3.220 \(\int \frac{\sqrt{x}}{(1+x)^{7/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{4 x^{3/2}}{15 (x+1)^{3/2}}+\frac{2 x^{3/2}}{5 (x+1)^{5/2}} \]

[Out]

(2*x^(3/2))/(5*(1 + x)^(5/2)) + (4*x^(3/2))/(15*(1 + x)^(3/2))

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Rubi [A]  time = 0.0030038, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {45, 37} \[ \frac{4 x^{3/2}}{15 (x+1)^{3/2}}+\frac{2 x^{3/2}}{5 (x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x)^(7/2),x]

[Out]

(2*x^(3/2))/(5*(1 + x)^(5/2)) + (4*x^(3/2))/(15*(1 + x)^(3/2))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{(1+x)^{7/2}} \, dx &=\frac{2 x^{3/2}}{5 (1+x)^{5/2}}+\frac{2}{5} \int \frac{\sqrt{x}}{(1+x)^{5/2}} \, dx\\ &=\frac{2 x^{3/2}}{5 (1+x)^{5/2}}+\frac{4 x^{3/2}}{15 (1+x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0062169, size = 21, normalized size = 0.64 \[ \frac{2 x^{3/2} (2 x+5)}{15 (x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + x)^(7/2),x]

[Out]

(2*x^(3/2)*(5 + 2*x))/(15*(1 + x)^(5/2))

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Maple [A]  time = 0.003, size = 16, normalized size = 0.5 \begin{align*}{\frac{10+4\,x}{15}{x}^{{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(1+x)^(7/2),x)

[Out]

2/15*x^(3/2)*(5+2*x)/(1+x)^(5/2)

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Maxima [A]  time = 0.943993, size = 27, normalized size = 0.82 \begin{align*} \frac{2 \, x^{\frac{5}{2}}{\left (\frac{5 \,{\left (x + 1\right )}}{x} - 3\right )}}{15 \,{\left (x + 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x)^(7/2),x, algorithm="maxima")

[Out]

2/15*x^(5/2)*(5*(x + 1)/x - 3)/(x + 1)^(5/2)

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Fricas [B]  time = 1.60976, size = 124, normalized size = 3.76 \begin{align*} \frac{2 \,{\left (2 \, x^{3} +{\left (2 \, x^{2} + 5 \, x\right )} \sqrt{x + 1} \sqrt{x} + 6 \, x^{2} + 6 \, x + 2\right )}}{15 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x)^(7/2),x, algorithm="fricas")

[Out]

2/15*(2*x^3 + (2*x^2 + 5*x)*sqrt(x + 1)*sqrt(x) + 6*x^2 + 6*x + 2)/(x^3 + 3*x^2 + 3*x + 1)

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Sympy [B]  time = 9.95409, size = 165, normalized size = 5. \begin{align*} \begin{cases} \frac{4 i \sqrt{-1 + \frac{1}{x + 1}}}{15} + \frac{2 i \sqrt{-1 + \frac{1}{x + 1}}}{15 \left (x + 1\right )} - \frac{2 i \sqrt{-1 + \frac{1}{x + 1}}}{5 \left (x + 1\right )^{2}} & \text{for}\: \frac{1}{\left |{x + 1}\right |} > 1 \\\frac{4 \sqrt{1 - \frac{1}{x + 1}} \left (x + 1\right )^{2}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac{2 \sqrt{1 - \frac{1}{x + 1}} \left (x + 1\right )}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac{8 \sqrt{1 - \frac{1}{x + 1}}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} + \frac{6 \sqrt{1 - \frac{1}{x + 1}}}{\left (x + 1\right ) \left (- 15 x + 15 \left (x + 1\right )^{2} - 15\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(1+x)**(7/2),x)

[Out]

Piecewise((4*I*sqrt(-1 + 1/(x + 1))/15 + 2*I*sqrt(-1 + 1/(x + 1))/(15*(x + 1)) - 2*I*sqrt(-1 + 1/(x + 1))/(5*(
x + 1)**2), 1/Abs(x + 1) > 1), (4*sqrt(1 - 1/(x + 1))*(x + 1)**2/(-15*x + 15*(x + 1)**2 - 15) - 2*sqrt(1 - 1/(
x + 1))*(x + 1)/(-15*x + 15*(x + 1)**2 - 15) - 8*sqrt(1 - 1/(x + 1))/(-15*x + 15*(x + 1)**2 - 15) + 6*sqrt(1 -
1/(x + 1))/((x + 1)*(-15*x + 15*(x + 1)**2 - 15)), True))

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Giac [B]  time = 1.11898, size = 89, normalized size = 2.7 \begin{align*} \frac{8 \,{\left (15 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{6} - 5 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{4} + 5 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} + 1\right )}}{15 \,{\left ({\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x)^(7/2),x, algorithm="giac")

[Out]

8/15*(15*(sqrt(x + 1) - sqrt(x))^6 - 5*(sqrt(x + 1) - sqrt(x))^4 + 5*(sqrt(x + 1) - sqrt(x))^2 + 1)/((sqrt(x +
1) - sqrt(x))^2 + 1)^5