3.2189 \(\int \frac{x^3}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{2 a^7 \log \left (a+b \sqrt{x}\right )}{b^8}+\frac{2 a^6 \sqrt{x}}{b^7}-\frac{a^5 x}{b^6}+\frac{2 a^4 x^{3/2}}{3 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{2 a^2 x^{5/2}}{5 b^3}-\frac{a x^3}{3 b^2}+\frac{2 x^{7/2}}{7 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.168375, antiderivative size = 107, 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 \[ -\frac{2 a^7 \log \left (a+b \sqrt{x}\right )}{b^8}+\frac{2 a^6 \sqrt{x}}{b^7}-\frac{a^5 x}{b^6}+\frac{2 a^4 x^{3/2}}{3 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{2 a^2 x^{5/2}}{5 b^3}-\frac{a x^3}{3 b^2}+\frac{2 x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*Sqrt[x]),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0385756, size = 99, normalized size = 0.93 \[ \frac{-420 a^7 \log \left (a+b \sqrt{x}\right )+420 a^6 b \sqrt{x}-210 a^5 b^2 x+140 a^4 b^3 x^{3/2}-105 a^3 b^4 x^2+84 a^2 b^5 x^{5/2}-70 a b^6 x^3+60 b^7 x^{7/2}}{210 b^8} \]

Antiderivative was successfully verified.

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

[Out]

(420*a^6*b*Sqrt[x] - 210*a^5*b^2*x + 140*a^4*b^3*x^(3/2) - 105*a^3*b^4*x^2 + 84*
a^2*b^5*x^(5/2) - 70*a*b^6*x^3 + 60*b^7*x^(7/2) - 420*a^7*Log[a + b*Sqrt[x]])/(2
10*b^8)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 88, normalized size = 0.8 \[ -{\frac{x{a}^{5}}{{b}^{6}}}+{\frac{2\,{a}^{4}}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}-{\frac{{x}^{2}{a}^{3}}{2\,{b}^{4}}}+{\frac{2\,{a}^{2}}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{2}{7\,b}{x}^{{\frac{7}{2}}}}-2\,{\frac{{a}^{7}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+2\,{\frac{{a}^{6}\sqrt{x}}{{b}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*x^(1/2)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44788, size = 174, normalized size = 1.63 \[ -\frac{2 \, a^{7} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{6} a}{3 \, b^{8}} + \frac{42 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{5 \, b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{2 \, b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )}^{3} a^{4}}{3 \, b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{2} a^{5}}{b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )} a^{6}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.240558, size = 119, normalized size = 1.11 \[ -\frac{70 \, a b^{6} x^{3} + 105 \, a^{3} b^{4} x^{2} + 210 \, a^{5} b^{2} x + 420 \, a^{7} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (15 \, b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 105 \, a^{6} b\right )} \sqrt{x}}{210 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/210*(70*a*b^6*x^3 + 105*a^3*b^4*x^2 + 210*a^5*b^2*x + 420*a^7*log(b*sqrt(x) +
a) - 4*(15*b^7*x^3 + 21*a^2*b^5*x^2 + 35*a^4*b^3*x + 105*a^6*b)*sqrt(x))/b^8

_______________________________________________________________________________________

Sympy [A]  time = 3.20987, size = 109, normalized size = 1.02 \[ \begin{cases} - \frac{2 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{8}} + \frac{2 a^{6} \sqrt{x}}{b^{7}} - \frac{a^{5} x}{b^{6}} + \frac{2 a^{4} x^{\frac{3}{2}}}{3 b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{2 a^{2} x^{\frac{5}{2}}}{5 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{2 x^{\frac{7}{2}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a**7*log(a/b + sqrt(x))/b**8 + 2*a**6*sqrt(x)/b**7 - a**5*x/b**6 +
2*a**4*x**(3/2)/(3*b**5) - a**3*x**2/(2*b**4) + 2*a**2*x**(5/2)/(5*b**3) - a*x*
*3/(3*b**2) + 2*x**(7/2)/(7*b), Ne(b, 0)), (x**4/(4*a), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214183, size = 120, normalized size = 1.12 \[ -\frac{2 \, a^{7}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{60 \, b^{6} x^{\frac{7}{2}} - 70 \, a b^{5} x^{3} + 84 \, a^{2} b^{4} x^{\frac{5}{2}} - 105 \, a^{3} b^{3} x^{2} + 140 \, a^{4} b^{2} x^{\frac{3}{2}} - 210 \, a^{5} b x + 420 \, a^{6} \sqrt{x}}{210 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a^7*ln(abs(b*sqrt(x) + a))/b^8 + 1/210*(60*b^6*x^(7/2) - 70*a*b^5*x^3 + 84*a^
2*b^4*x^(5/2) - 105*a^3*b^3*x^2 + 140*a^4*b^2*x^(3/2) - 210*a^5*b*x + 420*a^6*sq
rt(x))/b^7