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

Optimal. Leaf size=77 \[ \frac{a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x (A b-a B)}{b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^5}{5 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.137078, antiderivative size = 77, 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 \[ \frac{a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x (A b-a B)}{b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{5}}{5 b} + \frac{a^{\frac{3}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{x^{3} \left (A b - B a\right )}{3 b^{2}} - \frac{\left (A b - B a\right ) \int a\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0871806, size = 77, normalized size = 1. \[ -\frac{a^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{a x (a B-A b)}{b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 92, normalized size = 1.2 \[{\frac{B{x}^{5}}{5\,b}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{B{x}^{3}a}{3\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{Bx{a}^{2}}{{b}^{3}}}+{\frac{{a}^{2}A}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{B{a}^{3}}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^2+A)/(b*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233398, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, B b^{2} x^{5} - 10 \,{\left (B a b - A b^{2}\right )} x^{3} - 15 \,{\left (B a^{2} - A a b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 30 \,{\left (B a^{2} - A a b\right )} x}{30 \, b^{3}}, \frac{3 \, B b^{2} x^{5} - 5 \,{\left (B a b - A b^{2}\right )} x^{3} - 15 \,{\left (B a^{2} - A a b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 15 \,{\left (B a^{2} - A a b\right )} x}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(6*B*b^2*x^5 - 10*(B*a*b - A*b^2)*x^3 - 15*(B*a^2 - A*a*b)*sqrt(-a/b)*log(
(b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 30*(B*a^2 - A*a*b)*x)/b^3, 1/15*(3
*B*b^2*x^5 - 5*(B*a*b - A*b^2)*x^3 - 15*(B*a^2 - A*a*b)*sqrt(a/b)*arctan(x/sqrt(
a/b)) + 15*(B*a^2 - A*a*b)*x)/b^3]

_______________________________________________________________________________________

Sympy [A]  time = 1.94384, size = 150, normalized size = 1.95 \[ \frac{B x^{5}}{5 b} + \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right ) \log{\left (- \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right ) \log{\left (\frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.242373, size = 115, normalized size = 1.49 \[ -\frac{{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, B b^{4} x^{5} - 5 \, B a b^{3} x^{3} + 5 \, A b^{4} x^{3} + 15 \, B a^{2} b^{2} x - 15 \, A a b^{3} x}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(B*a^3 - A*a^2*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*B*b^4*x^5 - 5
*B*a*b^3*x^3 + 5*A*b^4*x^3 + 15*B*a^2*b^2*x - 15*A*a*b^3*x)/b^5