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

Optimal. Leaf size=116 \[ \frac{3 x \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{8 \left (c+d x^2\right )}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{5/2}}-\frac{x \left (a+b x^2\right ) (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.181628, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 x \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{8 \left (c+d x^2\right )}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{5/2}}-\frac{x \left (a+b x^2\right ) (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.0598, size = 105, normalized size = 0.91 \[ \frac{x \left (\frac{3 a^{2}}{8 c^{2}} - \frac{3 b^{2}}{8 d^{2}}\right )}{c + d x^{2}} + \frac{x \left (a + b x^{2}\right ) \left (a d - b c\right )}{4 c d \left (c + d x^{2}\right )^{2}} + \frac{\left (a d \left (3 a d + b c\right ) + b c \left (a d + 3 b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.171888, size = 121, normalized size = 1.04 \[ \frac{x \left (a^2 d^2 \left (5 c+3 d x^2\right )-2 a b c d \left (c-d x^2\right )-b^2 c^2 \left (3 c+5 d x^2\right )\right )}{8 c^2 d^2 \left (c+d x^2\right )^2}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 147, normalized size = 1.3 \[{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ({\frac{ \left ( 3\,{a}^{2}{d}^{2}+2\,abcd-5\,{b}^{2}{c}^{2} \right ){x}^{3}}{8\,{c}^{2}d}}+{\frac{ \left ( 5\,{a}^{2}{d}^{2}-2\,abcd-3\,{b}^{2}{c}^{2} \right ) x}{8\,{d}^{2}c}} \right ) }+{\frac{3\,{a}^{2}}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{ab}{4\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}}{8\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245758, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (3 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left ({\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x\right )} \sqrt{-c d}}{16 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{-c d}}, \frac{{\left (3 \, b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (3 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left ({\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x\right )} \sqrt{c d}}{8 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.6103, size = 223, normalized size = 1.92 \[ - \frac{\sqrt{- \frac{1}{c^{5} d^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- c^{3} d^{2} \sqrt{- \frac{1}{c^{5} d^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{5} d^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (c^{3} d^{2} \sqrt{- \frac{1}{c^{5} d^{5}}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{2} d^{3} + 2 a b c d^{2} - 5 b^{2} c^{2} d\right ) + x \left (5 a^{2} c d^{2} - 2 a b c^{2} d - 3 b^{2} c^{3}\right )}{8 c^{4} d^{2} + 16 c^{3} d^{3} x^{2} + 8 c^{2} d^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(c**5*d**5))*(3*a**2*d**2 + 2*a*b*c*d + 3*b**2*c**2)*log(-c**3*d**2*sqr
t(-1/(c**5*d**5)) + x)/16 + sqrt(-1/(c**5*d**5))*(3*a**2*d**2 + 2*a*b*c*d + 3*b*
*2*c**2)*log(c**3*d**2*sqrt(-1/(c**5*d**5)) + x)/16 + (x**3*(3*a**2*d**3 + 2*a*b
*c*d**2 - 5*b**2*c**2*d) + x*(5*a**2*c*d**2 - 2*a*b*c**2*d - 3*b**2*c**3))/(8*c*
*4*d**2 + 16*c**3*d**3*x**2 + 8*c**2*d**4*x**4)

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GIAC/XCAS [A]  time = 0.232259, size = 170, normalized size = 1.47 \[ \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{2} d^{2}} - \frac{5 \, b^{2} c^{2} d x^{3} - 2 \, a b c d^{2} x^{3} - 3 \, a^{2} d^{3} x^{3} + 3 \, b^{2} c^{3} x + 2 \, a b c^{2} d x - 5 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^2*d^2
) - 1/8*(5*b^2*c^2*d*x^3 - 2*a*b*c*d^2*x^3 - 3*a^2*d^3*x^3 + 3*b^2*c^3*x + 2*a*b
*c^2*d*x - 5*a^2*c*d^2*x)/((d*x^2 + c)^2*c^2*d^2)