3.2222 \(\int \frac{x^4}{\left (a+b \sqrt{x}\right )^8} \, dx\)

Optimal. Leaf size=172 \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]

[Out]

(2*a^9)/(7*b^10*(a + b*Sqrt[x])^7) - (3*a^8)/(b^10*(a + b*Sqrt[x])^6) + (72*a^7)
/(5*b^10*(a + b*Sqrt[x])^5) - (42*a^6)/(b^10*(a + b*Sqrt[x])^4) + (84*a^5)/(b^10
*(a + b*Sqrt[x])^3) - (126*a^4)/(b^10*(a + b*Sqrt[x])^2) + (168*a^3)/(b^10*(a +
b*Sqrt[x])) - (16*a*Sqrt[x])/b^9 + x/b^8 + (72*a^2*Log[a + b*Sqrt[x]])/b^10

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Rubi [A]  time = 0.330569, antiderivative size = 172, 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 \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^9)/(7*b^10*(a + b*Sqrt[x])^7) - (3*a^8)/(b^10*(a + b*Sqrt[x])^6) + (72*a^7)
/(5*b^10*(a + b*Sqrt[x])^5) - (42*a^6)/(b^10*(a + b*Sqrt[x])^4) + (84*a^5)/(b^10
*(a + b*Sqrt[x])^3) - (126*a^4)/(b^10*(a + b*Sqrt[x])^2) + (168*a^3)/(b^10*(a +
b*Sqrt[x])) - (16*a*Sqrt[x])/b^9 + x/b^8 + (72*a^2*Log[a + b*Sqrt[x]])/b^10

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{9}}{7 b^{10} \left (a + b \sqrt{x}\right )^{7}} - \frac{3 a^{8}}{b^{10} \left (a + b \sqrt{x}\right )^{6}} + \frac{72 a^{7}}{5 b^{10} \left (a + b \sqrt{x}\right )^{5}} - \frac{42 a^{6}}{b^{10} \left (a + b \sqrt{x}\right )^{4}} + \frac{84 a^{5}}{b^{10} \left (a + b \sqrt{x}\right )^{3}} - \frac{126 a^{4}}{b^{10} \left (a + b \sqrt{x}\right )^{2}} + \frac{168 a^{3}}{b^{10} \left (a + b \sqrt{x}\right )} + \frac{72 a^{2} \log{\left (a + b \sqrt{x} \right )}}{b^{10}} - \frac{16 a \sqrt{x}}{b^{9}} + \frac{2 \int ^{\sqrt{x}} x\, dx}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b*x**(1/2))**8,x)

[Out]

2*a**9/(7*b**10*(a + b*sqrt(x))**7) - 3*a**8/(b**10*(a + b*sqrt(x))**6) + 72*a**
7/(5*b**10*(a + b*sqrt(x))**5) - 42*a**6/(b**10*(a + b*sqrt(x))**4) + 84*a**5/(b
**10*(a + b*sqrt(x))**3) - 126*a**4/(b**10*(a + b*sqrt(x))**2) + 168*a**3/(b**10
*(a + b*sqrt(x))) + 72*a**2*log(a + b*sqrt(x))/b**10 - 16*a*sqrt(x)/b**9 + 2*Int
egral(x, (x, sqrt(x)))/b**8

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Mathematica [A]  time = 0.0693656, size = 150, normalized size = 0.87 \[ \frac{3349 a^9+20923 a^8 b \sqrt{x}+53949 a^7 b^2 x+72275 a^6 b^3 x^{3/2}+50225 a^5 b^4 x^2+12495 a^4 b^5 x^{5/2}-4655 a^3 b^6 x^3-3185 a^2 b^7 x^{7/2}+2520 a^2 \left (a+b \sqrt{x}\right )^7 \log \left (a+b \sqrt{x}\right )-315 a b^8 x^4+35 b^9 x^{9/2}}{35 b^{10} \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*Sqrt[x])^8,x]

[Out]

