3.2218 \(\int \frac{x}{(a+b \sqrt{x})^5} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4} \]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

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Rubi [A]  time = 0.002909, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^5,x]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b \sqrt{x}\right )^5} \, dx &=\frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.0040161, size = 21, normalized size = 1. \[ \frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^5,x]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

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Maple [B]  time = 0.006, size = 65, normalized size = 3.1 \begin{align*}{\frac{{a}^{3}}{2\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-4}}-2\,{\frac{{a}^{2}}{{b}^{4} \left ( a+b\sqrt{x} \right ) ^{3}}}+3\,{\frac{a}{{b}^{4} \left ( a+b\sqrt{x} \right ) ^{2}}}-2\,{\frac{1}{{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))^5,x)

[Out]

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

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Maxima [B]  time = 0.965936, size = 86, normalized size = 4.1 \begin{align*} -\frac{2}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{3 \, a}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} - \frac{2 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{3} b^{4}} + \frac{a^{3}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.24974, size = 216, normalized size = 10.29 \begin{align*} \frac{10 \, a b^{6} x^{3} - 5 \, a^{3} b^{4} x^{2} + 4 \, a^{5} b^{2} x - a^{7} - 4 \,{\left (b^{7} x^{3} + a^{2} b^{5} x^{2}\right )} \sqrt{x}}{2 \,{\left (b^{12} x^{4} - 4 \, a^{2} b^{10} x^{3} + 6 \, a^{4} b^{8} x^{2} - 4 \, a^{6} b^{6} x + a^{8} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

1/2*(10*a*b^6*x^3 - 5*a^3*b^4*x^2 + 4*a^5*b^2*x - a^7 - 4*(b^7*x^3 + a^2*b^5*x^2)*sqrt(x))/(b^12*x^4 - 4*a^2*b
^10*x^3 + 6*a^4*b^8*x^2 - 4*a^6*b^6*x + a^8*b^4)

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Sympy [A]  time = 2.47546, size = 65, normalized size = 3.1 \begin{align*} \begin{cases} \frac{x^{2}}{2 a^{5} + 8 a^{4} b \sqrt{x} + 12 a^{3} b^{2} x + 8 a^{2} b^{3} x^{\frac{3}{2}} + 2 a b^{4} x^{2}} & \text{for}\: a \neq 0 \\- \frac{2}{b^{5} \sqrt{x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((x**2/(2*a**5 + 8*a**4*b*sqrt(x) + 12*a**3*b**2*x + 8*a**2*b**3*x**(3/2) + 2*a*b**4*x**2), Ne(a, 0))
, (-2/(b**5*sqrt(x)), True))

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Giac [B]  time = 1.08762, size = 57, normalized size = 2.71 \begin{align*} -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

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