3.257 \(\int \frac{x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx\)

Optimal. Leaf size=19 \[ \frac{\tan ^{-1}\left (\frac{c x^3}{a+b x^2}\right )}{c} \]

[Out]

ArcTan[(c*x^3)/(a + b*x^2)]/c

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Rubi [A]  time = 0.174633, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\tan ^{-1}\left (\frac{c x^3}{a+b x^2}\right )}{c} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(3*a + b*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4 + c^2*x^6),x]

[Out]

ArcTan[(c*x^3)/(a + b*x^2)]/c

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Rubi in Sympy [A]  time = 53.0187, size = 14, normalized size = 0.74 \[ \frac{\operatorname{atan}{\left (\frac{c x^{3}}{a + b x^{2}} \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+3*a)/(c**2*x**6+b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

atan(c*x**3/(a + b*x**2))/c

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Mathematica [C]  time = 0.071569, size = 87, normalized size = 4.58 \[ \frac{1}{2} \text{RootSum}\left [\text{$\#$1}^6 c^2+\text{$\#$1}^4 b^2+2 \text{$\#$1}^2 a b+a^2\&,\frac{\text{$\#$1}^3 b \log (x-\text{$\#$1})+3 \text{$\#$1} a \log (x-\text{$\#$1})}{3 \text{$\#$1}^4 c^2+2 \text{$\#$1}^2 b^2+2 a b}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(3*a + b*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4 + c^2*x^6),x]

[Out]

RootSum[a^2 + 2*a*b*#1^2 + b^2*#1^4 + c^2*#1^6 & , (3*a*Log[x - #1]*#1 + b*Log[x
- #1]*#1^3)/(2*a*b + 2*b^2*#1^2 + 3*c^2*#1^4) & ]/2

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Maple [C]  time = 0.428, size = 75, normalized size = 4. \[{\frac{1}{2}\sum _{{\it \_R}={\it RootOf} \left ({c}^{2}{{\it \_Z}}^{6}+{b}^{2}{{\it \_Z}}^{4}+2\,ab{{\it \_Z}}^{2}+{a}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}b+3\,{{\it \_R}}^{2}a \right ) \ln \left ( x-{\it \_R} \right ) }{3\,{{\it \_R}}^{5}{c}^{2}+2\,{{\it \_R}}^{3}{b}^{2}+2\,{\it \_R}\,ab}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+3*a)/(c^2*x^6+b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

1/2*sum((_R^4*b+3*_R^2*a)/(3*_R^5*c^2+2*_R^3*b^2+2*_R*a*b)*ln(x-_R),_R=RootOf(_Z
^6*c^2+_Z^4*b^2+2*_Z^2*a*b+a^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + 3 \, a\right )} x^{2}}{c^{2} x^{6} + b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)*x^2/(c^2*x^6 + b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + 3*a)*x^2/(c^2*x^6 + b^2*x^4 + 2*a*b*x^2 + a^2), x)

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Fricas [A]  time = 0.264245, size = 112, normalized size = 5.89 \[ \frac{\arctan \left (\frac{c x}{b}\right ) - \arctan \left (\frac{b c^{2} x^{5} + a b^{2} x +{\left (b^{3} - a c^{2}\right )} x^{3}}{a^{2} c}\right ) + \arctan \left (\frac{b c^{2} x^{3} +{\left (b^{3} - a c^{2}\right )} x}{a b c}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)*x^2/(c^2*x^6 + b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="fricas")

[Out]

(arctan(c*x/b) - arctan((b*c^2*x^5 + a*b^2*x + (b^3 - a*c^2)*x^3)/(a^2*c)) + arc
tan((b*c^2*x^3 + (b^3 - a*c^2)*x)/(a*b*c)))/c

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Sympy [A]  time = 4.11852, size = 44, normalized size = 2.32 \[ \frac{- \frac{i \log{\left (- \frac{i a}{c} - \frac{i b x^{2}}{c} + x^{3} \right )}}{2} + \frac{i \log{\left (\frac{i a}{c} + \frac{i b x^{2}}{c} + x^{3} \right )}}{2}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+3*a)/(c**2*x**6+b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + 3 \, a\right )} x^{2}}{c^{2} x^{6} + b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)*x^2/(c^2*x^6 + b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)*x^2/(c^2*x^6 + b^2*x^4 + 2*a*b*x^2 + a^2), x)