3.188 \(\int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=126 \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a (6 b c-5 a d)}{3 c^3 x} \]

[Out]

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

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Rubi [A]  time = 0.336312, antiderivative size = 126, 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 \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a (6 b c-5 a d)}{3 c^3 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^4*(c + d*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 42.3322, size = 110, normalized size = 0.87 \[ - \frac{a^{2}}{3 c x^{3} \left (c + d x^{2}\right )} + \frac{a \left (5 a d - 6 b c\right )}{3 c^{3} x} + \frac{x \left (a d \left (5 a d - 6 b c\right ) + 3 b^{2} c^{2}\right )}{6 c^{3} \left (c + d x^{2}\right )} + \frac{\left (a d - b c\right ) \left (5 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{7}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**4/(d*x**2+c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.105999, size = 107, normalized size = 0.85 \[ \frac{\left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a^2}{3 c^2 x^3}+\frac{x (b c-a d)^2}{2 c^3 \left (c+d x^2\right )}+\frac{2 a (a d-b c)}{c^3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^2),x]

[Out]

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

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Maple [A]  time = 0.016, size = 161, normalized size = 1.3 \[ -{\frac{{a}^{2}}{3\,{c}^{2}{x}^{3}}}+2\,{\frac{{a}^{2}d}{{c}^{3}x}}-2\,{\frac{ab}{{c}^{2}x}}+{\frac{{a}^{2}{d}^{2}x}{2\,{c}^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{xabd}{{c}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{x{b}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}+{\frac{5\,{a}^{2}{d}^{2}}{2\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{abd}{{c}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*a^2/c^2/x^3+2*a^2/c^3/x*d-2*a/c^2/x*b+1/2/c^3*x/(d*x^2+c)*a^2*d^2-1/c^2*x/(
d*x^2+c)*a*b*d+1/2/c*x/(d*x^2+c)*b^2+5/2/c^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))
*a^2*d^2-3/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b*d+1/2/c/(c*d)^(1/2)*arcta
n(x*d/(c*d)^(1/2))*b^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/((d*x^2 + c)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228138, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \,{\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-c d}}{12 \,{\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \,{\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{c d}}{6 \,{\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^2*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.63194, size = 248, normalized size = 1.97 \[ - \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 18 a b c d + 3 b^{2} c^{2}\right ) + x^{2} \left (10 a^{2} c d - 12 a b c^{2}\right )}{6 c^{4} x^{3} + 6 c^{3} d x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**4/(d*x**2+c)**2,x)

[Out]

-sqrt(-1/(c**7*d))*(a*d - b*c)*(5*a*d - b*c)*log(-c**4*sqrt(-1/(c**7*d))*(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/(c**7*
d))*(a*d - b*c)*(5*a*d - b*c)*log(c**4*sqrt(-1/(c**7*d))*(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 + (-2*a**2*c**2 + x**4*(15*a**2*
d**2 - 18*a*b*c*d + 3*b**2*c**2) + x**2*(10*a**2*c*d - 12*a*b*c**2))/(6*c**4*x**
3 + 6*c**3*d*x**5)

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GIAC/XCAS [A]  time = 0.226281, size = 150, normalized size = 1.19 \[ \frac{{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} c^{3}} - \frac{6 \, a b c x^{2} - 6 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^2*x^4),x, algorithm="giac")

[Out]

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