3.2203 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^2 x^3} \, dx\)

Optimal. Leaf size=95 \[ -\frac{10 b^4 \log \left (a+b \sqrt{x}\right )}{a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{2 b^4}{a^5 \left (a+b \sqrt{x}\right )}+\frac{8 b^3}{a^5 \sqrt{x}}-\frac{3 b^2}{a^4 x}+\frac{4 b}{3 a^3 x^{3/2}}-\frac{1}{2 a^2 x^2} \]

[Out]

(2*b^4)/(a^5*(a + b*Sqrt[x])) - 1/(2*a^2*x^2) + (4*b)/(3*a^3*x^(3/2)) - (3*b^2)/
(a^4*x) + (8*b^3)/(a^5*Sqrt[x]) - (10*b^4*Log[a + b*Sqrt[x]])/a^6 + (5*b^4*Log[x
])/a^6

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Rubi [A]  time = 0.143036, antiderivative size = 95, 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 \[ -\frac{10 b^4 \log \left (a+b \sqrt{x}\right )}{a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{2 b^4}{a^5 \left (a+b \sqrt{x}\right )}+\frac{8 b^3}{a^5 \sqrt{x}}-\frac{3 b^2}{a^4 x}+\frac{4 b}{3 a^3 x^{3/2}}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^2*x^3),x]

[Out]

(2*b^4)/(a^5*(a + b*Sqrt[x])) - 1/(2*a^2*x^2) + (4*b)/(3*a^3*x^(3/2)) - (3*b^2)/
(a^4*x) + (8*b^3)/(a^5*Sqrt[x]) - (10*b^4*Log[a + b*Sqrt[x]])/a^6 + (5*b^4*Log[x
])/a^6

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Rubi in Sympy [A]  time = 20.3192, size = 97, normalized size = 1.02 \[ - \frac{1}{2 a^{2} x^{2}} + \frac{4 b}{3 a^{3} x^{\frac{3}{2}}} - \frac{3 b^{2}}{a^{4} x} + \frac{2 b^{4}}{a^{5} \left (a + b \sqrt{x}\right )} + \frac{8 b^{3}}{a^{5} \sqrt{x}} + \frac{10 b^{4} \log{\left (\sqrt{x} \right )}}{a^{6}} - \frac{10 b^{4} \log{\left (a + b \sqrt{x} \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*x**(1/2))**2,x)

[Out]

-1/(2*a**2*x**2) + 4*b/(3*a**3*x**(3/2)) - 3*b**2/(a**4*x) + 2*b**4/(a**5*(a + b
*sqrt(x))) + 8*b**3/(a**5*sqrt(x)) + 10*b**4*log(sqrt(x))/a**6 - 10*b**4*log(a +
b*sqrt(x))/a**6

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Mathematica [A]  time = 0.13143, size = 91, normalized size = 0.96 \[ \frac{\frac{a \left (-3 a^4+5 a^3 b \sqrt{x}-10 a^2 b^2 x+30 a b^3 x^{3/2}+60 b^4 x^2\right )}{x^2 \left (a+b \sqrt{x}\right )}-60 b^4 \log \left (a+b \sqrt{x}\right )+30 b^4 \log (x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^2*x^3),x]

[Out]

((a*(-3*a^4 + 5*a^3*b*Sqrt[x] - 10*a^2*b^2*x + 30*a*b^3*x^(3/2) + 60*b^4*x^2))/(
(a + b*Sqrt[x])*x^2) - 60*b^4*Log[a + b*Sqrt[x]] + 30*b^4*Log[x])/(6*a^6)

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Maple [A]  time = 0.016, size = 84, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}+{\frac{4\,b}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}+5\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{6}}}-10\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{6}}}+8\,{\frac{{b}^{3}}{{a}^{5}\sqrt{x}}}+2\,{\frac{{b}^{4}}{{a}^{5} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 1.44042, size = 119, normalized size = 1.25 \[ \frac{60 \, b^{4} x^{2} + 30 \, a b^{3} x^{\frac{3}{2}} - 10 \, a^{2} b^{2} x + 5 \, a^{3} b \sqrt{x} - 3 \, a^{4}}{6 \,{\left (a^{5} b x^{\frac{5}{2}} + a^{6} x^{2}\right )}} - \frac{10 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{6}} + \frac{5 \, b^{4} \log \left (x\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^3),x, algorithm="maxima")

