3.2211 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^3 x^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{20 b^3}{a^6 \sqrt{x}}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}-\frac{6 b^2}{a^5 x}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]

[Out]

b^4/(a^5*(a + b*Sqrt[x])^2) + (10*b^4)/(a^6*(a + b*Sqrt[x])) - 1/(2*a^3*x^2) + (
2*b)/(a^4*x^(3/2)) - (6*b^2)/(a^5*x) + (20*b^3)/(a^6*Sqrt[x]) - (30*b^4*Log[a +
b*Sqrt[x]])/a^7 + (15*b^4*Log[x])/a^7

_______________________________________________________________________________________

Rubi [A]  time = 0.179952, antiderivative size = 111, 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 \[ -\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{20 b^3}{a^6 \sqrt{x}}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}-\frac{6 b^2}{a^5 x}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

b^4/(a^5*(a + b*Sqrt[x])^2) + (10*b^4)/(a^6*(a + b*Sqrt[x])) - 1/(2*a^3*x^2) + (
2*b)/(a^4*x^(3/2)) - (6*b^2)/(a^5*x) + (20*b^3)/(a^6*Sqrt[x]) - (30*b^4*Log[a +
b*Sqrt[x]])/a^7 + (15*b^4*Log[x])/a^7

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.8141, size = 112, normalized size = 1.01 \[ - \frac{1}{2 a^{3} x^{2}} + \frac{2 b}{a^{4} x^{\frac{3}{2}}} + \frac{b^{4}}{a^{5} \left (a + b \sqrt{x}\right )^{2}} - \frac{6 b^{2}}{a^{5} x} + \frac{10 b^{4}}{a^{6} \left (a + b \sqrt{x}\right )} + \frac{20 b^{3}}{a^{6} \sqrt{x}} + \frac{30 b^{4} \log{\left (\sqrt{x} \right )}}{a^{7}} - \frac{30 b^{4} \log{\left (a + b \sqrt{x} \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a**3*x**2) + 2*b/(a**4*x**(3/2)) + b**4/(a**5*(a + b*sqrt(x))**2) - 6*b**2
/(a**5*x) + 10*b**4/(a**6*(a + b*sqrt(x))) + 20*b**3/(a**6*sqrt(x)) + 30*b**4*lo
g(sqrt(x))/a**7 - 30*b**4*log(a + b*sqrt(x))/a**7

_______________________________________________________________________________________

Mathematica [A]  time = 0.205424, size = 104, normalized size = 0.94 \[ \frac{\frac{a \left (-a^5+2 a^4 b \sqrt{x}-5 a^3 b^2 x+20 a^2 b^3 x^{3/2}+90 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^2 \left (a+b \sqrt{x}\right )^2}-60 b^4 \log \left (a+b \sqrt{x}\right )+30 b^4 \log (x)}{2 a^7} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-a^5 + 2*a^4*b*Sqrt[x] - 5*a^3*b^2*x + 20*a^2*b^3*x^(3/2) + 90*a*b^4*x^2 +
60*b^5*x^(5/2)))/((a + b*Sqrt[x])^2*x^2) - 60*b^4*Log[a + b*Sqrt[x]] + 30*b^4*Lo
g[x])/(2*a^7)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 100, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}{a}^{3}}}+2\,{\frac{b}{{a}^{4}{x}^{3/2}}}-6\,{\frac{{b}^{2}}{x{a}^{5}}}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+20\,{\frac{{b}^{3}}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{4}}{{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-2}}+10\,{\frac{{b}^{4}}{{a}^{6} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^(1/2))^3,x)

[Out]

-1/2/x^2/a^3+2*b/a^4/x^(3/2)-6*b^2/x/a^5+15*b^4*ln(x)/a^7-30*b^4*ln(a+b*x^(1/2))
/a^7+20*b^3/a^6/x^(1/2)+b^4/a^5/(a+b*x^(1/2))^2+10*b^4/a^6/(a+b*x^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.44644, size = 149, normalized size = 1.34 \[ \frac{60 \, b^{5} x^{\frac{5}{2}} + 90 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} - 5 \, a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} - a^{5}}{2 \,{\left (a^{6} b^{2} x^{3} + 2 \, a^{7} b x^{\frac{5}{2}} + a^{8} x^{2}\right )}} - \frac{30 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{4} \log \left (x\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(60*b^5*x^(5/2) + 90*a*b^4*x^2 + 20*a^2*b^3*x^(3/2) - 5*a^3*b^2*x + 2*a^4*b*
sqrt(x) - a^5)/(a^6*b^2*x^3 + 2*a^7*b*x^(5/2) + a^8*x^2) - 30*b^4*log(b*sqrt(x)
+ a)/a^7 + 15*b^4*log(x)/a^7

