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

Optimal. Leaf size=152 \[ -\frac{x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{a^2}{c x \left (c+d x^2\right )^2}+\frac{\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}+\frac{x \left (3 a d (2 b c-5 a d)+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )} \]

[Out]

-(a^2/(c*x*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(4*c^2*d*(c +
d*x^2)^2) + ((b^2*c^2 + 3*a*d*(2*b*c - 5*a*d))*x)/(8*c^3*d*(c + d*x^2)) + ((b^2
*c^2 + 3*a*d*(2*b*c - 5*a*d))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*d^(3/2))

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Rubi [A]  time = 0.269135, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{a^2}{c x \left (c+d x^2\right )^2}+\frac{x \left (\frac{3 a (2 b c-5 a d)}{c^2}+\frac{b^2}{d}\right )}{8 c \left (c+d x^2\right )}+\frac{\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2/(c*x*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(4*c^2*d*(c +
d*x^2)^2) + ((b^2/d + (3*a*(2*b*c - 5*a*d))/c^2)*x)/(8*c*(c + d*x^2)) + ((b^2*c
^2 + 3*a*d*(2*b*c - 5*a*d))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*d^(3/2))

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Rubi in Sympy [A]  time = 29.9501, size = 136, normalized size = 0.89 \[ - \frac{a^{2}}{c x \left (c + d x^{2}\right )^{2}} - \frac{x \left (a d \left (5 a d - 2 b c\right ) + b^{2} c^{2}\right )}{4 c^{2} d \left (c + d x^{2}\right )^{2}} + \frac{x \left (- 3 a d \left (5 a d - 2 b c\right ) + b^{2} c^{2}\right )}{8 c^{3} d \left (c + d x^{2}\right )} + \frac{\left (- 3 a d \left (5 a d - 2 b c\right ) + b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{7}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/(c*x*(c + d*x**2)**2) - x*(a*d*(5*a*d - 2*b*c) + b**2*c**2)/(4*c**2*d*(c +
d*x**2)**2) + x*(-3*a*d*(5*a*d - 2*b*c) + b**2*c**2)/(8*c**3*d*(c + d*x**2)) +
(-3*a*d*(5*a*d - 2*b*c) + b**2*c**2)*atan(sqrt(d)*x/sqrt(c))/(8*c**(7/2)*d**(3/2
))

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Mathematica [A]  time = 0.147895, size = 133, normalized size = 0.88 \[ \frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}+\frac{x \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )}-\frac{a^2}{c^3 x}-\frac{x (b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2/(c^3*x)) - ((b*c - a*d)^2*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2*c^2 + 6*a*b*c
*d - 7*a^2*d^2)*x)/(8*c^3*d*(c + d*x^2)) + ((b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*A
rcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*d^(3/2))

