3.182 \(\int \frac{x^{10}}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=98 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}+\frac{63 a^2 x}{8 b^5}-\frac{21 a x^3}{8 b^4}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{x^9}{4 b \left (a+b x^2\right )^2}+\frac{63 x^5}{40 b^3} \]

[Out]

(63*a^2*x)/(8*b^5) - (21*a*x^3)/(8*b^4) + (63*x^5)/(40*b^3) - x^9/(4*b*(a + b*x^
2)^2) - (9*x^7)/(8*b^2*(a + b*x^2)) - (63*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
8*b^(11/2))

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Rubi [A]  time = 0.119709, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}+\frac{63 a^2 x}{8 b^5}-\frac{21 a x^3}{8 b^4}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{x^9}{4 b \left (a+b x^2\right )^2}+\frac{63 x^5}{40 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a + b*x^2)^3,x]

[Out]

(63*a^2*x)/(8*b^5) - (21*a*x^3)/(8*b^4) + (63*x^5)/(40*b^3) - x^9/(4*b*(a + b*x^
2)^2) - (9*x^7)/(8*b^2*(a + b*x^2)) - (63*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
8*b^(11/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{63 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{11}{2}}} - \frac{21 a x^{3}}{8 b^{4}} - \frac{x^{9}}{4 b \left (a + b x^{2}\right )^{2}} - \frac{9 x^{7}}{8 b^{2} \left (a + b x^{2}\right )} + \frac{63 x^{5}}{40 b^{3}} + \frac{63 \int a^{2}\, dx}{8 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(b*x**2+a)**3,x)

[Out]

-63*a**(5/2)*atan(sqrt(b)*x/sqrt(a))/(8*b**(11/2)) - 21*a*x**3/(8*b**4) - x**9/(
4*b*(a + b*x**2)**2) - 9*x**7/(8*b**2*(a + b*x**2)) + 63*x**5/(40*b**3) + 63*Int
egral(a**2, x)/(8*b**5)

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Mathematica [A]  time = 0.0970531, size = 88, normalized size = 0.9 \[ \frac{315 a^4 x+525 a^3 b x^3+168 a^2 b^2 x^5-24 a b^3 x^7+8 b^4 x^9}{40 b^5 \left (a+b x^2\right )^2}-\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a + b*x^2)^3,x]

[Out]

(315*a^4*x + 525*a^3*b*x^3 + 168*a^2*b^2*x^5 - 24*a*b^3*x^7 + 8*b^4*x^9)/(40*b^5
*(a + b*x^2)^2) - (63*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(11/2))

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Maple [A]  time = 0.014, size = 88, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{b}^{3}}}-{\frac{a{x}^{3}}{{b}^{4}}}+6\,{\frac{{a}^{2}x}{{b}^{5}}}+{\frac{17\,{a}^{3}{x}^{3}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{a}^{3}}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(b*x^2+a)^3,x)

[Out]

1/5*x^5/b^3-a*x^3/b^4+6*a^2*x/b^5+17/8/b^4*a^3/(b*x^2+a)^2*x^3+15/8/b^5*a^4/(b*x
^2+a)^2*x-63/8/b^5*a^3/(a*b)^(1/2)*arctan(x*b/(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^10/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.20992, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, b^{4} x^{9} - 48 \, a b^{3} x^{7} + 336 \, a^{2} b^{2} x^{5} + 1050 \, a^{3} b x^{3} + 630 \, a^{4} x + 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{80 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac{8 \, b^{4} x^{9} - 24 \, a b^{3} x^{7} + 168 \, a^{2} b^{2} x^{5} + 525 \, a^{3} b x^{3} + 315 \, a^{4} x - 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{40 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[1/80*(16*b^4*x^9 - 48*a*b^3*x^7 + 336*a^2*b^2*x^5 + 1050*a^3*b*x^3 + 630*a^4*x
+ 315*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b)
- a)/(b*x^2 + a)))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5), 1/40*(8*b^4*x^9 - 24*a*b^
3*x^7 + 168*a^2*b^2*x^5 + 525*a^3*b*x^3 + 315*a^4*x - 315*(a^2*b^2*x^4 + 2*a^3*b
*x^2 + a^4)*sqrt(a/b)*arctan(x/sqrt(a/b)))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)]

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Sympy [A]  time = 2.2917, size = 144, normalized size = 1.47 \[ \frac{6 a^{2} x}{b^{5}} - \frac{a x^{3}}{b^{4}} + \frac{63 \sqrt{- \frac{a^{5}}{b^{11}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} - \frac{63 \sqrt{- \frac{a^{5}}{b^{11}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} + \frac{15 a^{4} x + 17 a^{3} b x^{3}}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac{x^{5}}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(b*x**2+a)**3,x)

[Out]

6*a**2*x/b**5 - a*x**3/b**4 + 63*sqrt(-a**5/b**11)*log(x - b**5*sqrt(-a**5/b**11
)/a**2)/16 - 63*sqrt(-a**5/b**11)*log(x + b**5*sqrt(-a**5/b**11)/a**2)/16 + (15*
a**4*x + 17*a**3*b*x**3)/(8*a**2*b**5 + 16*a*b**6*x**2 + 8*b**7*x**4) + x**5/(5*
b**3)

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GIAC/XCAS [A]  time = 0.231723, size = 113, normalized size = 1.15 \[ -\frac{63 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{5}} + \frac{17 \, a^{3} b x^{3} + 15 \, a^{4} x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{5}} + \frac{b^{12} x^{5} - 5 \, a b^{11} x^{3} + 30 \, a^{2} b^{10} x}{5 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

-63/8*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/8*(17*a^3*b*x^3 + 15*a^4*x)/
((b*x^2 + a)^2*b^5) + 1/5*(b^12*x^5 - 5*a*b^11*x^3 + 30*a^2*b^10*x)/b^15