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

Optimal. Leaf size=67 \[ \frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac{d x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.131405, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac{d x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.151, size = 60, normalized size = 0.9 \[ \frac{d x}{b^{2}} + \frac{x \left (a d - b c\right )}{2 b^{2} \left (a + b x^{2}\right )} - \frac{\left (3 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/b**2 + x*(a*d - b*c)/(2*b**2*(a + b*x**2)) - (3*a*d - b*c)*atan(sqrt(b)*x/sq
rt(a))/(2*sqrt(a)*b**(5/2))

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Mathematica [A]  time = 0.11306, size = 68, normalized size = 1.01 \[ -\frac{(3 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac{d x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 82, normalized size = 1.2 \[{\frac{dx}{{b}^{2}}}+{\frac{axd}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cx}{2\,b \left ( b{x}^{2}+a \right ) }}-{\frac{3\,ad}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{c}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((d*x^2 + c)*x^2/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227732, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (2 \, b d x^{3} -{\left (b c - 3 \, a d\right )} x\right )} \sqrt{-a b}}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-a b}}, \frac{{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, b d x^{3} -{\left (b c - 3 \, a d\right )} x\right )} \sqrt{a b}}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.58635, size = 114, normalized size = 1.7 \[ \frac{x \left (a d - b c\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{5}}} \left (3 a d - b c\right ) \log{\left (- a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a b^{5}}} \left (3 a d - b c\right ) \log{\left (a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} + \frac{d x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*d - b*c)/(2*a*b**2 + 2*b**3*x**2) + sqrt(-1/(a*b**5))*(3*a*d - b*c)*log(-a*
b**2*sqrt(-1/(a*b**5)) + x)/4 - sqrt(-1/(a*b**5))*(3*a*d - b*c)*log(a*b**2*sqrt(
-1/(a*b**5)) + x)/4 + d*x/b**2

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GIAC/XCAS [A]  time = 0.248684, size = 78, normalized size = 1.16 \[ \frac{d x}{b^{2}} + \frac{{\left (b c - 3 \, a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{b c x - a d x}{2 \,{\left (b x^{2} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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