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Maple [A]  time = 0.017, size = 199, normalized size = 1.3 \[ -{\frac{{a}^{2}}{{c}^{3}x}}-{\frac{7\,{x}^{3}{a}^{2}{d}^{2}}{8\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}abd}{4\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{x}^{3}}{8\,c \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,x{a}^{2}d}{8\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,abx}{4\,c \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{x{b}^{2}}{8\, \left ( d{x}^{2}+c \right ) ^{2}d}}-{\frac{15\,{a}^{2}d}{8\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,ab}{4\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{8\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a^2/c^3/x-7/8/c^3/(d*x^2+c)^2*x^3*a^2*d^2+3/4/c^2/(d*x^2+c)^2*x^3*a*b*d+1/8/c/(
d*x^2+c)^2*x^3*b^2-9/8/c^2/(d*x^2+c)^2*x*a^2*d+5/4/c/(d*x^2+c)^2*x*a*b-1/8/(d*x^
2+c)^2/d*x*b^2-15/8/c^3*d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2+3/4/c^2/(c*d)^
(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+1/8/c/d/(c*d)^(1/2)*arctan(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)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252564, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \,{\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} +{\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \log \left (-\frac{2 \, c d x -{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (8 \, a^{2} c^{2} d -{\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{4} +{\left (b^{2} c^{3} - 10 \, a b c^{2} d + 25 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{-c d}}{16 \,{\left (c^{3} d^{3} x^{5} + 2 \, c^{4} d^{2} x^{3} + c^{5} d x\right )} \sqrt{-c d}}, \frac{{\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \,{\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} +{\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (8 \, a^{2} c^{2} d -{\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{4} +{\left (b^{2} c^{3} - 10 \, a b c^{2} d + 25 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{c d}}{8 \,{\left (c^{3} d^{3} x^{5} + 2 \, c^{4} d^{2} x^{3} + c^{5} d x\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(((b^2*c^2*d^2 + 6*a*b*c*d^3 - 15*a^2*d^4)*x^5 + 2*(b^2*c^3*d + 6*a*b*c^2
*d^2 - 15*a^2*c*d^3)*x^3 + (b^2*c^4 + 6*a*b*c^3*d - 15*a^2*c^2*d^2)*x)*log(-(2*c
*d*x - (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) + 2*(8*a^2*c^2*d - (b^2*c^2*d + 6*a*
b*c*d^2 - 15*a^2*d^3)*x^4 + (b^2*c^3 - 10*a*b*c^2*d + 25*a^2*c*d^2)*x^2)*sqrt(-c
*d))/((c^3*d^3*x^5 + 2*c^4*d^2*x^3 + c^5*d*x)*sqrt(-c*d)), 1/8*(((b^2*c^2*d^2 +
6*a*b*c*d^3 - 15*a^2*d^4)*x^5 + 2*(b^2*c^3*d + 6*a*b*c^2*d^2 - 15*a^2*c*d^3)*x^3
+ (b^2*c^4 + 6*a*b*c^3*d - 15*a^2*c^2*d^2)*x)*arctan(sqrt(c*d)*x/c) - (8*a^2*c^
2*d - (b^2*c^2*d + 6*a*b*c*d^2 - 15*a^2*d^3)*x^4 + (b^2*c^3 - 10*a*b*c^2*d + 25*
a^2*c*d^2)*x^2)*sqrt(c*d))/((c^3*d^3*x^5 + 2*c^4*d^2*x^3 + c^5*d*x)*sqrt(c*d))]

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Sympy [A]  time = 5.60954, size = 224, normalized size = 1.47 \[ \frac{\sqrt{- \frac{1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log{\left (- c^{4} d \sqrt{- \frac{1}{c^{7} d^{3}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log{\left (c^{4} d \sqrt{- \frac{1}{c^{7} d^{3}}} + x \right )}}{16} - \frac{8 a^{2} c^{2} d + x^{4} \left (15 a^{2} d^{3} - 6 a b c d^{2} - b^{2} c^{2} d\right ) + x^{2} \left (25 a^{2} c d^{2} - 10 a b c^{2} d + b^{2} c^{3}\right )}{8 c^{5} d x + 16 c^{4} d^{2} x^{3} + 8 c^{3} d^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(c**7*d**3))*(15*a**2*d**2 - 6*a*b*c*d - b**2*c**2)*log(-c**4*d*sqrt(-1/
(c**7*d**3)) + x)/16 - sqrt(-1/(c**7*d**3))*(15*a**2*d**2 - 6*a*b*c*d - b**2*c**
2)*log(c**4*d*sqrt(-1/(c**7*d**3)) + x)/16 - (8*a**2*c**2*d + x**4*(15*a**2*d**3
- 6*a*b*c*d**2 - b**2*c**2*d) + x**2*(25*a**2*c*d**2 - 10*a*b*c**2*d + b**2*c**
3))/(8*c**5*d*x + 16*c**4*d**2*x**3 + 8*c**3*d**3*x**5)

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GIAC/XCAS [A]  time = 0.222535, size = 182, normalized size = 1.2 \[ -\frac{a^{2}}{c^{3} x} + \frac{{\left (b^{2} c^{2} + 6 \, a b c d - 15 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{3} d} + \frac{b^{2} c^{2} d x^{3} + 6 \, a b c d^{2} x^{3} - 7 \, a^{2} d^{3} x^{3} - b^{2} c^{3} x + 10 \, a b c^{2} d x - 9 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a^2/(c^3*x) + 1/8*(b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqr
t(c*d)*c^3*d) + 1/8*(b^2*c^2*d*x^3 + 6*a*b*c*d^2*x^3 - 7*a^2*d^3*x^3 - b^2*c^3*x
+ 10*a*b*c^2*d*x - 9*a^2*c*d^2*x)/((d*x^2 + c)^2*c^3*d)