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

Optimal. Leaf size=123 \[ -\frac{x}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 d x}{2 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(2 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{5/2}} \]

[Out]

(-3*d*x)/(2*(b*c - a*d)^2*Sqrt[c + d*x^2]) - x/(2*(b*c - a*d)*(a + b*x^2)*Sqrt[c
+ d*x^2]) + ((b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])
])/(2*Sqrt[a]*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.271164, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{x}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 d x}{2 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(2 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d*x)/(2*(b*c - a*d)^2*Sqrt[c + d*x^2]) - x/(2*(b*c - a*d)*(a + b*x^2)*Sqrt[c
+ d*x^2]) + ((b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])
])/(2*Sqrt[a]*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 45.0008, size = 104, normalized size = 0.85 \[ - \frac{3 d x}{2 \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{\left (a d + \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d*x/(2*sqrt(c + d*x**2)*(a*d - b*c)**2) + x/(2*(a + b*x**2)*sqrt(c + d*x**2)*
(a*d - b*c)) + (a*d + b*c/2)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))
/(sqrt(a)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.246196, size = 110, normalized size = 0.89 \[ \frac{1}{2} \left (\frac{(2 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{5/2}}-\frac{x \left (2 a d+b \left (c+3 d x^2\right )\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((x*(2*a*d + b*(c + 3*d*x^2)))/((b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2])) +
((b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*(
b*c - a*d)^(5/2)))/2

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Maple [B]  time = 0.022, size = 1453, normalized size = 11.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^2/(d*x^2+c)^(3/2),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)-3/4/b*d*(-a*b)^(1/2)/(a*d-b*c)^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)-
3/4/b*d^2*a/(a*d-b*c)^2/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+3/4/b*d*(-a*b)^(1/2)/(a*d-b*c)^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)))+3/4/b/(a*d-b*c)/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*d+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)+3/4/b*d*(-a*b)^(1/2)/(a*d-b*c)^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)-3/4/b*d^2*a/(a*d
-b*c)^2/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-3/4/b*d*(-a*b)^(1/2)/(a*d-b*c)^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)))+3/4/b/(a*d-b*c)/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*d-1/4/(-a*b)^(1/2)/(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)/(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)/(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)/(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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.594638, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b d x^{3} +{\left (b c + 2 \, a d\right )} x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d +{\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left (3 \, b d x^{3} +{\left (b c + 2 \, a d\right )} x\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d +{\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 3.35368, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x