3.2226 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^8} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]

[Out]

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

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Rubi [A]  time = 0.0512334, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^(-8),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.79497, size = 105, normalized size = 2.76 \[ \frac{2 x}{7 a \left (a + b \sqrt{x}\right )^{7}} + \frac{5 x}{21 a^{2} \left (a + b \sqrt{x}\right )^{6}} + \frac{4 x}{21 a^{3} \left (a + b \sqrt{x}\right )^{5}} + \frac{x}{7 a^{4} \left (a + b \sqrt{x}\right )^{4}} + \frac{2 x}{21 a^{5} \left (a + b \sqrt{x}\right )^{3}} + \frac{x}{21 a^{6} \left (a + b \sqrt{x}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.01407, size = 28, normalized size = 0.74 \[ -\frac{a+7 b \sqrt{x}}{21 b^2 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.102, size = 399, normalized size = 10.5 \[{\frac{1}{6\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-6}}-{\frac{1}{6\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{b}^{8} \left ( -{\frac{{a}^{2}}{{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-{\frac{2\,{a}^{6}}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-{\frac{6\,{a}^{4}}{5\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}}-{\frac{{a}^{8}}{7\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-{\frac{1}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}} \right ) -{\frac{{a}^{8}}{7\, \left ({b}^{2}x-{a}^{2} \right ) ^{7}{b}^{2}}}+{\frac{a}{7\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-7}}+{\frac{a}{7\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-7}}+28\,{a}^{2}{b}^{6} \left ( -1/4\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{8}}}-1/2\,{\frac{{a}^{4}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-3/5\,{\frac{{a}^{2}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}}-1/7\,{\frac{{a}^{6}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}} \right ) +70\,{a}^{4}{b}^{4} \left ( -1/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/5\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{5}{b}^{6}}}-1/7\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}} \right ) +28\,{a}^{6}{b}^{2} \left ( -1/6\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{6}{b}^{4}}}-1/7\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2))^8,x)

[Out]

1/6/b^2/(b*x^(1/2)-a)^6-1/6/b^2/(a+b*x^(1/2))^6+b^8*(-a^2/b^10/(b^2*x-a^2)^4-2/3
*a^6/b^10/(b^2*x-a^2)^6-6/5*a^4/b^10/(b^2*x-a^2)^5-1/7*a^8/b^10/(b^2*x-a^2)^7-1/
3/b^10/(b^2*x-a^2)^3)-1/7*a^8/(b^2*x-a^2)^7/b^2+1/7/b^2*a/(b*x^(1/2)-a)^7+1/7*a/
b^2/(a+b*x^(1/2))^7+28*a^2*b^6*(-1/4/(b^2*x-a^2)^4/b^8-1/2*a^4/b^8/(b^2*x-a^2)^6
-3/5*a^2/b^8/(b^2*x-a^2)^5-1/7*a^6/b^8/(b^2*x-a^2)^7)+70*a^4*b^4*(-1/3*a^2/b^6/(
b^2*x-a^2)^6-1/5/(b^2*x-a^2)^5/b^6-1/7*a^4/b^6/(b^2*x-a^2)^7)+28*a^6*b^2*(-1/6/(
b^2*x-a^2)^6/b^4-1/7*a^2/b^4/(b^2*x-a^2)^7)

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Maxima [A]  time = 1.44282, size = 41, normalized size = 1.08 \[ -\frac{1}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{2}} + \frac{2 \, a}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236788, size = 120, normalized size = 3.16 \[ -\frac{7 \, b \sqrt{x} + a}{21 \,{\left (7 \, a b^{8} x^{3} + 35 \, a^{3} b^{6} x^{2} + 21 \, a^{5} b^{4} x + a^{7} b^{2} +{\left (b^{9} x^{3} + 21 \, a^{2} b^{7} x^{2} + 35 \, a^{4} b^{5} x + 7 \, a^{6} b^{3}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 29.2737, size = 199, normalized size = 5.24 \[ \begin{cases} - \frac{a}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} - \frac{7 b \sqrt{x}}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{a^{8}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261001, size = 30, normalized size = 0.79 \[ -\frac{7 \, b \sqrt{x} + a}{21 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(7*b*sqrt(x) + a)/((b*sqrt(x) + a)^7*b^2)