3.2204 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^2 x^4} \, dx\)

Optimal. Leaf size=123 \[ -\frac{14 b^6 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{2 b^6}{a^7 \left (a+b \sqrt{x}\right )}+\frac{12 b^5}{a^7 \sqrt{x}}-\frac{5 b^4}{a^6 x}+\frac{8 b^3}{3 a^5 x^{3/2}}-\frac{3 b^2}{2 a^4 x^2}+\frac{4 b}{5 a^3 x^{5/2}}-\frac{1}{3 a^2 x^3} \]

[Out]

(2*b^6)/(a^7*(a + b*Sqrt[x])) - 1/(3*a^2*x^3) + (4*b)/(5*a^3*x^(5/2)) - (3*b^2)/
(2*a^4*x^2) + (8*b^3)/(3*a^5*x^(3/2)) - (5*b^4)/(a^6*x) + (12*b^5)/(a^7*Sqrt[x])
- (14*b^6*Log[a + b*Sqrt[x]])/a^8 + (7*b^6*Log[x])/a^8

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Rubi [A]  time = 0.182053, antiderivative size = 123, 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 \[ -\frac{14 b^6 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{2 b^6}{a^7 \left (a+b \sqrt{x}\right )}+\frac{12 b^5}{a^7 \sqrt{x}}-\frac{5 b^4}{a^6 x}+\frac{8 b^3}{3 a^5 x^{3/2}}-\frac{3 b^2}{2 a^4 x^2}+\frac{4 b}{5 a^3 x^{5/2}}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^2*x^4),x]

[Out]

(2*b^6)/(a^7*(a + b*Sqrt[x])) - 1/(3*a^2*x^3) + (4*b)/(5*a^3*x^(5/2)) - (3*b^2)/
(2*a^4*x^2) + (8*b^3)/(3*a^5*x^(3/2)) - (5*b^4)/(a^6*x) + (12*b^5)/(a^7*Sqrt[x])
- (14*b^6*Log[a + b*Sqrt[x]])/a^8 + (7*b^6*Log[x])/a^8

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Rubi in Sympy [A]  time = 26.8341, size = 126, normalized size = 1.02 \[ - \frac{1}{3 a^{2} x^{3}} + \frac{4 b}{5 a^{3} x^{\frac{5}{2}}} - \frac{3 b^{2}}{2 a^{4} x^{2}} + \frac{8 b^{3}}{3 a^{5} x^{\frac{3}{2}}} - \frac{5 b^{4}}{a^{6} x} + \frac{2 b^{6}}{a^{7} \left (a + b \sqrt{x}\right )} + \frac{12 b^{5}}{a^{7} \sqrt{x}} + \frac{14 b^{6} \log{\left (\sqrt{x} \right )}}{a^{8}} - \frac{14 b^{6} \log{\left (a + b \sqrt{x} \right )}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(a+b*x**(1/2))**2,x)

[Out]

-1/(3*a**2*x**3) + 4*b/(5*a**3*x**(5/2)) - 3*b**2/(2*a**4*x**2) + 8*b**3/(3*a**5
*x**(3/2)) - 5*b**4/(a**6*x) + 2*b**6/(a**7*(a + b*sqrt(x))) + 12*b**5/(a**7*sqr
t(x)) + 14*b**6*log(sqrt(x))/a**8 - 14*b**6*log(a + b*sqrt(x))/a**8

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Mathematica [A]  time = 0.17373, size = 115, normalized size = 0.93 \[ \frac{\frac{a \left (-10 a^6+14 a^5 b \sqrt{x}-21 a^4 b^2 x+35 a^3 b^3 x^{3/2}-70 a^2 b^4 x^2+210 a b^5 x^{5/2}+420 b^6 x^3\right )}{x^3 \left (a+b \sqrt{x}\right )}-420 b^6 \log \left (a+b \sqrt{x}\right )+210 b^6 \log (x)}{30 a^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^2*x^4),x]

[Out]

((a*(-10*a^6 + 14*a^5*b*Sqrt[x] - 21*a^4*b^2*x + 35*a^3*b^3*x^(3/2) - 70*a^2*b^4
*x^2 + 210*a*b^5*x^(5/2) + 420*b^6*x^3))/((a + b*Sqrt[x])*x^3) - 420*b^6*Log[a +
b*Sqrt[x]] + 210*b^6*Log[x])/(30*a^8)

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Maple [A]  time = 0.02, size = 106, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}}+{\frac{4\,b}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}+{\frac{8\,{b}^{3}}{3\,{a}^{5}}{x}^{-{\frac{3}{2}}}}-5\,{\frac{{b}^{4}}{{a}^{6}x}}+7\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{8}}}-14\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{8}}}+12\,{\frac{{b}^{5}}{{a}^{7}\sqrt{x}}}+2\,{\frac{{b}^{6}}{{a}^{7} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a+b*x^(1/2))^2,x)

[Out]

-1/3/x^3/a^2+4/5*b/a^3/x^(5/2)-3/2*b^2/a^4/x^2+8/3*b^3/a^5/x^(3/2)-5*b^4/a^6/x+7
*b^6*ln(x)/a^8-14*b^6*ln(a+b*x^(1/2))/a^8+12*b^5/a^7/x^(1/2)+2*b^6/a^7/(a+b*x^(1
/2))

