3.1841 \(\int \frac{x^6}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b^3 x}{a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]

[Out]

-((b^3*x)/a^4) + (b^2*x^3)/(3*a^3) - (b*x^5)/(5*a^2) + x^7/(7*a) + (b^(7/2)*ArcT
an[(Sqrt[a]*x)/Sqrt[b]])/a^(9/2)

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Rubi [A]  time = 0.0876424, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b^3 x}{a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b/x^2),x]

[Out]

-((b^3*x)/a^4) + (b^2*x^3)/(3*a^3) - (b*x^5)/(5*a^2) + x^7/(7*a) + (b^(7/2)*ArcT
an[(Sqrt[a]*x)/Sqrt[b]])/a^(9/2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - b^{3} \int \frac{1}{a^{4}}\, dx + \frac{x^{7}}{7 a} - \frac{b x^{5}}{5 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} + \frac{b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(a+b/x**2),x)

[Out]

-b**3*Integral(a**(-4), x) + x**7/(7*a) - b*x**5/(5*a**2) + b**2*x**3/(3*a**3) +
b**(7/2)*atan(sqrt(a)*x/sqrt(b))/a**(9/2)

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Mathematica [A]  time = 0.0486531, size = 68, normalized size = 1. \[ \frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b^3 x}{a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b/x^2),x]

[Out]

-((b^3*x)/a^4) + (b^2*x^3)/(3*a^3) - (b*x^5)/(5*a^2) + x^7/(7*a) + (b^(7/2)*ArcT
an[(Sqrt[a]*x)/Sqrt[b]])/a^(9/2)

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Maple [A]  time = 0.004, size = 60, normalized size = 0.9 \[{\frac{{x}^{7}}{7\,a}}-{\frac{b{x}^{5}}{5\,{a}^{2}}}+{\frac{{b}^{2}{x}^{3}}{3\,{a}^{3}}}-{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(a+b/x^2),x)

[Out]

1/7*x^7/a-1/5*b*x^5/a^2+1/3*b^2*x^3/a^3-b^3*x/a^4+b^4/a^4/(a*b)^(1/2)*arctan(a*x
/(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(x^6/(a + b/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232932, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, a^{3} x^{7} - 42 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, a^{4}}, \frac{15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) - 105 \, b^{3} x}{105 \, a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(a + b/x^2),x, algorithm="fricas")

[Out]

[1/210*(30*a^3*x^7 - 42*a^2*b*x^5 + 70*a*b^2*x^3 + 105*b^3*sqrt(-b/a)*log((a*x^2
+ 2*a*x*sqrt(-b/a) - b)/(a*x^2 + b)) - 210*b^3*x)/a^4, 1/105*(15*a^3*x^7 - 21*a
^2*b*x^5 + 35*a*b^2*x^3 + 105*b^3*sqrt(b/a)*arctan(x/sqrt(b/a)) - 105*b^3*x)/a^4
]

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Sympy [A]  time = 1.34158, size = 107, normalized size = 1.57 \[ - \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac{x^{7}}{7 a} - \frac{b x^{5}}{5 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} x}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(a+b/x**2),x)

[Out]

-sqrt(-b**7/a**9)*log(-a**4*sqrt(-b**7/a**9)/b**3 + x)/2 + sqrt(-b**7/a**9)*log(
a**4*sqrt(-b**7/a**9)/b**3 + x)/2 + x**7/(7*a) - b*x**5/(5*a**2) + b**2*x**3/(3*
a**3) - b**3*x/a**4

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GIAC/XCAS [A]  time = 0.230374, size = 88, normalized size = 1.29 \[ \frac{b^{4} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{15 \, a^{6} x^{7} - 21 \, a^{5} b x^{5} + 35 \, a^{4} b^{2} x^{3} - 105 \, a^{3} b^{3} x}{105 \, a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(a + b/x^2),x, algorithm="giac")

[Out]

b^4*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(15*a^6*x^7 - 21*a^5*b*x^5 + 3
5*a^4*b^2*x^3 - 105*a^3*b^3*x)/a^7