3.262 \(\int \frac{x^4 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.179617, 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 \[ -\frac{\sqrt{a} (3 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{a x (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{x (b c-2 a d)}{b^3}+\frac{d x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.2409, size = 80, normalized size = 0.92 \[ \frac{\sqrt{a} \left (5 a d - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{7}{2}}} - \frac{a x \left (a d - b c\right )}{2 b^{3} \left (a + b x^{2}\right )} + \frac{d x^{3}}{3 b^{2}} - \frac{x \left (2 a d - b c\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*(5*a*d - 3*b*c)*atan(sqrt(b)*x/sqrt(a))/(2*b**(7/2)) - a*x*(a*d - b*c)/(
2*b**3*(a + b*x**2)) + d*x**3/(3*b**2) - x*(2*a*d - b*c)/b**3

_______________________________________________________________________________________

Mathematica [A]  time = 0.119615, size = 89, normalized size = 1.02 \[ \frac{x \left (a b c-a^2 d\right )}{2 b^3 \left (a+b x^2\right )}+\frac{\sqrt{a} (5 a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (b c-2 a d)}{b^3}+\frac{d x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 105, normalized size = 1.2 \[{\frac{d{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{adx}{{b}^{3}}}+{\frac{cx}{{b}^{2}}}-{\frac{x{a}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{acx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}d}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,ac}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.234294, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, b^{2} d x^{5} + 4 \,{\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \,{\left (3 \, a b c - 5 \, a^{2} d +{\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \,{\left (3 \, a b c - 5 \, a^{2} d\right )} x}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, \frac{2 \, b^{2} d x^{5} + 2 \,{\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \,{\left (3 \, a b c - 5 \, a^{2} d +{\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 3 \,{\left (3 \, a b c - 5 \, a^{2} d\right )} x}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.07707, size = 128, normalized size = 1.47 \[ - \frac{x \left (a^{2} d - a b c\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{\sqrt{- \frac{a}{b^{7}}} \left (5 a d - 3 b c\right ) \log{\left (- b^{3} \sqrt{- \frac{a}{b^{7}}} + x \right )}}{4} + \frac{\sqrt{- \frac{a}{b^{7}}} \left (5 a d - 3 b c\right ) \log{\left (b^{3} \sqrt{- \frac{a}{b^{7}}} + x \right )}}{4} + \frac{d x^{3}}{3 b^{2}} - \frac{x \left (2 a d - b c\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.235402, size = 119, normalized size = 1.37 \[ -\frac{{\left (3 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} + \frac{a b c x - a^{2} d x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{b^{4} d x^{3} + 3 \, b^{4} c x - 6 \, a b^{3} d x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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