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Maxima [A]  time = 1.44335, size = 149, normalized size = 1.21 \[ \frac{420 \, b^{6} x^{3} + 210 \, a b^{5} x^{\frac{5}{2}} - 70 \, a^{2} b^{4} x^{2} + 35 \, a^{3} b^{3} x^{\frac{3}{2}} - 21 \, a^{4} b^{2} x + 14 \, a^{5} b \sqrt{x} - 10 \, a^{6}}{30 \,{\left (a^{7} b x^{\frac{7}{2}} + a^{8} x^{3}\right )}} - \frac{14 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^4),x, algorithm="maxima")

[Out]

1/30*(420*b^6*x^3 + 210*a*b^5*x^(5/2) - 70*a^2*b^4*x^2 + 35*a^3*b^3*x^(3/2) - 21
*a^4*b^2*x + 14*a^5*b*sqrt(x) - 10*a^6)/(a^7*b*x^(7/2) + a^8*x^3) - 14*b^6*log(b
*sqrt(x) + a)/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 0.245178, size = 182, normalized size = 1.48 \[ \frac{420 \, a b^{6} x^{3} - 70 \, a^{3} b^{4} x^{2} - 21 \, a^{5} b^{2} x - 10 \, a^{7} - 420 \,{\left (b^{7} x^{\frac{7}{2}} + a b^{6} x^{3}\right )} \log \left (b \sqrt{x} + a\right ) + 420 \,{\left (b^{7} x^{\frac{7}{2}} + a b^{6} x^{3}\right )} \log \left (\sqrt{x}\right ) + 7 \,{\left (30 \, a^{2} b^{5} x^{2} + 5 \, a^{4} b^{3} x + 2 \, a^{6} b\right )} \sqrt{x}}{30 \,{\left (a^{8} b x^{\frac{7}{2}} + a^{9} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^4),x, algorithm="fricas")

[Out]

1/30*(420*a*b^6*x^3 - 70*a^3*b^4*x^2 - 21*a^5*b^2*x - 10*a^7 - 420*(b^7*x^(7/2)
+ a*b^6*x^3)*log(b*sqrt(x) + a) + 420*(b^7*x^(7/2) + a*b^6*x^3)*log(sqrt(x)) + 7
*(30*a^2*b^5*x^2 + 5*a^4*b^3*x + 2*a^6*b)*sqrt(x))/(a^8*b*x^(7/2) + a^9*x^3)

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Sympy [A]  time = 19.7215, size = 396, normalized size = 3.22 \[ \begin{cases} \frac{\tilde{\infty }}{x^{4}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{4 b^{2} x^{4}} & \text{for}\: a = 0 \\- \frac{1}{3 a^{2} x^{3}} & \text{for}\: b = 0 \\- \frac{10 a^{7} \sqrt{x}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{14 a^{6} b x}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{21 a^{5} b^{2} x^{\frac{3}{2}}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{35 a^{4} b^{3} x^{2}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{70 a^{3} b^{4} x^{\frac{5}{2}}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 a^{2} b^{5} x^{3}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 a b^{6} x^{\frac{7}{2}} \log{\left (x \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 a b^{6} x^{\frac{7}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 b^{7} x^{4} \log{\left (x \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 b^{7} x^{4} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 b^{7} x^{4}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(a+b*x**(1/2))**2,x)

[Out]

Piecewise((zoo/x**4, Eq(a, 0) & Eq(b, 0)), (-1/(4*b**2*x**4), Eq(a, 0)), (-1/(3*
a**2*x**3), Eq(b, 0)), (-10*a**7*sqrt(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 1
4*a**6*b*x/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 21*a**5*b**2*x**(3/2)/(30*a**9*
x**(7/2) + 30*a**8*b*x**4) + 35*a**4*b**3*x**2/(30*a**9*x**(7/2) + 30*a**8*b*x**
4) - 70*a**3*b**4*x**(5/2)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 210*a**2*b**5*x
**3/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 210*a*b**6*x**(7/2)*log(x)/(30*a**9*x*
*(7/2) + 30*a**8*b*x**4) - 420*a*b**6*x**(7/2)*log(a/b + sqrt(x))/(30*a**9*x**(7
/2) + 30*a**8*b*x**4) + 210*b**7*x**4*log(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4)
- 420*b**7*x**4*log(a/b + sqrt(x))/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 420*b*
*7*x**4/(30*a**9*x**(7/2) + 30*a**8*b*x**4), True))

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GIAC/XCAS [A]  time = 0.243208, size = 151, normalized size = 1.23 \[ -\frac{14 \, b^{6}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{8}} + \frac{7 \, b^{6}{\rm ln}\left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{3} + 210 \, a^{2} b^{5} x^{\frac{5}{2}} - 70 \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x^{\frac{3}{2}} - 21 \, a^{5} b^{2} x + 14 \, a^{6} b \sqrt{x} - 10 \, a^{7}}{30 \,{\left (b \sqrt{x} + a\right )} a^{8} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^4),x, algorithm="giac")

[Out]

-14*b^6*ln(abs(b*sqrt(x) + a))/a^8 + 7*b^6*ln(abs(x))/a^8 + 1/30*(420*a*b^6*x^3
+ 210*a^2*b^5*x^(5/2) - 70*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) - 21*a^5*b^2*x + 14*
a^6*b*sqrt(x) - 10*a^7)/((b*sqrt(x) + a)*a^8*x^3)