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

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

[Out]

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

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Rubi [A]  time = 0.180542, 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 \[ \frac{3 x \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{8 \left (a+b x^2\right )}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{x \left (c+d x^2\right ) (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.4706, size = 105, normalized size = 0.91 \[ \frac{x \left (- \frac{3 d^{2}}{8 b^{2}} + \frac{3 c^{2}}{8 a^{2}}\right )}{a + b x^{2}} - \frac{x \left (c + d x^{2}\right ) \left (a d - b c\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{\left (a d \left (3 a d + b c\right ) + b c \left (a d + 3 b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.163337, size = 124, normalized size = 1.07 \[ \frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{x \left (-3 a^3 d^2-a^2 b d \left (2 c+5 d x^2\right )+a b^2 c \left (5 c+2 d x^2\right )+3 b^3 c^2 x^2\right )}{8 a^2 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^2/(b*x^2+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21489, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (3 \, b^{3} c^{2} + 2 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \sqrt{-a b}}, \frac{{\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (3 \, b^{3} c^{2} + 2 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.72889, size = 223, normalized size = 1.92 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{16} - \frac{x^{3} \left (5 a^{2} b d^{2} - 2 a b^{2} c d - 3 b^{3} c^{2}\right ) + x \left (3 a^{3} d^{2} + 2 a^{2} b c d - 5 a b^{2} c^{2}\right )}{8 a^{4} b^{2} + 16 a^{3} b^{3} x^{2} + 8 a^{2} b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232621, size = 170, normalized size = 1.47 \[ \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{2}} + \frac{3 \, b^{3} c^{2} x^{3} + 2 \, a b^{2} c d x^{3} - 5 \, a^{2} b d^{2} x^{3} + 5 \, a b^{2} c^{2} x - 2 \, a^{2} b c d x - 3 \, a^{3} d^{2} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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