3.2213 \(\int \frac{x^4}{\left (a+b \sqrt{x}\right )^5} \, dx\)

Optimal. Leaf size=155 \[ \frac{a^9}{2 b^{10} \left (a+b \sqrt{x}\right )^4}-\frac{6 a^8}{b^{10} \left (a+b \sqrt{x}\right )^3}+\frac{36 a^7}{b^{10} \left (a+b \sqrt{x}\right )^2}-\frac{168 a^6}{b^{10} \left (a+b \sqrt{x}\right )}-\frac{252 a^5 \log \left (a+b \sqrt{x}\right )}{b^{10}}+\frac{140 a^4 \sqrt{x}}{b^9}-\frac{35 a^3 x}{b^8}+\frac{10 a^2 x^{3/2}}{b^7}-\frac{5 a x^2}{2 b^6}+\frac{2 x^{5/2}}{5 b^5} \]

[Out]

a^9/(2*b^10*(a + b*Sqrt[x])^4) - (6*a^8)/(b^10*(a + b*Sqrt[x])^3) + (36*a^7)/(b^
10*(a + b*Sqrt[x])^2) - (168*a^6)/(b^10*(a + b*Sqrt[x])) + (140*a^4*Sqrt[x])/b^9
- (35*a^3*x)/b^8 + (10*a^2*x^(3/2))/b^7 - (5*a*x^2)/(2*b^6) + (2*x^(5/2))/(5*b^
5) - (252*a^5*Log[a + b*Sqrt[x]])/b^10

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Rubi [A]  time = 0.309515, antiderivative size = 155, 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 \[ \frac{a^9}{2 b^{10} \left (a+b \sqrt{x}\right )^4}-\frac{6 a^8}{b^{10} \left (a+b \sqrt{x}\right )^3}+\frac{36 a^7}{b^{10} \left (a+b \sqrt{x}\right )^2}-\frac{168 a^6}{b^{10} \left (a+b \sqrt{x}\right )}-\frac{252 a^5 \log \left (a+b \sqrt{x}\right )}{b^{10}}+\frac{140 a^4 \sqrt{x}}{b^9}-\frac{35 a^3 x}{b^8}+\frac{10 a^2 x^{3/2}}{b^7}-\frac{5 a x^2}{2 b^6}+\frac{2 x^{5/2}}{5 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*Sqrt[x])^5,x]

[Out]

a^9/(2*b^10*(a + b*Sqrt[x])^4) - (6*a^8)/(b^10*(a + b*Sqrt[x])^3) + (36*a^7)/(b^
10*(a + b*Sqrt[x])^2) - (168*a^6)/(b^10*(a + b*Sqrt[x])) + (140*a^4*Sqrt[x])/b^9
- (35*a^3*x)/b^8 + (10*a^2*x^(3/2))/b^7 - (5*a*x^2)/(2*b^6) + (2*x^(5/2))/(5*b^
5) - (252*a^5*Log[a + b*Sqrt[x]])/b^10

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{9}}{2 b^{10} \left (a + b \sqrt{x}\right )^{4}} - \frac{6 a^{8}}{b^{10} \left (a + b \sqrt{x}\right )^{3}} + \frac{36 a^{7}}{b^{10} \left (a + b \sqrt{x}\right )^{2}} - \frac{168 a^{6}}{b^{10} \left (a + b \sqrt{x}\right )} - \frac{252 a^{5} \log{\left (a + b \sqrt{x} \right )}}{b^{10}} + \frac{140 a^{4} \sqrt{x}}{b^{9}} - \frac{70 a^{3} \int ^{\sqrt{x}} x\, dx}{b^{8}} + \frac{10 a^{2} x^{\frac{3}{2}}}{b^{7}} - \frac{5 a x^{2}}{2 b^{6}} + \frac{2 x^{\frac{5}{2}}}{5 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b*x**(1/2))**5,x)

[Out]

a**9/(2*b**10*(a + b*sqrt(x))**4) - 6*a**8/(b**10*(a + b*sqrt(x))**3) + 36*a**7/
(b**10*(a + b*sqrt(x))**2) - 168*a**6/(b**10*(a + b*sqrt(x))) - 252*a**5*log(a +
b*sqrt(x))/b**10 + 140*a**4*sqrt(x)/b**9 - 70*a**3*Integral(x, (x, sqrt(x)))/b*
*8 + 10*a**2*x**(3/2)/b**7 - 5*a*x**2/(2*b**6) + 2*x**(5/2)/(5*b**5)

