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

Optimal. Leaf size=92 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x (a d+3 b c)}{8 a^2 b \left (a+b x^2\right )}+\frac{x (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0850816, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x (a d+3 b c)}{8 a^2 b \left (a+b x^2\right )}+\frac{x (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.6024, size = 78, normalized size = 0.85 \[ - \frac{x \left (a d - b c\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{x \left (a d + 3 b c\right )}{8 a^{2} b \left (a + b x^{2}\right )} + \frac{\left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.104814, size = 84, normalized size = 0.91 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x \left (a^2 (-d)+a b \left (5 c+d x^2\right )+3 b^2 c x^2\right )}{8 a^2 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 89, normalized size = 1. \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( ad+3\,bc \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{ \left ( ad-5\,bc \right ) x}{8\,ab}} \right ) }+{\frac{d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(b*x^2+a)^3,x)

[Out]

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

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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)/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208923, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, b^{3} c + a b^{2} d\right )} x^{4} + 3 \, a^{2} b c + a^{3} d + 2 \,{\left (3 \, a b^{2} c + a^{2} b d\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (3 \, b^{2} c + a b d\right )} x^{3} +{\left (5 \, a b c - a^{2} d\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{-a b}}, \frac{{\left ({\left (3 \, b^{3} c + a b^{2} d\right )} x^{4} + 3 \, a^{2} b c + a^{3} d + 2 \,{\left (3 \, a b^{2} c + a^{2} b d\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (3 \, b^{2} c + a b d\right )} x^{3} +{\left (5 \, a b c - a^{2} d\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.7968, size = 150, normalized size = 1.63 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d + 3 b c\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d + 3 b c\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{x^{3} \left (a b d + 3 b^{2} c\right ) + x \left (- a^{2} d + 5 a b c\right )}{8 a^{4} b + 16 a^{3} b^{2} x^{2} + 8 a^{2} b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231321, size = 105, normalized size = 1.14 \[ \frac{{\left (3 \, b c + a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b} + \frac{3 \, b^{2} c x^{3} + a b d x^{3} + 5 \, a b c x - a^{2} d x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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