3.910 \(\int \frac{1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx\)

Optimal. Leaf size=89 \[ -\frac{1}{2 x^2 (a+b)}+\frac{\sqrt{a} (a-b) \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a+b)^2}+\frac{a \log \left (a x^4+2 a x^2+a+b\right )}{2 (a+b)^2}-\frac{2 a \log (x)}{(a+b)^2} \]

[Out]

-1/(2*(a + b)*x^2) + (Sqrt[a]*(a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*Sq
rt[b]*(a + b)^2) - (2*a*Log[x])/(a + b)^2 + (a*Log[a + b + 2*a*x^2 + a*x^4])/(2*
(a + b)^2)

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Rubi [A]  time = 0.294809, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{1}{2 x^2 (a+b)}+\frac{\sqrt{a} (a-b) \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a+b)^2}+\frac{a \log \left (a x^4+2 a x^2+a+b\right )}{2 (a+b)^2}-\frac{2 a \log (x)}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-1/(2*(a + b)*x^2) + (Sqrt[a]*(a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*Sq
rt[b]*(a + b)^2) - (2*a*Log[x])/(a + b)^2 + (a*Log[a + b + 2*a*x^2 + a*x^4])/(2*
(a + b)^2)

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Rubi in Sympy [A]  time = 43.8031, size = 83, normalized size = 0.93 \[ \frac{\sqrt{a} \left (a - b\right ) \operatorname{atan}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{b} \left (a + b\right )^{2}} - \frac{a \log{\left (x^{2} \right )}}{\left (a + b\right )^{2}} + \frac{a \log{\left (a x^{4} + 2 a x^{2} + a + b \right )}}{2 \left (a + b\right )^{2}} - \frac{1}{2 x^{2} \left (a + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a*x**4+2*a*x**2+a+b),x)

[Out]

sqrt(a)*(a - b)*atan(sqrt(a)*(x**2 + 1)/sqrt(b))/(2*sqrt(b)*(a + b)**2) - a*log(
x**2)/(a + b)**2 + a*log(a*x**4 + 2*a*x**2 + a + b)/(2*(a + b)**2) - 1/(2*x**2*(
a + b))

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Mathematica [C]  time = 0.170537, size = 163, normalized size = 1.83 \[ \frac{\left (2 a^{3/2} \sqrt{b}-i a^2+i a b\right ) \log \left (\sqrt{a} x^2+\sqrt{a}-i \sqrt{b}\right )}{4 \sqrt{a} \sqrt{b} (a+b)^2}+\frac{\left (2 a^{3/2} \sqrt{b}+i a^2-i a b\right ) \log \left (\sqrt{a} x^2+\sqrt{a}+i \sqrt{b}\right )}{4 \sqrt{a} \sqrt{b} (a+b)^2}-\frac{1}{2 x^2 (a+b)}-\frac{2 a \log (x)}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-1/(2*(a + b)*x^2) - (2*a*Log[x])/(a + b)^2 + (((-I)*a^2 + 2*a^(3/2)*Sqrt[b] + I
*a*b)*Log[Sqrt[a] - I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2) + ((
I*a^2 + 2*a^(3/2)*Sqrt[b] - I*a*b)*Log[Sqrt[a] + I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sq
rt[a]*Sqrt[b]*(a + b)^2)

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Maple [A]  time = 0.012, size = 110, normalized size = 1.2 \[{\frac{a\ln \left ( a{x}^{4}+2\,a{x}^{2}+a+b \right ) }{2\, \left ( a+b \right ) ^{2}}}+{\frac{{a}^{2}}{2\, \left ( a+b \right ) ^{2}}\arctan \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ab}{2\, \left ( a+b \right ) ^{2}}\arctan \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{ \left ( 2\,a+2\,b \right ){x}^{2}}}-2\,{\frac{a\ln \left ( x \right ) }{ \left ( a+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a*x^4+2*a*x^2+a+b),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290071, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a - b\right )} x^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{-\frac{a}{b}} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right ) - 2 \, a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 8 \, a x^{2} \log \left (x\right ) + 2 \, a + 2 \, b}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}, -\frac{{\left (a - b\right )} x^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a x^{2} + a}\right ) - a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, a x^{2} \log \left (x\right ) + a + b}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a + b)*x^3),x, algorithm="fricas")

[Out]

[-1/4*((a - b)*x^2*sqrt(-a/b)*log((a*x^4 + 2*a*x^2 - 2*(b*x^2 + b)*sqrt(-a/b) +
a - b)/(a*x^4 + 2*a*x^2 + a + b)) - 2*a*x^2*log(a*x^4 + 2*a*x^2 + a + b) + 8*a*x
^2*log(x) + 2*a + 2*b)/((a^2 + 2*a*b + b^2)*x^2), -1/2*((a - b)*x^2*sqrt(a/b)*ar
ctan(b*sqrt(a/b)/(a*x^2 + a)) - a*x^2*log(a*x^4 + 2*a*x^2 + a + b) + 4*a*x^2*log
(x) + a + b)/((a^2 + 2*a*b + b^2)*x^2)]

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Sympy [A]  time = 17.2805, size = 386, normalized size = 4.34 \[ - \frac{2 a \log{\left (x \right )}}{\left (a + b\right )^{2}} + \left (\frac{a}{2 \left (a + b\right )^{2}} - \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log{\left (x^{2} + \frac{4 a^{2} b \left (\frac{a}{2 \left (a + b\right )^{2}} - \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac{a}{2 \left (a + b\right )^{2}} - \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac{a}{2 \left (a + b\right )^{2}} - \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} + \left (\frac{a}{2 \left (a + b\right )^{2}} + \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log{\left (x^{2} + \frac{4 a^{2} b \left (\frac{a}{2 \left (a + b\right )^{2}} + \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac{a}{2 \left (a + b\right )^{2}} + \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac{a}{2 \left (a + b\right )^{2}} + \frac{\sqrt{- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} - \frac{1}{x^{2} \left (2 a + 2 b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.537401, size = 169, normalized size = 1.9 \[ \frac{a{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{a{\rm ln}\left (x^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac{{\left (a^{2} - a b\right )} \arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a b}} + \frac{2 \, a x^{2} - a - b}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a + b)*x^3),x, algorithm="giac")

[Out]

1/2*a*ln(a*x^4 + 2*a*x^2 + a + b)/(a^2 + 2*a*b + b^2) - a*ln(x^2)/(a^2 + 2*a*b +
b^2) + 1/2*(a^2 - a*b)*arctan((a*x^2 + a)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(
a*b)) + 1/2*(2*a*x^2 - a - b)/((a^2 + 2*a*b + b^2)*x^2)