3.2225 \(\int \frac{x}{\left (a+b \sqrt{x}\right )^8} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 a^3}{7 b^4 \left (a+b \sqrt{x}\right )^7}-\frac{a^2}{b^4 \left (a+b \sqrt{x}\right )^6}+\frac{6 a}{5 b^4 \left (a+b \sqrt{x}\right )^5}-\frac{1}{2 b^4 \left (a+b \sqrt{x}\right )^4} \]

[Out]

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

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Rubi [A]  time = 0.10931, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 a^3}{7 b^4 \left (a+b \sqrt{x}\right )^7}-\frac{a^2}{b^4 \left (a+b \sqrt{x}\right )^6}+\frac{6 a}{5 b^4 \left (a+b \sqrt{x}\right )^5}-\frac{1}{2 b^4 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[x])^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.1514, size = 71, normalized size = 0.91 \[ \frac{2 a^{3}}{7 b^{4} \left (a + b \sqrt{x}\right )^{7}} - \frac{a^{2}}{b^{4} \left (a + b \sqrt{x}\right )^{6}} + \frac{6 a}{5 b^{4} \left (a + b \sqrt{x}\right )^{5}} - \frac{1}{2 b^{4} \left (a + b \sqrt{x}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0208027, size = 50, normalized size = 0.64 \[ -\frac{a^3+7 a^2 b \sqrt{x}+21 a b^2 x+35 b^3 x^{3/2}}{70 b^4 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3 + 7*a^2*b*Sqrt[x] + 21*a*b^2*x + 35*b^3*x^(3/2))/(70*b^4*(a + b*Sqrt[x])^7
)

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Maple [A]  time = 0.009, size = 65, normalized size = 0.8 \[{\frac{2\,{a}^{3}}{7\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{{a}^{2}}{{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{6\,a}{5\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-5}}-{\frac{1}{2\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/2))^8,x)

[Out]

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

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Maxima [A]  time = 1.44373, size = 86, normalized size = 1.1 \[ -\frac{1}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} + \frac{6 \, a}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{4}} - \frac{a^{2}}{{\left (b \sqrt{x} + a\right )}^{6} b^{4}} + \frac{2 \, a^{3}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227216, size = 147, normalized size = 1.88 \[ -\frac{21 \, a b^{2} x + a^{3} + 7 \,{\left (5 \, b^{3} x + a^{2} b\right )} \sqrt{x}}{70 \,{\left (7 \, a b^{10} x^{3} + 35 \, a^{3} b^{8} x^{2} + 21 \, a^{5} b^{6} x + a^{7} b^{4} +{\left (b^{11} x^{3} + 21 \, a^{2} b^{9} x^{2} + 35 \, a^{4} b^{7} x + 7 \, a^{6} b^{5}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/70*(21*a*b^2*x + a^3 + 7*(5*b^3*x + a^2*b)*sqrt(x))/(7*a*b^10*x^3 + 35*a^3*b^
8*x^2 + 21*a^5*b^6*x + a^7*b^4 + (b^11*x^3 + 21*a^2*b^9*x^2 + 35*a^4*b^7*x + 7*a
^6*b^5)*sqrt(x))

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Sympy [A]  time = 30.2573, size = 410, normalized size = 5.26 \[ \begin{cases} - \frac{a^{3}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt{x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac{3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac{5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac{7}{2}}} - \frac{7 a^{2} b \sqrt{x}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt{x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac{3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac{5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac{7}{2}}} - \frac{21 a b^{2} x}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt{x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac{3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac{5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac{7}{2}}} - \frac{35 b^{3} x^{\frac{3}{2}}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt{x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac{3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac{5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac{7}{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{8}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**3/(70*a**7*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x + 2450
*a**4*b**7*x**(3/2) + 2450*a**3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a*b**1
0*x**3 + 70*b**11*x**(7/2)) - 7*a**2*b*sqrt(x)/(70*a**7*b**4 + 490*a**6*b**5*sqr
t(x) + 1470*a**5*b**6*x + 2450*a**4*b**7*x**(3/2) + 2450*a**3*b**8*x**2 + 1470*a
**2*b**9*x**(5/2) + 490*a*b**10*x**3 + 70*b**11*x**(7/2)) - 21*a*b**2*x/(70*a**7
*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x + 2450*a**4*b**7*x**(3/2) + 245
0*a**3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a*b**10*x**3 + 70*b**11*x**(7/2
)) - 35*b**3*x**(3/2)/(70*a**7*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x +
2450*a**4*b**7*x**(3/2) + 2450*a**3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a
*b**10*x**3 + 70*b**11*x**(7/2)), Ne(b, 0)), (x**2/(2*a**8), True))

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GIAC/XCAS [A]  time = 0.224605, size = 57, normalized size = 0.73 \[ -\frac{35 \, b^{3} x^{\frac{3}{2}} + 21 \, a b^{2} x + 7 \, a^{2} b \sqrt{x} + a^{3}}{70 \,{\left (b \sqrt{x} + a\right )}^{7} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/70*(35*b^3*x^(3/2) + 21*a*b^2*x + 7*a^2*b*sqrt(x) + a^3)/((b*sqrt(x) + a)^7*b
^4)