3.75 \(\int \frac{x^2 \left (d+e x^2\right )}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=145 \[ \frac{x \left (a+b x^2\right ) (b d-a e)}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.266099, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{x \left (a+b x^2\right ) (b d-a e)}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x**2+d)/((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0860841, size = 80, normalized size = 0.55 \[ \frac{\left (a+b x^2\right ) \left (\sqrt{b} x \left (-3 a e+3 b d+b e x^2\right )+3 \sqrt{a} (a e-b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{3 b^{5/2} \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

((a + b*x^2)*(Sqrt[b]*x*(3*b*d - 3*a*e + b*e*x^2) + 3*Sqrt[a]*(-(b*d) + a*e)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.013, size = 90, normalized size = 0.6 \[{\frac{b{x}^{2}+a}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}be-3\,\sqrt{ab}xae+3\,\sqrt{ab}xbd+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{2}e-3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) abd \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260384, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b e x^{3} - 3 \,{\left (b d - a e\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \,{\left (b d - a e\right )} x}{6 \, b^{2}}, \frac{b e x^{3} - 3 \,{\left (b d - a e\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 3 \,{\left (b d - a e\right )} x}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.76036, size = 90, normalized size = 0.62 \[ - \frac{\sqrt{- \frac{a}{b^{5}}} \left (a e - b d\right ) \log{\left (- b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{a}{b^{5}}} \left (a e - b d\right ) \log{\left (b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{e x^{3}}{3 b} - \frac{x \left (a e - b d\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(e*x**2+d)/((b*x**2+a)**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.265498, size = 136, normalized size = 0.94 \[ -\frac{{\left (a b d{\rm sign}\left (b x^{2} + a\right ) - a^{2} e{\rm sign}\left (b x^{2} + a\right )\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} x^{3} e{\rm sign}\left (b x^{2} + a\right ) + 3 \, b^{2} d x{\rm sign}\left (b x^{2} + a\right ) - 3 \, a b x e{\rm sign}\left (b x^{2} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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