(3349*a^9 + 20923*a^8*b*Sqrt[x] + 53949*a^7*b^2*x + 72275*a^6*b^3*x^(3/2) + 5022
5*a^5*b^4*x^2 + 12495*a^4*b^5*x^(5/2) - 4655*a^3*b^6*x^3 - 3185*a^2*b^7*x^(7/2)
- 315*a*b^8*x^4 + 35*b^9*x^(9/2) + 2520*a^2*(a + b*Sqrt[x])^7*Log[a + b*Sqrt[x]]
)/(35*b^10*(a + b*Sqrt[x])^7)

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Maple [A]  time = 0.016, size = 151, normalized size = 0.9 \[{\frac{x}{{b}^{8}}}+72\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{10}}}-16\,{\frac{a\sqrt{x}}{{b}^{9}}}+{\frac{2\,{a}^{9}}{7\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-7}}-3\,{\frac{{a}^{8}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{6}}}+{\frac{72\,{a}^{7}}{5\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-5}}-42\,{\frac{{a}^{6}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{4}}}+84\,{\frac{{a}^{5}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{3}}}-126\,{\frac{{a}^{4}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}+168\,{\frac{{a}^{3}}{{b}^{10} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b*x^(1/2))^8,x)

[Out]

x/b^8+72*a^2*ln(a+b*x^(1/2))/b^10-16*a*x^(1/2)/b^9+2/7*a^9/b^10/(a+b*x^(1/2))^7-
3*a^8/b^10/(a+b*x^(1/2))^6+72/5*a^7/b^10/(a+b*x^(1/2))^5-42*a^6/b^10/(a+b*x^(1/2
))^4+84*a^5/b^10/(a+b*x^(1/2))^3-126*a^4/b^10/(a+b*x^(1/2))^2+168*a^3/b^10/(a+b*
x^(1/2))

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Maxima [A]  time = 1.44218, size = 219, normalized size = 1.27 \[ \frac{72 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{10}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )} a}{b^{10}} + \frac{168 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{10}} - \frac{126 \, a^{4}}{{\left (b \sqrt{x} + a\right )}^{2} b^{10}} + \frac{84 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{3} b^{10}} - \frac{42 \, a^{6}}{{\left (b \sqrt{x} + a\right )}^{4} b^{10}} + \frac{72 \, a^{7}}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{10}} - \frac{3 \, a^{8}}{{\left (b \sqrt{x} + a\right )}^{6} b^{10}} + \frac{2 \, a^{9}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="maxima")

[Out]

72*a^2*log(b*sqrt(x) + a)/b^10 + (b*sqrt(x) + a)^2/b^10 - 18*(b*sqrt(x) + a)*a/b
^10 + 168*a^3/((b*sqrt(x) + a)*b^10) - 126*a^4/((b*sqrt(x) + a)^2*b^10) + 84*a^5
/((b*sqrt(x) + a)^3*b^10) - 42*a^6/((b*sqrt(x) + a)^4*b^10) + 72/5*a^7/((b*sqrt(
x) + a)^5*b^10) - 3*a^8/((b*sqrt(x) + a)^6*b^10) + 2/7*a^9/((b*sqrt(x) + a)^7*b^
10)

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Fricas [A]  time = 0.244276, size = 356, normalized size = 2.07 \[ -\frac{315 \, a b^{8} x^{4} + 4655 \, a^{3} b^{6} x^{3} - 50225 \, a^{5} b^{4} x^{2} - 53949 \, a^{7} b^{2} x - 3349 \, a^{9} - 2520 \,{\left (7 \, a^{3} b^{6} x^{3} + 35 \, a^{5} b^{4} x^{2} + 21 \, a^{7} b^{2} x + a^{9} +{\left (a^{2} b^{7} x^{3} + 21 \, a^{4} b^{5} x^{2} + 35 \, a^{6} b^{3} x + 7 \, a^{8} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) - 7 \,{\left (5 \, b^{9} x^{4} - 455 \, a^{2} b^{7} x^{3} + 1785 \, a^{4} b^{5} x^{2} + 10325 \, a^{6} b^{3} x + 2989 \, a^{8} b\right )} \sqrt{x}}{35 \,{\left (7 \, a b^{16} x^{3} + 35 \, a^{3} b^{14} x^{2} + 21 \, a^{5} b^{12} x + a^{7} b^{10} +{\left (b^{17} x^{3} + 21 \, a^{2} b^{15} x^{2} + 35 \, a^{4} b^{13} x + 7 \, a^{6} b^{11}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="fricas")

