3.76 \(\int \frac{x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{a x (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^3}{3 b^2} \]

[Out]

((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + (a*(A*b - a*B)*x)/(2*b^3*(a + b*x^2))
- (Sqrt[a]*(3*A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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Rubi [A]  time = 0.174415, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{a x (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + (a*(A*b - a*B)*x)/(2*b^3*(a + b*x^2))
- (Sqrt[a]*(3*A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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Rubi in Sympy [A]  time = 43.9716, size = 80, normalized size = 0.92 \[ \frac{B x^{3}}{3 b^{2}} - \frac{\sqrt{a} \left (3 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{7}{2}}} + \frac{a x \left (A b - B a\right )}{2 b^{3} \left (a + b x^{2}\right )} + \frac{x \left (A b - 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**3/(3*b**2) - sqrt(a)*(3*A*b - 5*B*a)*atan(sqrt(b)*x/sqrt(a))/(2*b**(7/2)) +
a*x*(A*b - B*a)/(2*b**3*(a + b*x**2)) + x*(A*b - 2*B*a)/b**3

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Mathematica [A]  time = 0.118742, size = 89, normalized size = 1.02 \[ \frac{x \left (a A b-a^2 B\right )}{2 b^3 \left (a+b x^2\right )}+\frac{\sqrt{a} (5 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + ((a*A*b - a^2*B)*x)/(2*b^3*(a + b*x^2)
) + (Sqrt[a]*(-3*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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Maple [A]  time = 0.012, size = 105, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{Bxa}{{b}^{3}}}+{\frac{aAx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{Bx{a}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,Aa}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}B}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*x^3/b^2+1/b^2*A*x-2/b^3*B*x*a+1/2*a/b^2*x/(b*x^2+a)*A-1/2*a^2/b^3*x/(b*x^2
+a)*B-3/2*a/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A+5/2*a^2/b^3/(a*b)^(1/2)*ar
ctan(x*b/(a*b)^(1/2))*B

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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*x^2 + A)*x^4/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237634, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, B b^{2} x^{5} - 4 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x^{3} - 3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \,{\left (5 \, B a^{2} - 3 \, A a b\right )} x}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, \frac{2 \, B b^{2} x^{5} - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 3 \,{\left (5 \, B a^{2} - 3 \, A a b\right )} x}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*B*b^2*x^5 - 4*(5*B*a*b - 3*A*b^2)*x^3 - 3*(5*B*a^2 - 3*A*a*b + (5*B*a*b
- 3*A*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 6*
(5*B*a^2 - 3*A*a*b)*x)/(b^4*x^2 + a*b^3), 1/6*(2*B*b^2*x^5 - 2*(5*B*a*b - 3*A*b^
2)*x^3 + 3*(5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x^2)*sqrt(a/b)*arctan(x/sqrt
(a/b)) - 3*(5*B*a^2 - 3*A*a*b)*x)/(b^4*x^2 + a*b^3)]

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Sympy [A]  time = 3.07614, size = 128, normalized size = 1.47 \[ \frac{B x^{3}}{3 b^{2}} - \frac{x \left (- A a b + B a^{2}\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{\sqrt{- \frac{a}{b^{7}}} \left (- 3 A b + 5 B a\right ) \log{\left (- b^{3} \sqrt{- \frac{a}{b^{7}}} + x \right )}}{4} + \frac{\sqrt{- \frac{a}{b^{7}}} \left (- 3 A b + 5 B a\right ) \log{\left (b^{3} \sqrt{- \frac{a}{b^{7}}} + x \right )}}{4} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**3/(3*b**2) - x*(-A*a*b + B*a**2)/(2*a*b**3 + 2*b**4*x**2) - sqrt(-a/b**7)*(
-3*A*b + 5*B*a)*log(-b**3*sqrt(-a/b**7) + x)/4 + sqrt(-a/b**7)*(-3*A*b + 5*B*a)*
log(b**3*sqrt(-a/b**7) + x)/4 - x*(-A*b + 2*B*a)/b**3

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GIAC/XCAS [A]  time = 0.223948, size = 119, normalized size = 1.37 \[ \frac{{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} - \frac{B a^{2} x - A a b x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{B b^{4} x^{3} - 6 \, B a b^{3} x + 3 \, A b^{4} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(5*B*a^2 - 3*A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/2*(B*a^2*x - A
*a*b*x)/((b*x^2 + a)*b^3) + 1/3*(B*b^4*x^3 - 6*B*a*b^3*x + 3*A*b^4*x)/b^6