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

Optimal. Leaf size=58 \[ -\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (b c-a d)}{b^2}+\frac{d x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.101515, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (b c-a d)}{b^2}+\frac{d x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.1229, size = 49, normalized size = 0.84 \[ \frac{\sqrt{a} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{d x^{3}}{3 b} - \frac{x \left (a d - b c\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0703124, size = 57, normalized size = 0.98 \[ \frac{\sqrt{a} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (b c-a d)}{b^2}+\frac{d x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 68, normalized size = 1.2 \[{\frac{d{x}^{3}}{3\,b}}-{\frac{adx}{{b}^{2}}}+{\frac{cx}{b}}+{\frac{{a}^{2}d}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ac}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x^2+c)/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246286, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, b d x^{3} - 3 \,{\left (b c - a d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \,{\left (b c - a d\right )} x}{6 \, b^{2}}, \frac{b d x^{3} - 3 \,{\left (b c - a d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 3 \,{\left (b c - a d\right )} x}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(2*b*d*x^3 - 3*(b*c - a*d)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b
*x^2 + a)) + 6*(b*c - a*d)*x)/b^2, 1/3*(b*d*x^3 - 3*(b*c - a*d)*sqrt(a/b)*arctan
(x/sqrt(a/b)) + 3*(b*c - a*d)*x)/b^2]

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Sympy [A]  time = 1.78137, size = 90, normalized size = 1.55 \[ - \frac{\sqrt{- \frac{a}{b^{5}}} \left (a d - b c\right ) \log{\left (- b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{a}{b^{5}}} \left (a d - b c\right ) \log{\left (b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{d x^{3}}{3 b} - \frac{x \left (a d - b c\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-a/b**5)*(a*d - b*c)*log(-b**2*sqrt(-a/b**5) + x)/2 + sqrt(-a/b**5)*(a*d -
b*c)*log(b**2*sqrt(-a/b**5) + x)/2 + d*x**3/(3*b) - x*(a*d - b*c)/b**2

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GIAC/XCAS [A]  time = 0.227073, size = 78, normalized size = 1.34 \[ -\frac{{\left (a b c - a^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} d x^{3} + 3 \, b^{2} c x - 3 \, a b d x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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