3.2197 \(\int \frac{x^3}{\left (a+b \sqrt{x}\right )^2} \, dx\)

Optimal. Leaf size=111 \[ \frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.193152, 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 \[ \frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*Sqrt[x])^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{7}}{b^{8} \left (a + b \sqrt{x}\right )} + \frac{14 a^{6} \log{\left (a + b \sqrt{x} \right )}}{b^{8}} - \frac{12 a^{5} \sqrt{x}}{b^{7}} + \frac{10 a^{4} \int ^{\sqrt{x}} x\, dx}{b^{6}} - \frac{8 a^{3} x^{\frac{3}{2}}}{3 b^{5}} + \frac{3 a^{2} x^{2}}{2 b^{4}} - \frac{4 a x^{\frac{5}{2}}}{5 b^{3}} + \frac{x^{3}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**7/(b**8*(a + b*sqrt(x))) + 14*a**6*log(a + b*sqrt(x))/b**8 - 12*a**5*sqrt(x
)/b**7 + 10*a**4*Integral(x, (x, sqrt(x)))/b**6 - 8*a**3*x**(3/2)/(3*b**5) + 3*a
**2*x**2/(2*b**4) - 4*a*x**(5/2)/(5*b**3) + x**3/(3*b**2)

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Mathematica [A]  time = 0.0544409, size = 102, normalized size = 0.92 \[ \frac{\frac{60 a^7}{a+b \sqrt{x}}+420 a^6 \log \left (a+b \sqrt{x}\right )-360 a^5 b \sqrt{x}+150 a^4 b^2 x-80 a^3 b^3 x^{3/2}+45 a^2 b^4 x^2-24 a b^5 x^{5/2}+10 b^6 x^3}{30 b^8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*Sqrt[x])^2,x]

[Out]

((60*a^7)/(a + b*Sqrt[x]) - 360*a^5*b*Sqrt[x] + 150*a^4*b^2*x - 80*a^3*b^3*x^(3/
2) + 45*a^2*b^4*x^2 - 24*a*b^5*x^(5/2) + 10*b^6*x^3 + 420*a^6*Log[a + b*Sqrt[x]]
)/(30*b^8)

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Maple [A]  time = 0.011, size = 94, normalized size = 0.9 \[ 5\,{\frac{{a}^{4}x}{{b}^{6}}}-{\frac{8\,{a}^{3}}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{4\,a}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}+14\,{\frac{{a}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}-12\,{\frac{{a}^{5}\sqrt{x}}{{b}^{7}}}+2\,{\frac{{a}^{7}}{{b}^{8} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 1.4498, size = 174, normalized size = 1.57 \[ \frac{14 \, a^{6} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{{\left (b \sqrt{x} + a\right )}^{6}}{3 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a}{5 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac{70 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4}}{b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )} a^{5}}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^2,x, algorithm="maxima")

[Out]

14*a^6*log(b*sqrt(x) + a)/b^8 + 1/3*(b*sqrt(x) + a)^6/b^8 - 14/5*(b*sqrt(x) + a)
^5*a/b^8 + 21/2*(b*sqrt(x) + a)^4*a^2/b^8 - 70/3*(b*sqrt(x) + a)^3*a^3/b^8 + 35*
(b*sqrt(x) + a)^2*a^4/b^8 - 42*(b*sqrt(x) + a)*a^5/b^8 + 2*a^7/((b*sqrt(x) + a)*
b^8)

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Fricas [A]  time = 0.237738, size = 154, normalized size = 1.39 \[ -\frac{14 \, a b^{6} x^{3} + 35 \, a^{3} b^{4} x^{2} + 210 \, a^{5} b^{2} x - 60 \, a^{7} - 420 \,{\left (a^{6} b \sqrt{x} + a^{7}\right )} \log \left (b \sqrt{x} + a\right ) -{\left (10 \, b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 70 \, a^{4} b^{3} x - 360 \, a^{6} b\right )} \sqrt{x}}{30 \,{\left (b^{9} \sqrt{x} + a b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^2,x, algorithm="fricas")

[Out]

-1/30*(14*a*b^6*x^3 + 35*a^3*b^4*x^2 + 210*a^5*b^2*x - 60*a^7 - 420*(a^6*b*sqrt(
x) + a^7)*log(b*sqrt(x) + a) - (10*b^7*x^3 + 21*a^2*b^5*x^2 + 70*a^4*b^3*x - 360
*a^6*b)*sqrt(x))/(b^9*sqrt(x) + a*b^8)

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Sympy [A]  time = 6.46221, size = 272, normalized size = 2.45 \[ \begin{cases} \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{70 a^{4} b^{3} x^{\frac{3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{21 a^{2} b^{5} x^{\frac{5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{10 b^{7} x^{\frac{7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((420*a**7*log(a/b + sqrt(x))/(30*a*b**8 + 30*b**9*sqrt(x)) + 420*a**7/
(30*a*b**8 + 30*b**9*sqrt(x)) + 420*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(30*a*b**8
+ 30*b**9*sqrt(x)) - 210*a**5*b**2*x/(30*a*b**8 + 30*b**9*sqrt(x)) + 70*a**4*b*
*3*x**(3/2)/(30*a*b**8 + 30*b**9*sqrt(x)) - 35*a**3*b**4*x**2/(30*a*b**8 + 30*b*
*9*sqrt(x)) + 21*a**2*b**5*x**(5/2)/(30*a*b**8 + 30*b**9*sqrt(x)) - 14*a*b**6*x*
*3/(30*a*b**8 + 30*b**9*sqrt(x)) + 10*b**7*x**(7/2)/(30*a*b**8 + 30*b**9*sqrt(x)
), Ne(b, 0)), (x**4/(4*a**2), True))

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GIAC/XCAS [A]  time = 0.245738, size = 135, normalized size = 1.22 \[ \frac{14 \, a^{6}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac{5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac{3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt{x}}{30 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^2,x, algorithm="giac")

[Out]

14*a^6*ln(abs(b*sqrt(x) + a))/b^8 + 2*a^7/((b*sqrt(x) + a)*b^8) + 1/30*(10*b^10*
x^3 - 24*a*b^9*x^(5/2) + 45*a^2*b^8*x^2 - 80*a^3*b^7*x^(3/2) + 150*a^4*b^6*x - 3
60*a^5*b^5*sqrt(x))/b^12