3.179 \(\int \frac{a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=42 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (d*Log[a + b*x^2])/(2*b)

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Rubi [A]  time = 0.0679989, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[(a^2*c + a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3 + b^2*c*x^4 + b^2*d*x^5)/(a + b*x^2)^3,x]

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (d*Log[a + b*x^2])/(2*b)

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Rubi in Sympy [A]  time = 34.4324, size = 37, normalized size = 0.88 \[ \frac{d \log{\left (a + b x^{2} \right )}}{2 b} + \frac{c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*d*x**5+b**2*c*x**4+2*a*b*d*x**3+2*a*b*c*x**2+a**2*d*x+a**2*c)/(b*x**2+a)**3,x)

[Out]

d*log(a + b*x**2)/(2*b) + c*atan(sqrt(b)*x/sqrt(a))/(sqrt(a)*sqrt(b))

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Mathematica [A]  time = 0.0247769, size = 42, normalized size = 1. \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2*c + a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3 + b^2*c*x^4 + b^2*d*x^5)/(a + b*x^2)^3,x]

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (d*Log[a + b*x^2])/(2*b)

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Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \[{\frac{d\ln \left ( b{x}^{2}+a \right ) }{2\,b}}+{c\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(b*x^2+a)^3,x)

[Out]

1/2*d*ln(b*x^2+a)/b+c/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2561, size = 1, normalized size = 0.02 \[ \left [\frac{b c \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + \sqrt{-a b} d \log \left (b x^{2} + a\right )}{2 \, \sqrt{-a b} b}, \frac{2 \, b c \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + \sqrt{a b} d \log \left (b x^{2} + a\right )}{2 \, \sqrt{a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*d*x^5 + b^2*c*x^4 + 2*a*b*d*x^3 + 2*a*b*c*x^2 + a^2*d*x + a^2*c)/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.813131, size = 124, normalized size = 2.95 \[ \left (\frac{d}{2 b} - \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) \log{\left (x + \frac{2 a b \left (\frac{d}{2 b} - \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} + \left (\frac{d}{2 b} + \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) \log{\left (x + \frac{2 a b \left (\frac{d}{2 b} + \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*d*x**5+b**2*c*x**4+2*a*b*d*x**3+2*a*b*c*x**2+a**2*d*x+a**2*c)/(b*x**2+a)**3,x)

[Out]

(d/(2*b) - c*sqrt(-a*b**3)/(2*a*b**2))*log(x + (2*a*b*(d/(2*b) - c*sqrt(-a*b**3)
/(2*a*b**2)) - a*d)/(b*c)) + (d/(2*b) + c*sqrt(-a*b**3)/(2*a*b**2))*log(x + (2*a
*b*(d/(2*b) + c*sqrt(-a*b**3)/(2*a*b**2)) - a*d)/(b*c))

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GIAC/XCAS [A]  time = 0.265637, size = 42, normalized size = 1. \[ \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} + \frac{d{\rm ln}\left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*d*x^5 + b^2*c*x^4 + 2*a*b*d*x^3 + 2*a*b*c*x^2 + a^2*d*x + a^2*c)/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

c*arctan(b*x/sqrt(a*b))/sqrt(a*b) + 1/2*d*ln(b*x^2 + a)/b