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 = 0.0138937, 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 \[ \frac{4 x^{3/2}}{15 (x+1)^{3/2}}+\frac{2 x^{3/2}}{5 (x+1)^{5/2}} \]

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))

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Rubi in Sympy [A]  time = 1.22108, size = 29, normalized size = 0.88 \[ \frac{4 x^{\frac{3}{2}}}{15 \left (x + 1\right )^{\frac{3}{2}}} + \frac{2 x^{\frac{3}{2}}}{5 \left (x + 1\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(1+x)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.014717, 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 = 0.002, size = 16, normalized size = 0.5 \[{\frac{4\,x+10}{15}{x}^{{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{5}{2}}}} \]

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

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Maxima [A]  time = 1.32939, size = 27, normalized size = 0.82 \[ \frac{2 \, x^{\frac{5}{2}}{\left (\frac{5 \,{\left (x + 1\right )}}{x} - 3\right )}}{15 \,{\left (x + 1\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1)^(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 [A]  time = 0.203264, size = 127, normalized size = 3.85 \[ \frac{2 \,{\left (60 \, x^{3} - 5 \,{\left (12 \, x^{2} + 11 \, x + 2\right )} \sqrt{x + 1} \sqrt{x} + 85 \, x^{2} + 30 \, x + 2\right )}}{15 \,{\left (16 \, x^{5} + 60 \, x^{4} + 85 \, x^{3} -{\left (16 \, x^{4} + 52 \, x^{3} + 61 \, x^{2} + 30 \, x + 5\right )} \sqrt{x + 1} \sqrt{x} + 55 \, x^{2} + 15 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1)^(7/2),x, algorithm="fricas")

[Out]

2/15*(60*x^3 - 5*(12*x^2 + 11*x + 2)*sqrt(x + 1)*sqrt(x) + 85*x^2 + 30*x + 2)/(1
6*x^5 + 60*x^4 + 85*x^3 - (16*x^4 + 52*x^3 + 61*x^2 + 30*x + 5)*sqrt(x + 1)*sqrt
(x) + 55*x^2 + 15*x + 1)

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Sympy [A]  time = 72.8128, size = 165, normalized size = 5. \[ \begin{cases} \frac{4 i \sqrt{-1 + \frac{1}{x + 1}}}{15} + \frac{2 i \sqrt{-1 + \frac{1}{x + 1}}}{15 \left (x + 1\right )} - \frac{2 i \sqrt{-1 + \frac{1}{x + 1}}}{5 \left (x + 1\right )^{2}} & \text{for}\: \left |{\frac{1}{x + 1}}\right | > 1 \\\frac{4 \sqrt{1 - \frac{1}{x + 1}} \left (x + 1\right )^{2}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac{2 \sqrt{1 - \frac{1}{x + 1}} \left (x + 1\right )}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac{8 \sqrt{1 - \frac{1}{x + 1}}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} + \frac{6 \sqrt{1 - \frac{1}{x + 1}}}{\left (x + 1\right ) \left (- 15 x + 15 \left (x + 1\right )^{2} - 15\right )} & \text{otherwise} \end{cases} \]

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), Abs(1/(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/XCAS [A]  time = 0.204204, size = 89, normalized size = 2.7 \[ \frac{8 \,{\left (15 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{6} - 5 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{4} + 5 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} + 1\right )}}{15 \,{\left ({\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} + 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1)^(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