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Mathematica [A]  time = 0.064673, size = 150, normalized size = 0.97 \[ \frac{-1375 a^9-2980 a^8 b \sqrt{x}+570 a^7 b^2 x+5420 a^6 b^3 x^{3/2}+3875 a^5 b^4 x^2-2520 a^5 \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )+504 a^4 b^5 x^{5/2}-84 a^3 b^6 x^3+24 a^2 b^7 x^{7/2}-9 a b^8 x^4+4 b^9 x^{9/2}}{10 b^{10} \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*Sqrt[x])^5,x]

[Out]

(-1375*a^9 - 2980*a^8*b*Sqrt[x] + 570*a^7*b^2*x + 5420*a^6*b^3*x^(3/2) + 3875*a^
5*b^4*x^2 + 504*a^4*b^5*x^(5/2) - 84*a^3*b^6*x^3 + 24*a^2*b^7*x^(7/2) - 9*a*b^8*
x^4 + 4*b^9*x^(9/2) - 2520*a^5*(a + b*Sqrt[x])^4*Log[a + b*Sqrt[x]])/(10*b^10*(a
+ b*Sqrt[x])^4)

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Maple [A]  time = 0.015, size = 134, normalized size = 0.9 \[ -35\,{\frac{{a}^{3}x}{{b}^{8}}}+10\,{\frac{{a}^{2}{x}^{3/2}}{{b}^{7}}}-{\frac{5\,a{x}^{2}}{2\,{b}^{6}}}+{\frac{2}{5\,{b}^{5}}{x}^{{\frac{5}{2}}}}-252\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{10}}}+140\,{\frac{{a}^{4}\sqrt{x}}{{b}^{9}}}+{\frac{{a}^{9}}{2\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-4}}-6\,{\frac{{a}^{8}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{3}}}+36\,{\frac{{a}^{7}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}-168\,{\frac{{a}^{6}}{{b}^{10} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b*x^(1/2))^5,x)

[Out]

-35*a^3*x/b^8+10*a^2*x^(3/2)/b^7-5/2*a*x^2/b^6+2/5*x^(5/2)/b^5-252*a^5*ln(a+b*x^
(1/2))/b^10+140*a^4*x^(1/2)/b^9+1/2*a^9/b^10/(a+b*x^(1/2))^4-6*a^8/b^10/(a+b*x^(
1/2))^3+36*a^7/b^10/(a+b*x^(1/2))^2-168*a^6/b^10/(a+b*x^(1/2))

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Maxima [A]  time = 1.42974, size = 220, normalized size = 1.42 \[ -\frac{252 \, a^{5} \log \left (b \sqrt{x} + a\right )}{b^{10}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{10}} - \frac{9 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{10}} + \frac{24 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{10}} + \frac{252 \,{\left (b \sqrt{x} + a\right )} a^{4}}{b^{10}} - \frac{168 \, a^{6}}{{\left (b \sqrt{x} + a\right )} b^{10}} + \frac{36 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{10}} - \frac{6 \, a^{8}}{{\left (b \sqrt{x} + a\right )}^{3} b^{10}} + \frac{a^{9}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^5,x, algorithm="maxima")

[Out]

-252*a^5*log(b*sqrt(x) + a)/b^10 + 2/5*(b*sqrt(x) + a)^5/b^10 - 9/2*(b*sqrt(x) +
a)^4*a/b^10 + 24*(b*sqrt(x) + a)^3*a^2/b^10 - 84*(b*sqrt(x) + a)^2*a^3/b^10 + 2
52*(b*sqrt(x) + a)*a^4/b^10 - 168*a^6/((b*sqrt(x) + a)*b^10) + 36*a^7/((b*sqrt(x
) + a)^2*b^10) - 6*a^8/((b*sqrt(x) + a)^3*b^10) + 1/2*a^9/((b*sqrt(x) + a)^4*b^1
0)

