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

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

[Out]

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

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Rubi [A]  time = 0.138298, 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 \[ \frac{a^{3/2} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x (b c-a d)}{b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.086598, size = 77, normalized size = 1. \[ -\frac{a^{3/2} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{a x (a d-b c)}{b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 92, normalized size = 1.2 \[{\frac{d{x}^{5}}{5\,b}}-{\frac{{x}^{3}ad}{3\,{b}^{2}}}+{\frac{c{x}^{3}}{3\,b}}+{\frac{x{a}^{2}d}{{b}^{3}}}-{\frac{acx}{{b}^{2}}}-{\frac{{a}^{3}d}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{a}^{2}c}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x^2+c)/(b*x^2+a),x)

[Out]

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

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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^4/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23309, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, b^{2} d x^{5} + 10 \,{\left (b^{2} c - a b d\right )} x^{3} - 15 \,{\left (a b c - a^{2} d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 30 \,{\left (a b c - a^{2} d\right )} x}{30 \, b^{3}}, \frac{3 \, b^{2} d x^{5} + 5 \,{\left (b^{2} c - a b d\right )} x^{3} + 15 \,{\left (a b c - a^{2} d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 15 \,{\left (a b c - a^{2} d\right )} x}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.91, size = 150, normalized size = 1.95 \[ \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right ) \log{\left (- \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right ) \log{\left (\frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} + \frac{d x^{5}}{5 b} - \frac{x^{3} \left (a d - b c\right )}{3 b^{2}} + \frac{x \left (a^{2} d - a b c\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218848, size = 113, normalized size = 1.47 \[ \frac{{\left (a^{2} b c - a^{3} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} d x^{5} + 5 \, b^{4} c x^{3} - 5 \, a b^{3} d x^{3} - 15 \, a b^{3} c x + 15 \, a^{2} b^{2} d x}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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