3.2208 \(\int \frac{x^2}{(a+b \sqrt{x})^3} \, dx\)

Optimal. Leaf size=88 \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]

[Out]

a^5/(b^6*(a + b*Sqrt[x])^2) - (10*a^4)/(b^6*(a + b*Sqrt[x])) + (12*a^2*Sqrt[x])/b^5 - (3*a*x)/b^4 + (2*x^(3/2)
)/(3*b^3) - (20*a^3*Log[a + b*Sqrt[x]])/b^6

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Rubi [A]  time = 0.0627423, antiderivative size = 88, 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, Rules used = {266, 43} \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sqrt[x])^3,x]

[Out]

a^5/(b^6*(a + b*Sqrt[x])^2) - (10*a^4)/(b^6*(a + b*Sqrt[x])) + (12*a^2*Sqrt[x])/b^5 - (3*a*x)/b^4 + (2*x^(3/2)
)/(3*b^3) - (20*a^3*Log[a + b*Sqrt[x]])/b^6

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b \sqrt{x}\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{6 a^2}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{b^3}-\frac{a^5}{b^5 (a+b x)^3}+\frac{5 a^4}{b^5 (a+b x)^2}-\frac{10 a^3}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0632485, size = 83, normalized size = 0.94 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt{x}\right )^2}-\frac{30 a^4}{a+b \sqrt{x}}+36 a^2 b \sqrt{x}-60 a^3 \log \left (a+b \sqrt{x}\right )-9 a b^2 x+2 b^3 x^{3/2}}{3 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Sqrt[x])^3,x]

[Out]

((3*a^5)/(a + b*Sqrt[x])^2 - (30*a^4)/(a + b*Sqrt[x]) + 36*a^2*b*Sqrt[x] - 9*a*b^2*x + 2*b^3*x^(3/2) - 60*a^3*
Log[a + b*Sqrt[x]])/(3*b^6)

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Maple [A]  time = 0.008, size = 77, normalized size = 0.9 \begin{align*} -3\,{\frac{ax}{{b}^{4}}}+{\frac{2}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-20\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+12\,{\frac{{a}^{2}\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}}-10\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*x^(1/2))^3,x)

[Out]

-3*a*x/b^4+2/3*x^(3/2)/b^3-20*a^3*ln(a+b*x^(1/2))/b^6+12*a^2*x^(1/2)/b^5+a^5/b^6/(a+b*x^(1/2))^2-10*a^4/b^6/(a
+b*x^(1/2))

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Maxima [A]  time = 1.00541, size = 127, normalized size = 1.44 \begin{align*} -\frac{20 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{6}} - \frac{10 \, a^{4}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.27419, size = 296, normalized size = 3.36 \begin{align*} -\frac{9 \, a b^{6} x^{3} - 18 \, a^{3} b^{4} x^{2} - 24 \, a^{5} b^{2} x + 27 \, a^{7} + 60 \,{\left (a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{7} x^{3} + 16 \, a^{2} b^{5} x^{2} - 50 \, a^{4} b^{3} x + 30 \, a^{6} b\right )} \sqrt{x}}{3 \,{\left (b^{10} x^{2} - 2 \, a^{2} b^{8} x + a^{4} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="fricas")

[Out]

-1/3*(9*a*b^6*x^3 - 18*a^3*b^4*x^2 - 24*a^5*b^2*x + 27*a^7 + 60*(a^3*b^4*x^2 - 2*a^5*b^2*x + a^7)*log(b*sqrt(x
) + a) - 2*(b^7*x^3 + 16*a^2*b^5*x^2 - 50*a^4*b^3*x + 30*a^6*b)*sqrt(x))/(b^10*x^2 - 2*a^2*b^8*x + a^4*b^6)

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Sympy [A]  time = 1.2593, size = 332, normalized size = 3.77 \begin{align*} \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{30 a^{5}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{120 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{60 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{60 a^{3} b^{2} x}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{5 a b^{4} x^{2}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{2 b^{5} x^{\frac{5}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-60*a**5*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 30*a**5/(3*a**2*b**6 + 6*
a*b**7*sqrt(x) + 3*b**8*x) - 120*a**4*b*sqrt(x)*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x)
- 60*a**3*b**2*x*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) + 60*a**3*b**2*x/(3*a**2*b**6
+ 6*a*b**7*sqrt(x) + 3*b**8*x) + 20*a**2*b**3*x**(3/2)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 5*a*b**4
*x**2/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) + 2*b**5*x**(5/2)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x
), Ne(b, 0)), (x**3/(3*a**3), True))

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Giac [A]  time = 1.11369, size = 107, normalized size = 1.22 \begin{align*} -\frac{20 \, a^{3} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} - \frac{10 \, a^{4} b \sqrt{x} + 9 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{\frac{3}{2}} - 9 \, a b^{5} x + 36 \, a^{2} b^{4} \sqrt{x}}{3 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="giac")

[Out]

-20*a^3*log(abs(b*sqrt(x) + a))/b^6 - (10*a^4*b*sqrt(x) + 9*a^5)/((b*sqrt(x) + a)^2*b^6) + 1/3*(2*b^6*x^(3/2)
- 9*a*b^5*x + 36*a^2*b^4*sqrt(x))/b^9