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

Optimal. Leaf size=159 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 \sqrt{c}}-\frac{b \sqrt{c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac{\sqrt{c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]

[Out]

-((b*Sqrt[c + d*x^2])/(a^2*(a + b*x^2))) - Sqrt[c + d*x^2]/(2*a*x^2*(a + b*x^2))
+ ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*Sqrt[c]) - (Sqrt[b]*(
4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*Sqrt[b
*c - a*d])

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Rubi [A]  time = 0.618938, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 \sqrt{c}}-\frac{b \sqrt{c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac{\sqrt{c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*Sqrt[c + d*x^2])/(a^2*(a + b*x^2))) - Sqrt[c + d*x^2]/(2*a*x^2*(a + b*x^2))
+ ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*Sqrt[c]) - (Sqrt[b]*(
4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*Sqrt[b
*c - a*d])

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Rubi in Sympy [A]  time = 61.2622, size = 133, normalized size = 0.84 \[ \frac{\sqrt{c + d x^{2}}}{2 a x^{2} \left (a + b x^{2}\right )} - \frac{\sqrt{c + d x^{2}}}{a^{2} x^{2}} - \frac{\sqrt{b} \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} \sqrt{a d - b c}} - \frac{\left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3} \sqrt{c}} \]

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)**2,x)

[Out]

sqrt(c + d*x**2)/(2*a*x**2*(a + b*x**2)) - sqrt(c + d*x**2)/(a**2*x**2) - sqrt(b
)*(3*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(2*a**3*sqrt(a*
d - b*c)) - (a*d - 4*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*a**3*sqrt(c))

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Mathematica [C]  time = 1.98927, size = 343, normalized size = 2.16 \[ -\frac{\frac{\sqrt{b} (4 b c-3 a d) \log \left (\frac{4 a^3 \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\sqrt{b} \left (\sqrt{b} x+i \sqrt{a}\right ) (4 b c-3 a d) \sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}+\frac{\sqrt{b} (4 b c-3 a d) \log \left (\frac{4 i a^3 \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\sqrt{b} \left (\sqrt{a}+i \sqrt{b} x\right ) (4 b c-3 a d) \sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}+\frac{2 a \left (a+2 b x^2\right ) \sqrt{c+d x^2}}{x^2 \left (a+b x^2\right )}-\frac{2 (4 b c-a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}+\frac{2 \log (x) (4 b c-a d)}{\sqrt{c}}}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a*(a + 2*b*x^2)*Sqrt[c + d*x^2])/(x^2*(a + b*x^2)) + (2*(4*b*c - a*d)*Log[x
])/Sqrt[c] - (2*(4*b*c - a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/Sqrt[c] + (Sqrt[
b]*(4*b*c - 3*a*d)*Log[(4*a^3*(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[
c + d*x^2]))/(Sqrt[b]*(4*b*c - 3*a*d)*Sqrt[b*c - a*d]*(I*Sqrt[a] + Sqrt[b]*x))])
/Sqrt[b*c - a*d] + (Sqrt[b]*(4*b*c - 3*a*d)*Log[((4*I)*a^3*(Sqrt[b]*c + I*Sqrt[a
]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b]*(4*b*c - 3*a*d)*Sqrt[b*c - a*
d]*(Sqrt[a] + I*Sqrt[b]*x))])/Sqrt[b*c - a*d])/(4*a^3)

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Maple [B]  time = 0.025, size = 2669, normalized size = 16.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.441971, 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^3),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.243367, size = 258, normalized size = 1.62 \[ -\frac{1}{2} \, d^{3}{\left (\frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b - 2 \, \sqrt{d x^{2} + c} b c + \sqrt{d x^{2} + c} a d}{{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )} a^{2} d^{2}} - \frac{{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} d^{3}} + \frac{{\left (4 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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