*x**2)/(2*a*(a + b*x**2)) - sqrt(c)*atanh(sqrt(c + d*x**2)/sqrt(c))/a*
*2 + (a*d/2 - b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(a**2*sqrt(b)*
sqrt(a*d - b*c))
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Mathematica [C] time = 1.02889, size = 313, normalized size = 2.63 \[ \frac{\frac{(2 b c-a d) \log \left (-\frac{4 a^2 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) \sqrt{b c-a d} (2 b c-a d)}\right )}{\sqrt{b} \sqrt{b c-a d}}+\frac{(2 b c-a d) \log \left (-\frac{4 a^2 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) \sqrt{b c-a d} (2 b c-a d)}\right )}{\sqrt{b} \sqrt{b c-a d}}+\frac{2 a \sqrt{c+d x^2}}{a+b x^2}-4 \sqrt{c} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+4 \sqrt{c} \log (x)}{4 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]
[Out]
((2*a*Sqrt[c + d*x^2])/(a + b*x^2) + 4*Sqrt[c]*Log[x] - 4*Sqrt[c]*Log[c + Sqrt[c
]*Sqrt[c + d*x^2]] + ((2*b*c - a*d)*Log[(-4*a^2*Sqrt[b]*(Sqrt[b]*c - I*Sqrt[a]*d
*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b*c - a*d]*(2*b*c - a*d)*(I*Sqrt[a]
+ Sqrt[b]*x))])/(Sqrt[b]*Sqrt[b*c - a*d]) + ((2*b*c - a*d)*Log[(-4*a^2*Sqrt[b]*
(Sqrt[b]*c + I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b*c - a*d]*
(2*b*c - a*d)*((-I)*Sqrt[a] + Sqrt[b]*x))])/(Sqrt[b]*Sqrt[b*c - a*d]))/(4*a^2)
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Maple [B] time = 0.023, size = 2585, normalized size = 21.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(1/2)/x/(b*x^2+a)^2,x)
[Out]
-1/a^2*c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/a^2*(d*x^2+c)^(1/2)-1/2/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)/b*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/b/(-(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/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/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)/b*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/b/(-(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/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-1/4/(-a*b)^(1/2)/a/(a*d-b*c)*b/(x-
1/b*(-a*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)^(3/2)+1/4/a*d/(a*d-b*c)*((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)-1/4/(-a*b)^(1/2)*d^(3/2)/(a*d-b*c)
*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))+1/4*d^2/(a*d-b*c)/b/
(-(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)))-1/4/a*d/(a*d-b*c)/(-(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+1/4/(-a*b)^(1/2)/a*d/(a*d-b*c)*b*((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+1/4/(-a*b)^(1/2)/a*d^(1/2)/(a*d-b*c)*b*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))*c+1/4/(-a*b)^(1/2)/a/(a*d-b*c)*b/(x+1/b*(-a*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)^(3/2)+1
/4/a*d/(a*d-b*c)*((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)+1/4/(-a*b)^(1/2)*d^(3/2)/(a*d-b*c)*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))+1/4*d^2/(a*d-b*c)/b/(-(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)))-1/4/a*d/(a*d-b*c)/(-(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-1/4/(-a*b)^(1/2)/a*d/(a*d-b*c)*b*((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-1/4/(-a*b)^(1/2)/a*d^
(1/2)/(a*d-b*c)*b*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))*c
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x), x)
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Fricas [A] time = 0.360969, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="fricas")
[Out]
[1/8*(4*sqrt(b^2*c - a*b*d)*(b*x^2 + a)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*
sqrt(c) + 2*c)/x^2) + 4*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c)*a - (2*a*b*c - a^2*d
+ (2*b^2*c - a*b*d)*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)*sqrt(b^2*c - a*b*d) - 4*(2*b^3*c^2 - 3*a*b^2*c*d
+ a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x^2)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 +
a^2)))/((a^2*b*x^2 + a^3)*sqrt(b^2*c - a*b*d)), -1/8*(8*sqrt(b^2*c - a*b*d)*(b*
x^2 + a)*sqrt(-c)*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - 4*sqrt(b^2*c - a*b*d)*s
qrt(d*x^2 + c)*a + (2*a*b*c - a^2*d + (2*b^2*c - a*b*d)*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)*sqrt(b^2*c - a
*b*d) - 4*(2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x^2)*sqrt
(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((a^2*b*x^2 + a^3)*sqrt(b^2*c - a*b*d
)), 1/4*(2*sqrt(-b^2*c + a*b*d)*(b*x^2 + a)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 +
c)*sqrt(c) + 2*c)/x^2) + 2*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)*a - (2*a*b*c -
a^2*d + (2*b^2*c - a*b*d)*x^2)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c +
a*b*d)/((b^2*c - a*b*d)*sqrt(d*x^2 + c))))/((a^2*b*x^2 + a^3)*sqrt(-b^2*c + a*b
*d)), -1/4*(4*sqrt(-b^2*c + a*b*d)*(b*x^2 + a)*sqrt(-c)*arctan(c/(sqrt(d*x^2 + c
)*sqrt(-c))) - 2*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)*a + (2*a*b*c - a^2*d + (2*
b^2*c - a*b*d)*x^2)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)/((b
^2*c - a*b*d)*sqrt(d*x^2 + c))))/((a^2*b*x^2 + a^3)*sqrt(-b^2*c + a*b*d))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{x \left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(1/2)/x/(b*x**2+a)**2,x)
[Out]
Integral(sqrt(c + d*x**2)/(x*(a + b*x**2)**2), x)
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GIAC/XCAS [A] time = 0.249757, size = 170, normalized size = 1.43 \[ \frac{1}{2} \, d^{2}{\left (\frac{\sqrt{d x^{2} + c}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a d} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} + \frac{2 \, c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="giac")
[Out]
1/2*d^2*(sqrt(d*x^2 + c)/(((d*x^2 + c)*b - b*c + a*d)*a*d) - (2*b*c - a*d)*arcta
n(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*c*a
rctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)*d^2))