3.2224 \(\int \frac{x^2}{\left (a+b \sqrt{x}\right )^8} \, dx\)

Optimal. Leaf size=43 \[ \frac{x^3}{21 a^2 \left (a+b \sqrt{x}\right )^6}+\frac{2 x^3}{7 a \left (a+b \sqrt{x}\right )^7} \]

[Out]

(2*x^3)/(7*a*(a + b*Sqrt[x])^7) + x^3/(21*a^2*(a + b*Sqrt[x])^6)

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Rubi [A]  time = 0.0479277, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^3}{21 a^2 \left (a+b \sqrt{x}\right )^6}+\frac{2 x^3}{7 a \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*Sqrt[x])^8,x]

[Out]

(2*x^3)/(7*a*(a + b*Sqrt[x])^7) + x^3/(21*a^2*(a + b*Sqrt[x])^6)

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Rubi in Sympy [A]  time = 6.58194, size = 36, normalized size = 0.84 \[ \frac{2 x^{3}}{7 a \left (a + b \sqrt{x}\right )^{7}} + \frac{x^{3}}{21 a^{2} \left (a + b \sqrt{x}\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b*x**(1/2))**8,x)

[Out]

2*x**3/(7*a*(a + b*sqrt(x))**7) + x**3/(21*a**2*(a + b*sqrt(x))**6)

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Mathematica [A]  time = 0.0353904, size = 74, normalized size = 1.72 \[ -\frac{a^5+7 a^4 b \sqrt{x}+21 a^3 b^2 x+35 a^2 b^3 x^{3/2}+35 a b^4 x^2+21 b^5 x^{5/2}}{21 b^6 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*Sqrt[x])^8,x]

[Out]

-(a^5 + 7*a^4*b*Sqrt[x] + 21*a^3*b^2*x + 35*a^2*b^3*x^(3/2) + 35*a*b^4*x^2 + 21*
b^5*x^(5/2))/(21*b^6*(a + b*Sqrt[x])^7)

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Maple [B]  time = 0.01, size = 99, normalized size = 2.3 \[{\frac{10\,a}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}+4\,{\frac{{a}^{3}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{5}}}-{\frac{1}{{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}}-{\frac{5\,{a}^{4}}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-6}}-5\,{\frac{{a}^{2}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{4}}}+{\frac{2\,{a}^{5}}{7\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/2))^8,x)

[Out]

10/3*a/b^6/(a+b*x^(1/2))^3+4*a^3/b^6/(a+b*x^(1/2))^5-1/b^6/(a+b*x^(1/2))^2-5/3*a
^4/b^6/(a+b*x^(1/2))^6-5*a^2/b^6/(a+b*x^(1/2))^4+2/7*a^5/b^6/(a+b*x^(1/2))^7

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Maxima [A]  time = 1.4415, size = 132, normalized size = 3.07 \[ -\frac{1}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} + \frac{10 \, a}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{6}} - \frac{5 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{4} b^{6}} + \frac{4 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{5} b^{6}} - \frac{5 \, a^{4}}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{6}} + \frac{2 \, a^{5}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^8,x, algorithm="maxima")

[Out]

-1/((b*sqrt(x) + a)^2*b^6) + 10/3*a/((b*sqrt(x) + a)^3*b^6) - 5*a^2/((b*sqrt(x)
+ a)^4*b^6) + 4*a^3/((b*sqrt(x) + a)^5*b^6) - 5/3*a^4/((b*sqrt(x) + a)^6*b^6) +
2/7*a^5/((b*sqrt(x) + a)^7*b^6)

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Fricas [A]  time = 0.238638, size = 177, normalized size = 4.12 \[ -\frac{35 \, a b^{4} x^{2} + 21 \, a^{3} b^{2} x + a^{5} + 7 \,{\left (3 \, b^{5} x^{2} + 5 \, a^{2} b^{3} x + a^{4} b\right )} \sqrt{x}}{21 \,{\left (7 \, a b^{12} x^{3} + 35 \, a^{3} b^{10} x^{2} + 21 \, a^{5} b^{8} x + a^{7} b^{6} +{\left (b^{13} x^{3} + 21 \, a^{2} b^{11} x^{2} + 35 \, a^{4} b^{9} x + 7 \, a^{6} b^{7}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^8,x, algorithm="fricas")

[Out]

-1/21*(35*a*b^4*x^2 + 21*a^3*b^2*x + a^5 + 7*(3*b^5*x^2 + 5*a^2*b^3*x + a^4*b)*s
qrt(x))/(7*a*b^12*x^3 + 35*a^3*b^10*x^2 + 21*a^5*b^8*x + a^7*b^6 + (b^13*x^3 + 2
1*a^2*b^11*x^2 + 35*a^4*b^9*x + 7*a^6*b^7)*sqrt(x))

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Sympy [A]  time = 30.2126, size = 619, normalized size = 14.4 \[ \begin{cases} - \frac{a^{5}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} - \frac{7 a^{4} b \sqrt{x}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} - \frac{21 a^{3} b^{2} x}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} - \frac{35 a^{2} b^{3} x^{\frac{3}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} - \frac{35 a b^{4} x^{2}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} - \frac{21 b^{5} x^{\frac{5}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt{x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac{3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac{5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac{7}{2}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{8}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((-a**5/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a
**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*
x**3 + 21*b**13*x**(7/2)) - 7*a**4*b*sqrt(x)/(21*a**7*b**6 + 147*a**6*b**7*sqrt(
x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b
**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 21*a**3*b**2*x/(21*a**7*
b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a*
*3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2))
- 35*a**2*b**3*x**(3/2)/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x
+ 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a
*b**12*x**3 + 21*b**13*x**(7/2)) - 35*a*b**4*x**2/(21*a**7*b**6 + 147*a**6*b**7*
sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a
**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 21*b**5*x**(5/2)/(2
1*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) +
735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**
(7/2)), Ne(b, 0)), (x**3/(3*a**8), True))

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GIAC/XCAS [A]  time = 0.260071, size = 86, normalized size = 2. \[ -\frac{21 \, b^{5} x^{\frac{5}{2}} + 35 \, a b^{4} x^{2} + 35 \, a^{2} b^{3} x^{\frac{3}{2}} + 21 \, a^{3} b^{2} x + 7 \, a^{4} b \sqrt{x} + a^{5}}{21 \,{\left (b \sqrt{x} + a\right )}^{7} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^8,x, algorithm="giac")

[Out]

-1/21*(21*b^5*x^(5/2) + 35*a*b^4*x^2 + 35*a^2*b^3*x^(3/2) + 21*a^3*b^2*x + 7*a^4
*b*sqrt(x) + a^5)/((b*sqrt(x) + a)^7*b^6)