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

Optimal. Leaf size=89 \[ \frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{3/2}}-\frac{x \sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.15905, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{3/2}}-\frac{x \sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4845, size = 73, normalized size = 0.82 \[ \frac{x \sqrt{c + d x^{2}}}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(c + d*x**2)/(2*(a + b*x**2)*(a*d - b*c)) - c*atanh(x*sqrt(a*d - b*c)/(sqr
t(a)*sqrt(c + d*x**2)))/(2*sqrt(a)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.151971, size = 86, normalized size = 0.97 \[ \frac{\frac{x \sqrt{c+d x^2}}{a+b x^2}-\frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{2 a d-2 b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 817, normalized size = 9.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.341371, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} x +{\left (b c x^{2} + a c\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x -{\left (b c x^{2} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 2.00856, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x