3.2209 \(\int \frac{x}{(a+b \sqrt{x})^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0369614, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^3,x]

[Out]

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

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}{\left (a+b \sqrt{x}\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}+\frac{2 \sqrt{x}}{b^3}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0405811, size = 57, normalized size = 0.89 \[ \frac{\frac{a^3}{\left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{a+b \sqrt{x}}-6 a \log \left (a+b \sqrt{x}\right )+2 b \sqrt{x}}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \begin{align*} -6\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{\sqrt{x}}{{b}^{3}}}+{\frac{{a}^{3}}{{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-2}}-6\,{\frac{{a}^{2}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0219, size = 81, normalized size = 1.27 \begin{align*} -\frac{6 \, a \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{4}} - \frac{6 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28328, size = 213, normalized size = 3.33 \begin{align*} \frac{7 \, a^{3} b^{2} x - 5 \, a^{5} - 6 \,{\left (a b^{4} x^{2} - 2 \, a^{3} b^{2} x + a^{5}\right )} \log \left (b \sqrt{x} + a\right ) + 2 \,{\left (b^{5} x^{2} - 5 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt{x}}{b^{8} x^{2} - 2 \, a^{2} b^{6} x + a^{4} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.13798, size = 264, normalized size = 4.12 \begin{align*} \begin{cases} - \frac{6 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{8 a^{3}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{12 a^{2} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{10 a^{2} b \sqrt{x}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{6 a b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{a b^{2} x}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{2 b^{3} x^{\frac{3}{2}}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*a**3*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 8*a**3/(a**2*b**4 + 2*a*b**5*s
qrt(x) + b**6*x) - 12*a**2*b*sqrt(x)*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 10*a**2*b*sq
rt(x)/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 6*a*b**2*x*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) +
b**6*x) + a*b**2*x/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) + 2*b**3*x**(3/2)/(a**2*b**4 + 2*a*b**5*sqrt(x) + b
**6*x), Ne(b, 0)), (x**2/(2*a**3), True))

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Giac [A]  time = 1.09492, size = 72, normalized size = 1.12 \begin{align*} -\frac{6 \, a \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, \sqrt{x}}{b^{3}} - \frac{6 \, a^{2} b \sqrt{x} + 5 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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