3.518 \(\int \frac{x^9}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac{4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac{2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5} \]

[Out]

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

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Rubi [A]  time = 0.14519, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac{4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac{2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^2)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.1032, size = 87, normalized size = 0.93 \[ - \frac{a^{4}}{7 b^{5} \left (a + b x^{2}\right )^{\frac{7}{2}}} + \frac{4 a^{3}}{5 b^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{2 a^{2}}{b^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{4 a}{b^{5} \sqrt{a + b x^{2}}} + \frac{\sqrt{a + b x^{2}}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**2+a)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.040633, size = 61, normalized size = 0.65 \[ \frac{128 a^4+448 a^3 b x^2+560 a^2 b^2 x^4+280 a b^3 x^6+35 b^4 x^8}{35 b^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^2)^(9/2),x]

[Out]

(128*a^4 + 448*a^3*b*x^2 + 560*a^2*b^2*x^4 + 280*a*b^3*x^6 + 35*b^4*x^8)/(35*b^5
*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.008, size = 58, normalized size = 0.6 \[{\frac{35\,{x}^{8}{b}^{4}+280\,a{x}^{6}{b}^{3}+560\,{a}^{2}{x}^{4}{b}^{2}+448\,{a}^{3}{x}^{2}b+128\,{a}^{4}}{35\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^2+a)^(9/2),x)

[Out]

1/35*(35*b^4*x^8+280*a*b^3*x^6+560*a^2*b^2*x^4+448*a^3*b*x^2+128*a^4)/(b*x^2+a)^
(7/2)/b^5

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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^9/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25747, size = 138, normalized size = 1.47 \[ \frac{{\left (35 \, b^{4} x^{8} + 280 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 448 \, a^{3} b x^{2} + 128 \, a^{4}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

1/35*(35*b^4*x^8 + 280*a*b^3*x^6 + 560*a^2*b^2*x^4 + 448*a^3*b*x^2 + 128*a^4)*sq
rt(b*x^2 + a)/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)

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Sympy [A]  time = 20.2494, size = 454, normalized size = 4.83 \[ \begin{cases} \frac{128 a^{4}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{448 a^{3} b x^{2}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{560 a^{2} b^{2} x^{4}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{280 a b^{3} x^{6}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} + \frac{35 b^{4} x^{8}}{35 a^{3} b^{5} \sqrt{a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt{a + b x^{2}} + 105 a b^{7} x^{4} \sqrt{a + b x^{2}} + 35 b^{8} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{10}}{10 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**2+a)**(9/2),x)

[Out]

Piecewise((128*a**4/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a +
b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 4
48*a**3*b*x**2/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x*
*2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 560*a*
*2*b**2*x**4/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2
) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 280*a*b*
*3*x**6/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 1
05*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 35*b**4*x**8/
(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**
7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**10/(10*
a**(9/2)), True))

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GIAC/XCAS [A]  time = 0.215655, size = 96, normalized size = 1.02 \[ \frac{35 \, \sqrt{b x^{2} + a} + \frac{140 \,{\left (b x^{2} + a\right )}^{3} a - 70 \,{\left (b x^{2} + a\right )}^{2} a^{2} + 28 \,{\left (b x^{2} + a\right )} a^{3} - 5 \, a^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}}{35 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

1/35*(35*sqrt(b*x^2 + a) + (140*(b*x^2 + a)^3*a - 70*(b*x^2 + a)^2*a^2 + 28*(b*x
^2 + a)*a^3 - 5*a^4)/(b*x^2 + a)^(7/2))/b^5