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

Optimal. Leaf size=118 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d) (5 b c-a d)}{2 c d^3}+\frac{x^3 (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.292757, antiderivative size = 118, 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 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d) (5 b c-a d)}{2 c d^3}+\frac{x^3 (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.0846, size = 100, normalized size = 0.85 \[ \frac{b^{2} x^{3}}{3 d^{2}} + \frac{x^{3} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{x \left (a d - 5 b c\right ) \left (a d - b c\right )}{2 c d^{3}} + \frac{\left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{c} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.120253, size = 105, normalized size = 0.89 \[ \frac{\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{2 b x (b c-a d)}{d^3}+\frac{b^2 x^3}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 156, normalized size = 1.3 \[{\frac{{b}^{2}{x}^{3}}{3\,{d}^{2}}}+2\,{\frac{abx}{{d}^{2}}}-2\,{\frac{x{b}^{2}c}{{d}^{3}}}-{\frac{{a}^{2}x}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{xabc}{{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}{c}^{2}x}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{abc}{{d}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{5\,{b}^{2}{c}^{2}}{2\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248417, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} +{\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 6 \, a b d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{-c d}}{12 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} +{\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (2 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 6 \, a b d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{c d}}{6 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.94537, size = 245, normalized size = 2.08 \[ \frac{b^{2} x^{3}}{3 d^{2}} - \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c d^{3} + 2 d^{4} x^{2}} - \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (- \frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (\frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{x \left (2 a b d - 2 b^{2} c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227917, size = 154, normalized size = 1.31 \[ \frac{{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} d^{3}} - \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{b^{2} d^{4} x^{3} - 6 \, b^{2} c d^{3} x + 6 \, a b d^{4} x}{3 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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