3.2217 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^5} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{2 b^2 \left (a+b \sqrt{x}\right )^4}-\frac{2}{3 b^2 \left (a+b \sqrt{x}\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.0522011, 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 \[ \frac{a}{2 b^2 \left (a+b \sqrt{x}\right )^4}-\frac{2}{3 b^2 \left (a+b \sqrt{x}\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^(-5),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.36315, size = 48, normalized size = 1.26 \[ \frac{x}{2 a \left (a + b \sqrt{x}\right )^{4}} + \frac{x}{3 a^{2} \left (a + b \sqrt{x}\right )^{3}} + \frac{x}{6 a^{3} \left (a + b \sqrt{x}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**(1/2))**5,x)

[Out]

x/(2*a*(a + b*sqrt(x))**4) + x/(3*a**2*(a + b*sqrt(x))**3) + x/(6*a**3*(a + b*sq
rt(x))**2)

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Mathematica [A]  time = 0.0159688, size = 28, normalized size = 0.74 \[ -\frac{a+4 b \sqrt{x}}{6 b^2 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^(-5),x]

[Out]

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

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Maple [B]  time = 0.063, size = 200, normalized size = 5.3 \[ -{\frac{1}{3\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-3}}-{\frac{a}{4\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-4}}-{\frac{1}{3\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{a}{4\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{{a}^{5}}{4\, \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{2}}}-5\,a{b}^{4} \left ( -1/4\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-2/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}}-1/2\,{\frac{1}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{2}}} \right ) -10\,{a}^{3}{b}^{2} \left ( -1/4\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-1/3\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{3}{b}^{4}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2))^5,x)

[Out]

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

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Maxima [A]  time = 1.43151, size = 41, normalized size = 1.08 \[ -\frac{2}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{2}} + \frac{a}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239021, size = 74, normalized size = 1.95 \[ -\frac{4 \, b \sqrt{x} + a}{6 \,{\left (b^{6} x^{2} + 6 \, a^{2} b^{4} x + a^{4} b^{2} + 4 \,{\left (a b^{5} x + a^{3} b^{3}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.15061, size = 121, normalized size = 3.18 \[ \begin{cases} - \frac{a}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt{x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac{3}{2}} + 6 b^{6} x^{2}} - \frac{4 b \sqrt{x}}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt{x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac{3}{2}} + 6 b^{6} x^{2}} & \text{for}\: b \neq 0 \\\frac{x}{a^{5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((-a/(6*a**4*b**2 + 24*a**3*b**3*sqrt(x) + 36*a**2*b**4*x + 24*a*b**5*x
**(3/2) + 6*b**6*x**2) - 4*b*sqrt(x)/(6*a**4*b**2 + 24*a**3*b**3*sqrt(x) + 36*a*
*2*b**4*x + 24*a*b**5*x**(3/2) + 6*b**6*x**2), Ne(b, 0)), (x/a**5, True))

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GIAC/XCAS [A]  time = 0.263334, size = 30, normalized size = 0.79 \[ -\frac{4 \, b \sqrt{x} + a}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-5),x, algorithm="giac")

[Out]

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