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Fricas [A]  time = 0.246303, size = 263, normalized size = 1.7 \[ -\frac{9 \, a b^{8} x^{4} + 84 \, a^{3} b^{6} x^{3} - 3875 \, a^{5} b^{4} x^{2} - 570 \, a^{7} b^{2} x + 1375 \, a^{9} + 2520 \,{\left (a^{5} b^{4} x^{2} + 6 \, a^{7} b^{2} x + a^{9} + 4 \,{\left (a^{6} b^{3} x + a^{8} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (b^{9} x^{4} + 6 \, a^{2} b^{7} x^{3} + 126 \, a^{4} b^{5} x^{2} + 1355 \, a^{6} b^{3} x - 745 \, a^{8} b\right )} \sqrt{x}}{10 \,{\left (b^{14} x^{2} + 6 \, a^{2} b^{12} x + a^{4} b^{10} + 4 \,{\left (a b^{13} x + a^{3} b^{11}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^5,x, algorithm="fricas")

[Out]

-1/10*(9*a*b^8*x^4 + 84*a^3*b^6*x^3 - 3875*a^5*b^4*x^2 - 570*a^7*b^2*x + 1375*a^
9 + 2520*(a^5*b^4*x^2 + 6*a^7*b^2*x + a^9 + 4*(a^6*b^3*x + a^8*b)*sqrt(x))*log(b
*sqrt(x) + a) - 4*(b^9*x^4 + 6*a^2*b^7*x^3 + 126*a^4*b^5*x^2 + 1355*a^6*b^3*x -
745*a^8*b)*sqrt(x))/(b^14*x^2 + 6*a^2*b^12*x + a^4*b^10 + 4*(a*b^13*x + a^3*b^11
)*sqrt(x))

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Sympy [A]  time = 26.4734, size = 1013, normalized size = 6.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((-2520*a**9*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x)
+ 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 5439*a**9/(10*a**4*b*
*10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x
**2) - 10080*a**8*b*sqrt(x)*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sq
rt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 19236*a**8*b*sq
rt(x)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(
3/2) + 10*b**14*x**2) - 15120*a**7*b**2*x*log(a/b + sqrt(x))/(10*a**4*b**10 + 40
*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 2
3814*a**7*b**2*x/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a
*b**13*x**(3/2) + 10*b**14*x**2) - 10080*a**6*b**3*x**(3/2)*log(a/b + sqrt(x))/(
10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) +
10*b**14*x**2) - 10836*a**6*b**3*x**(3/2)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x)
+ 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 2520*a**5*b**4*x**2*
log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40
*a*b**13*x**(3/2) + 10*b**14*x**2) - 189*a**5*b**4*x**2/(10*a**4*b**10 + 40*a**3
*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 504*a*
*4*b**5*x**(5/2)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a
*b**13*x**(3/2) + 10*b**14*x**2) - 84*a**3*b**6*x**3/(10*a**4*b**10 + 40*a**3*b*
*11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 24*a**2*b
**7*x**(7/2)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**
13*x**(3/2) + 10*b**14*x**2) - 9*a*b**8*x**4/(10*a**4*b**10 + 40*a**3*b**11*sqrt
(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 4*b**9*x**(9/2)/(
10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) +
10*b**14*x**2), Ne(b, 0)), (x**5/(5*a**5), True))

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GIAC/XCAS [A]  time = 0.259332, size = 163, normalized size = 1.05 \[ -\frac{252 \, a^{5}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{10}} - \frac{336 \, a^{6} b^{3} x^{\frac{3}{2}} + 936 \, a^{7} b^{2} x + 876 \, a^{8} b \sqrt{x} + 275 \, a^{9}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{10}} + \frac{4 \, b^{20} x^{\frac{5}{2}} - 25 \, a b^{19} x^{2} + 100 \, a^{2} b^{18} x^{\frac{3}{2}} - 350 \, a^{3} b^{17} x + 1400 \, a^{4} b^{16} \sqrt{x}}{10 \, b^{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*sqrt(x) + a)^5,x, algorithm="giac")

[Out]

-252*a^5*ln(abs(b*sqrt(x) + a))/b^10 - 1/2*(336*a^6*b^3*x^(3/2) + 936*a^7*b^2*x
+ 876*a^8*b*sqrt(x) + 275*a^9)/((b*sqrt(x) + a)^4*b^10) + 1/10*(4*b^20*x^(5/2) -
25*a*b^19*x^2 + 100*a^2*b^18*x^(3/2) - 350*a^3*b^17*x + 1400*a^4*b^16*sqrt(x))/
b^25