[Out]

-1/35*(315*a*b^8*x^4 + 4655*a^3*b^6*x^3 - 50225*a^5*b^4*x^2 - 53949*a^7*b^2*x -
3349*a^9 - 2520*(7*a^3*b^6*x^3 + 35*a^5*b^4*x^2 + 21*a^7*b^2*x + a^9 + (a^2*b^7*
x^3 + 21*a^4*b^5*x^2 + 35*a^6*b^3*x + 7*a^8*b)*sqrt(x))*log(b*sqrt(x) + a) - 7*(
5*b^9*x^4 - 455*a^2*b^7*x^3 + 1785*a^4*b^5*x^2 + 10325*a^6*b^3*x + 2989*a^8*b)*s
qrt(x))/(7*a*b^16*x^3 + 35*a^3*b^14*x^2 + 21*a^5*b^12*x + a^7*b^10 + (b^17*x^3 +
21*a^2*b^15*x^2 + 35*a^4*b^13*x + 7*a^6*b^11)*sqrt(x))

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Sympy [A]  time = 34.9145, size = 1945, normalized size = 11.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((2520*a**9*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x)
+ 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*
b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 5589*a**9/(35*a**7*b**1
0 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*
a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)
) + 17640*a**8*b*sqrt(x)*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt
(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a
**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 36603*a**8*b*sqrt(x
)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x
**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35
*b**17*x**(7/2)) + 52920*a**7*b**2*x*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**
6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*
x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 100989*
a**7*b**2*x/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a*
*4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16
*x**3 + 35*b**17*x**(7/2)) + 88200*a**6*b**3*x**(3/2)*log(a/b + sqrt(x))/(35*a**
7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) +
1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x*
*(7/2)) + 150675*a**6*b**3*x**(3/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 73
5*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**1
5*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 88200*a**5*b**4*x**2*log(a/
b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a
**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**1
6*x**3 + 35*b**17*x**(7/2)) + 128625*a**5*b**4*x**2/(35*a**7*b**10 + 245*a**6*b*
*11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2
+ 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 52920*a**4*
b**5*x**(5/2)*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a
**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x
**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 59535*a**4*b**5*x**(5/2)/(35*a
**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2)
+ 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*
x**(7/2)) + 17640*a**3*b**6*x**3*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b*
*11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2
+ 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 11025*a**3*
b**6*x**3/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4
*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x
**3 + 35*b**17*x**(7/2)) + 2520*a**2*b**7*x**(7/2)*log(a/b + sqrt(x))/(35*a**7*b
**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 12
25*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7
/2)) - 945*a**2*b**7*x**(7/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5
*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(
5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 315*a*b**8*x**4/(35*a**7*b**10 +
245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3
*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) +
35*b**9*x**(9/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 12
25*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*
b**16*x**3 + 35*b**17*x**(7/2)), Ne(b, 0)), (x**5/(5*a**8), True))

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GIAC/XCAS [A]  time = 0.219014, size = 161, normalized size = 0.94 \[ \frac{72 \, a^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{10}} + \frac{b^{8} x - 16 \, a b^{7} \sqrt{x}}{b^{16}} + \frac{5880 \, a^{3} b^{6} x^{3} + 30870 \, a^{4} b^{5} x^{\frac{5}{2}} + 69090 \, a^{5} b^{4} x^{2} + 83790 \, a^{6} b^{3} x^{\frac{3}{2}} + 57834 \, a^{7} b^{2} x + 21483 \, a^{8} b \sqrt{x} + 3349 \, a^{9}}{35 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="giac")

[Out]

72*a^2*ln(abs(b*sqrt(x) + a))/b^10 + (b^8*x - 16*a*b^7*sqrt(x))/b^16 + 1/35*(588
0*a^3*b^6*x^3 + 30870*a^4*b^5*x^(5/2) + 69090*a^5*b^4*x^2 + 83790*a^6*b^3*x^(3/2
) + 57834*a^7*b^2*x + 21483*a^8*b*sqrt(x) + 3349*a^9)/((b*sqrt(x) + a)^7*b^10)