3.2219 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^5 x^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{10 b}{a^6 \sqrt{x}}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}-\frac{1}{a^5 x}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.209466, antiderivative size = 126, 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 \[ -\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{10 b}{a^6 \sqrt{x}}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}-\frac{1}{a^5 x}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^5*x^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.4844, size = 124, normalized size = 0.98 \[ \frac{b^{2}}{2 a^{3} \left (a + b \sqrt{x}\right )^{4}} + \frac{2 b^{2}}{a^{4} \left (a + b \sqrt{x}\right )^{3}} + \frac{6 b^{2}}{a^{5} \left (a + b \sqrt{x}\right )^{2}} - \frac{1}{a^{5} x} + \frac{20 b^{2}}{a^{6} \left (a + b \sqrt{x}\right )} + \frac{10 b}{a^{6} \sqrt{x}} + \frac{30 b^{2} \log{\left (\sqrt{x} \right )}}{a^{7}} - \frac{30 b^{2} \log{\left (a + b \sqrt{x} \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.129912, size = 104, normalized size = 0.83 \[ \frac{\frac{a \left (-2 a^5+12 a^4 b \sqrt{x}+125 a^3 b^2 x+260 a^2 b^3 x^{3/2}+210 a b^4 x^2+60 b^5 x^{5/2}\right )}{x \left (a+b \sqrt{x}\right )^4}-60 b^2 \log \left (a+b \sqrt{x}\right )+30 b^2 \log (x)}{2 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^5*x^2),x]

[Out]

((a*(-2*a^5 + 12*a^4*b*Sqrt[x] + 125*a^3*b^2*x + 260*a^2*b^3*x^(3/2) + 210*a*b^4
*x^2 + 60*b^5*x^(5/2)))/((a + b*Sqrt[x])^4*x) - 60*b^2*Log[a + b*Sqrt[x]] + 30*b
^2*Log[x])/(2*a^7)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 113, normalized size = 0.9 \[ -{\frac{1}{x{a}^{5}}}+15\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+10\,{\frac{b}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{2}}{2\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-4}}+2\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\sqrt{x} \right ) ^{3}}}+6\,{\frac{{b}^{2}}{{a}^{5} \left ( a+b\sqrt{x} \right ) ^{2}}}+20\,{\frac{{b}^{2}}{{a}^{6} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^(1/2))^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44138, size = 176, normalized size = 1.4 \[ \frac{60 \, b^{5} x^{\frac{5}{2}} + 210 \, a b^{4} x^{2} + 260 \, a^{2} b^{3} x^{\frac{3}{2}} + 125 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt{x} - 2 \, a^{5}}{2 \,{\left (a^{6} b^{4} x^{3} + 4 \, a^{7} b^{3} x^{\frac{5}{2}} + 6 \, a^{8} b^{2} x^{2} + 4 \, a^{9} b x^{\frac{3}{2}} + a^{10} x\right )}} - \frac{30 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{2} \log \left (x\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^2),x, algorithm="maxima")

[Out]

1/2*(60*b^5*x^(5/2) + 210*a*b^4*x^2 + 260*a^2*b^3*x^(3/2) + 125*a^3*b^2*x + 12*a
^4*b*sqrt(x) - 2*a^5)/(a^6*b^4*x^3 + 4*a^7*b^3*x^(5/2) + 6*a^8*b^2*x^2 + 4*a^9*b
*x^(3/2) + a^10*x) - 30*b^2*log(b*sqrt(x) + a)/a^7 + 15*b^2*log(x)/a^7

_______________________________________________________________________________________

Fricas [A]  time = 0.246633, size = 301, normalized size = 2.39 \[ \frac{210 \, a^{2} b^{4} x^{2} + 125 \, a^{4} b^{2} x - 2 \, a^{6} - 60 \,{\left (b^{6} x^{3} + 6 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x + 4 \,{\left (a b^{5} x^{2} + a^{3} b^{3} x\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 60 \,{\left (b^{6} x^{3} + 6 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x + 4 \,{\left (a b^{5} x^{2} + a^{3} b^{3} x\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 4 \,{\left (15 \, a b^{5} x^{2} + 65 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{4} x^{3} + 6 \, a^{9} b^{2} x^{2} + a^{11} x + 4 \,{\left (a^{8} b^{3} x^{2} + a^{10} b x\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^2),x, algorithm="fricas")

[Out]

1/2*(210*a^2*b^4*x^2 + 125*a^4*b^2*x - 2*a^6 - 60*(b^6*x^3 + 6*a^2*b^4*x^2 + a^4
*b^2*x + 4*(a*b^5*x^2 + a^3*b^3*x)*sqrt(x))*log(b*sqrt(x) + a) + 60*(b^6*x^3 + 6
*a^2*b^4*x^2 + a^4*b^2*x + 4*(a*b^5*x^2 + a^3*b^3*x)*sqrt(x))*log(sqrt(x)) + 4*(
15*a*b^5*x^2 + 65*a^3*b^3*x + 3*a^5*b)*sqrt(x))/(a^7*b^4*x^3 + 6*a^9*b^2*x^2 + a
^11*x + 4*(a^8*b^3*x^2 + a^10*b*x)*sqrt(x))

_______________________________________________________________________________________

Sympy [A]  time = 55.2021, size = 1232, normalized size = 9.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b**5*x**(7/2)), Eq(a, 0)),
(-1/(a**5*x), Eq(b, 0)), (-2*a**6*sqrt(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 +
12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 12*a**5*b*x/(
2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2
*a**7*b**4*x**(7/2)) + 30*a**4*b**2*x**(3/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*
b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 60*a
**4*b**2*x**(3/2)*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**
9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 125*a**4*b**2*x**(3
/2)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**
3 + 2*a**7*b**4*x**(7/2)) + 120*a**3*b**3*x**2*log(x)/(2*a**11*x**(3/2) + 8*a**1
0*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 24
0*a**3*b**3*x**2*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9
*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 260*a**3*b**3*x**2/(
2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2
*a**7*b**4*x**(7/2)) + 180*a**2*b**4*x**(5/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10
*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 360
*a**2*b**4*x**(5/2)*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a
**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 210*a**2*b**4*x**
(5/2)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x
**3 + 2*a**7*b**4*x**(7/2)) + 120*a*b**5*x**3*log(x)/(2*a**11*x**(3/2) + 8*a**10
*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 240
*a*b**5*x**3*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**
2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 60*a*b**5*x**3/(2*a**11*
x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b*
*4*x**(7/2)) + 30*b**6*x**(7/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a
**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 60*b**6*x**(7/2)*
log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) +
8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253098, size = 136, normalized size = 1.08 \[ -\frac{30 \, b^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 210 \, a^{2} b^{4} x^{2} + 260 \, a^{3} b^{3} x^{\frac{3}{2}} + 125 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt{x} - 2 \, a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} a^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^2),x, algorithm="giac")

[Out]

-30*b^2*ln(abs(b*sqrt(x) + a))/a^7 + 15*b^2*ln(abs(x))/a^7 + 1/2*(60*a*b^5*x^(5/
2) + 210*a^2*b^4*x^2 + 260*a^3*b^3*x^(3/2) + 125*a^4*b^2*x + 12*a^5*b*sqrt(x) -
2*a^6)/((b*sqrt(x) + a)^4*a^7*x)