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

Optimal. Leaf size=80 \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{2 b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.161995, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.6286, size = 65, normalized size = 0.81 \[ - \frac{\sqrt{c + d x^{2}}}{2 b \left (a + b x^{2}\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0854687, size = 80, normalized size = 1. \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 1617, normalized size = 20.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(-a*b)^(1/2)/a/b/(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*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)/b^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*a/b^2*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*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/b*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*(-a*b)^(1/2)/a/b*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*(-a*b)^(1/2
)/a/b/(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*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
)/b^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*a/b^2*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*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/b*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*(-a*b)^(1/2)/a/b*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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248809, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b d x^{2} + a d\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{8 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b d x^{2} + a d\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right ) - 2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}{4 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*((b*d*x^2 + 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)*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
)) - 4*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/((b^2*x^2 + a*b)*sqrt(b^2*c - a*b*d)
), 1/4*((b*d*x^2 + a*d)*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))) - 2*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c))/((
b^2*x^2 + a*b)*sqrt(-b^2*c + a*b*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238177, size = 107, normalized size = 1.34 \[ \frac{1}{2} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x^{2} + c}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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