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

Optimal. Leaf size=116 \[ \frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{7/2}}-\frac{x (b c-5 a d) (b c-a d)}{2 a b^3}+\frac{x^3 (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x^3}{3 b^2} \]

[Out]

-((b*c - 5*a*d)*(b*c - a*d)*x)/(2*a*b^3) + (d^2*x^3)/(3*b^2) + ((b*c - a*d)^2*x^
3)/(2*a*b^2*(a + b*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*Sqrt[a]*b^(7/2))

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Rubi [A]  time = 0.288539, antiderivative size = 116, 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 \[ \frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{7/2}}-\frac{x (b c-5 a d) (b c-a d)}{2 a b^3}+\frac{x^3 (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - 5*a*d)*(b*c - a*d)*x)/(2*a*b^3) + (d^2*x^3)/(3*b^2) + ((b*c - a*d)^2*x^
3)/(2*a*b^2*(a + b*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*Sqrt[a]*b^(7/2))

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Rubi in Sympy [A]  time = 45.041, size = 100, normalized size = 0.86 \[ \frac{d^{2} x^{3}}{3 b^{2}} + \frac{x^{3} \left (a d - b c\right )^{2}}{2 a b^{2} \left (a + b x^{2}\right )} - \frac{x \left (a d - b c\right ) \left (5 a d - b c\right )}{2 a b^{3}} + \frac{\left (a d - b c\right ) \left (5 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x**3/(3*b**2) + x**3*(a*d - b*c)**2/(2*a*b**2*(a + b*x**2)) - x*(a*d - b*c)
*(5*a*d - b*c)/(2*a*b**3) + (a*d - b*c)*(5*a*d - b*c)*atan(sqrt(b)*x/sqrt(a))/(2
*sqrt(a)*b**(7/2))

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Mathematica [A]  time = 0.113475, size = 105, normalized size = 0.91 \[ \frac{\left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{7/2}}-\frac{x (b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{2 d x (b c-a d)}{b^3}+\frac{d^2 x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(b*c - a*d)*x)/b^3 + (d^2*x^3)/(3*b^2) - ((b*c - a*d)^2*x)/(2*b^3*(a + b*x^
2)) + ((b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]
*b^(7/2))

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Maple [A]  time = 0.012, size = 156, normalized size = 1.3 \[{\frac{{d}^{2}{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{a{d}^{2}x}{{b}^{3}}}+2\,{\frac{dxc}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}x}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{acxd}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{x{c}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}{d}^{2}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{acd}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{c}^{2}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^2*x^3/b^2-2*d^2/b^3*a*x+2*d/b^2*x*c-1/2/b^3*x/(b*x^2+a)*a^2*d^2+1/b^2*x/(b
*x^2+a)*c*a*d-1/2/b*x/(b*x^2+a)*c^2+5/2/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*
a^2*d^2-3/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c*a*d+1/2/b/(a*b)^(1/2)*arctan
(x*b/(a*b)^(1/2))*c^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)^2*x^2/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248215, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{5} + 2 \,{\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{3} - 3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x\right )} \sqrt{-a b}}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, b^{2} d^{2} x^{5} + 2 \,{\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{3} - 3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x\right )} \sqrt{a b}}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d^2 + (b^3*c^2 - 6*a*b^2*c*d + 5*a^2*b
*d^2)*x^2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(2*b^2*d^2*x^
5 + 2*(6*b^2*c*d - 5*a*b*d^2)*x^3 - 3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x)*sqrt(
-a*b))/((b^4*x^2 + a*b^3)*sqrt(-a*b)), 1/6*(3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d
^2 + (b^3*c^2 - 6*a*b^2*c*d + 5*a^2*b*d^2)*x^2)*arctan(sqrt(a*b)*x/a) + (2*b^2*d
^2*x^5 + 2*(6*b^2*c*d - 5*a*b*d^2)*x^3 - 3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x)*
sqrt(a*b))/((b^4*x^2 + a*b^3)*sqrt(a*b))]

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Sympy [A]  time = 4.02083, size = 245, normalized size = 2.11 \[ - \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{\sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (- \frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (\frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{d^{2} x^{3}}{3 b^{2}} - \frac{x \left (2 a d^{2} - 2 b c d\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(2*a*b**3 + 2*b**4*x**2) - sqrt(-1/(a*b**
7))*(a*d - b*c)*(5*a*d - b*c)*log(-a*b**3*sqrt(-1/(a*b**7))*(a*d - b*c)*(5*a*d -
b*c)/(5*a**2*d**2 - 6*a*b*c*d + b**2*c**2) + x)/4 + sqrt(-1/(a*b**7))*(a*d - b*
c)*(5*a*d - b*c)*log(a*b**3*sqrt(-1/(a*b**7))*(a*d - b*c)*(5*a*d - b*c)/(5*a**2*
d**2 - 6*a*b*c*d + b**2*c**2) + x)/4 + d**2*x**3/(3*b**2) - x*(2*a*d**2 - 2*b*c*
d)/b**3

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GIAC/XCAS [A]  time = 0.23674, size = 154, normalized size = 1.33 \[ \frac{{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} - \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{b^{4} d^{2} x^{3} + 6 \, b^{4} c d x - 6 \, a b^{3} d^{2} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/
2*(b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^2 + a)*b^3) + 1/3*(b^4*d^2*x^3 + 6
*b^4*c*d*x - 6*a*b^3*d^2*x)/b^6