[Out]

1/6*(60*b^4*x^2 + 30*a*b^3*x^(3/2) - 10*a^2*b^2*x + 5*a^3*b*sqrt(x) - 3*a^4)/(a^
5*b*x^(5/2) + a^6*x^2) - 10*b^4*log(b*sqrt(x) + a)/a^6 + 5*b^4*log(x)/a^6

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Fricas [A]  time = 0.246651, size = 151, normalized size = 1.59 \[ \frac{60 \, a b^{4} x^{2} - 10 \, a^{3} b^{2} x - 3 \, a^{5} - 60 \,{\left (b^{5} x^{\frac{5}{2}} + a b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) + 60 \,{\left (b^{5} x^{\frac{5}{2}} + a b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) + 5 \,{\left (6 \, a^{2} b^{3} x + a^{4} b\right )} \sqrt{x}}{6 \,{\left (a^{6} b x^{\frac{5}{2}} + a^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^3),x, algorithm="fricas")

[Out]

1/6*(60*a*b^4*x^2 - 10*a^3*b^2*x - 3*a^5 - 60*(b^5*x^(5/2) + a*b^4*x^2)*log(b*sq
rt(x) + a) + 60*(b^5*x^(5/2) + a*b^4*x^2)*log(sqrt(x)) + 5*(6*a^2*b^3*x + a^4*b)
*sqrt(x))/(a^6*b*x^(5/2) + a^7*x^2)

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Sympy [A]  time = 15.0083, size = 333, normalized size = 3.51 \[ \begin{cases} \frac{\tilde{\infty }}{x^{3}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 a^{2} x^{2}} & \text{for}\: b = 0 \\- \frac{1}{3 b^{2} x^{3}} & \text{for}\: a = 0 \\- \frac{3 a^{5} \sqrt{x}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{5 a^{4} b x}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{10 a^{3} b^{2} x^{\frac{3}{2}}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 a^{2} b^{3} x^{2}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 a b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{60 a b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{60 a b^{4} x^{\frac{5}{2}}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 b^{5} x^{3} \log{\left (x \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{60 b^{5} x^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*x**(1/2))**2,x)

[Out]

Piecewise((zoo/x**3, Eq(a, 0) & Eq(b, 0)), (-1/(2*a**2*x**2), Eq(b, 0)), (-1/(3*
b**2*x**3), Eq(a, 0)), (-3*a**5*sqrt(x)/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 5*a*
*4*b*x/(6*a**7*x**(5/2) + 6*a**6*b*x**3) - 10*a**3*b**2*x**(3/2)/(6*a**7*x**(5/2
) + 6*a**6*b*x**3) + 30*a**2*b**3*x**2/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 30*a*
b**4*x**(5/2)*log(x)/(6*a**7*x**(5/2) + 6*a**6*b*x**3) - 60*a*b**4*x**(5/2)*log(
a/b + sqrt(x))/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 60*a*b**4*x**(5/2)/(6*a**7*x*
*(5/2) + 6*a**6*b*x**3) + 30*b**5*x**3*log(x)/(6*a**7*x**(5/2) + 6*a**6*b*x**3)
- 60*b**5*x**3*log(a/b + sqrt(x))/(6*a**7*x**(5/2) + 6*a**6*b*x**3), True))

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GIAC/XCAS [A]  time = 0.272017, size = 122, normalized size = 1.28 \[ -\frac{10 \, b^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{6}} + \frac{5 \, b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, a b^{4} x^{2} + 30 \, a^{2} b^{3} x^{\frac{3}{2}} - 10 \, a^{3} b^{2} x + 5 \, a^{4} b \sqrt{x} - 3 \, a^{5}}{6 \,{\left (b \sqrt{x} + a\right )} a^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^3),x, algorithm="giac")

[Out]

-10*b^4*ln(abs(b*sqrt(x) + a))/a^6 + 5*b^4*ln(abs(x))/a^6 + 1/6*(60*a*b^4*x^2 +
30*a^2*b^3*x^(3/2) - 10*a^3*b^2*x + 5*a^4*b*sqrt(x) - 3*a^5)/((b*sqrt(x) + a)*a^
6*x^2)