_______________________________________________________________________________________

Fricas [A]  time = 0.240411, size = 211, normalized size = 1.9 \[ \frac{90 \, a^{2} b^{4} x^{2} - 5 \, a^{4} b^{2} x - a^{6} - 60 \,{\left (b^{6} x^{3} + 2 \, a b^{5} x^{\frac{5}{2}} + a^{2} b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) + 60 \,{\left (b^{6} x^{3} + 2 \, a b^{5} x^{\frac{5}{2}} + a^{2} b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) + 2 \,{\left (30 \, a b^{5} x^{2} + 10 \, a^{3} b^{3} x + a^{5} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{2} x^{3} + 2 \, a^{8} b x^{\frac{5}{2}} + a^{9} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(90*a^2*b^4*x^2 - 5*a^4*b^2*x - a^6 - 60*(b^6*x^3 + 2*a*b^5*x^(5/2) + a^2*b^
4*x^2)*log(b*sqrt(x) + a) + 60*(b^6*x^3 + 2*a*b^5*x^(5/2) + a^2*b^4*x^2)*log(sqr
t(x)) + 2*(30*a*b^5*x^2 + 10*a^3*b^3*x + a^5*b)*sqrt(x))/(a^7*b^2*x^3 + 2*a^8*b*
x^(5/2) + a^9*x^2)

_______________________________________________________________________________________

Sympy [A]  time = 38.4782, size = 612, normalized size = 5.51 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b^{3} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{1}{2 a^{3} x^{2}} & \text{for}\: b = 0 \\- \frac{a^{6} \sqrt{x}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{2 a^{5} b x}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{5 a^{4} b^{2} x^{\frac{3}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{20 a^{3} b^{3} x^{2}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{90 a^{2} b^{4} x^{\frac{5}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{120 a b^{5} x^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 b^{6} x^{\frac{7}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 b^{6} x^{\frac{7}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b**3*x**(7/2)), Eq(a, 0)),
(-1/(2*a**3*x**2), Eq(b, 0)), (-a**6*sqrt(x)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 +
2*a**7*b**2*x**(7/2)) + 2*a**5*b*x/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b*
*2*x**(7/2)) - 5*a**4*b**2*x**(3/2)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b*
*2*x**(7/2)) + 20*a**3*b**3*x**2/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*
x**(7/2)) + 30*a**2*b**4*x**(5/2)*log(x)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a*
*7*b**2*x**(7/2)) - 60*a**2*b**4*x**(5/2)*log(a/b + sqrt(x))/(2*a**9*x**(5/2) +
4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 90*a**2*b**4*x**(5/2)/(2*a**9*x**(5/2) +
4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 60*a*b**5*x**3*log(x)/(2*a**9*x**(5/2)
+ 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) - 120*a*b**5*x**3*log(a/b + sqrt(x))/(2*
a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 60*a*b**5*x**3/(2*a**9*x
**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 30*b**6*x**(7/2)*log(x)/(2*a**
9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) - 60*b**6*x**(7/2)*log(a/b +
sqrt(x))/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295352, size = 136, normalized size = 1.23 \[ -\frac{30 \, b^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 90 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac{3}{2}} - 5 \, a^{4} b^{2} x + 2 \, a^{5} b \sqrt{x} - a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{2} a^{7} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30*b^4*ln(abs(b*sqrt(x) + a))/a^7 + 15*b^4*ln(abs(x))/a^7 + 1/2*(60*a*b^5*x^(5/
2) + 90*a^2*b^4*x^2 + 20*a^3*b^3*x^(3/2) - 5*a^4*b^2*x + 2*a^5*b*sqrt(x) - a^6)/
((b*sqrt(x) + a)^2*a^7*x^2)