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

Optimal. Leaf size=76 \[ \frac{512 \left (2 \sqrt{x}+1\right )}{405 \sqrt{x+\sqrt{x}+1}}+\frac{64 \left (2 \sqrt{x}+1\right )}{135 \left (x+\sqrt{x}+1\right )^{3/2}}+\frac{4 \left (2 \sqrt{x}+1\right )}{15 \left (x+\sqrt{x}+1\right )^{5/2}} \]

[Out]

(4*(1 + 2*Sqrt[x]))/(15*(1 + Sqrt[x] + x)^(5/2)) + (64*(1 + 2*Sqrt[x]))/(135*(1
+ Sqrt[x] + x)^(3/2)) + (512*(1 + 2*Sqrt[x]))/(405*Sqrt[1 + Sqrt[x] + x])

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Rubi [A]  time = 0.049974, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{512 \left (2 \sqrt{x}+1\right )}{405 \sqrt{x+\sqrt{x}+1}}+\frac{64 \left (2 \sqrt{x}+1\right )}{135 \left (x+\sqrt{x}+1\right )^{3/2}}+\frac{4 \left (2 \sqrt{x}+1\right )}{15 \left (x+\sqrt{x}+1\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]

[Out]

(4*(1 + 2*Sqrt[x]))/(15*(1 + Sqrt[x] + x)^(5/2)) + (64*(1 + 2*Sqrt[x]))/(135*(1
+ Sqrt[x] + x)^(3/2)) + (512*(1 + 2*Sqrt[x]))/(405*Sqrt[1 + Sqrt[x] + x])

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Rubi in Sympy [A]  time = 2.48128, size = 65, normalized size = 0.86 \[ \frac{64 \left (2 \sqrt{x} + 1\right )}{135 \left (\sqrt{x} + x + 1\right )^{\frac{3}{2}}} + \frac{4 \left (2 \sqrt{x} + 1\right )}{15 \left (\sqrt{x} + x + 1\right )^{\frac{5}{2}}} + \frac{256 \left (4 \sqrt{x} + 2\right )}{405 \sqrt{\sqrt{x} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64*(2*sqrt(x) + 1)/(135*(sqrt(x) + x + 1)**(3/2)) + 4*(2*sqrt(x) + 1)/(15*(sqrt(
x) + x + 1)**(5/2)) + 256*(4*sqrt(x) + 2)/(405*sqrt(sqrt(x) + x + 1))

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Mathematica [A]  time = 0.0275061, size = 49, normalized size = 0.64 \[ \frac{4 \left (2 \sqrt{x}+1\right ) \left (256 x^{3/2}+128 x^2+432 x+304 \sqrt{x}+203\right )}{405 \left (x+\sqrt{x}+1\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]

[Out]

(4*(1 + 2*Sqrt[x])*(203 + 304*Sqrt[x] + 432*x + 256*x^(3/2) + 128*x^2))/(405*(1
+ Sqrt[x] + x)^(5/2))

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Maple [A]  time = 0.003, size = 53, normalized size = 0.7 \[{\frac{4}{15} \left ( 1+2\,\sqrt{x} \right ) \left ( 1+x+\sqrt{x} \right ) ^{-{\frac{5}{2}}}}+{\frac{64}{135} \left ( 1+2\,\sqrt{x} \right ) \left ( 1+x+\sqrt{x} \right ) ^{-{\frac{3}{2}}}}+{\frac{512}{405} \left ( 1+2\,\sqrt{x} \right ){\frac{1}{\sqrt{1+x+\sqrt{x}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(1+x+x^(1/2))^(7/2),x)

[Out]

4/15*(1+2*x^(1/2))/(1+x+x^(1/2))^(5/2)+64/135*(1+2*x^(1/2))/(1+x+x^(1/2))^(3/2)+
512/405*(1+2*x^(1/2))/(1+x+x^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x + \sqrt{x} + 1\right )}^{\frac{7}{2}} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.300592, size = 128, normalized size = 1.68 \[ -\frac{4 \,{\left (128 \, x^{5} + 272 \, x^{4} + 455 \, x^{3} + 232 \, x^{2} -{\left (256 \, x^{5} + 736 \, x^{4} + 1366 \, x^{3} + 1427 \, x^{2} + 839 \, x + 101\right )} \sqrt{x} - 128 \, x - 203\right )} \sqrt{x + \sqrt{x} + 1}}{405 \,{\left (x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/405*(128*x^5 + 272*x^4 + 455*x^3 + 232*x^2 - (256*x^5 + 736*x^4 + 1366*x^3 +
1427*x^2 + 839*x + 101)*sqrt(x) - 128*x - 203)*sqrt(x + sqrt(x) + 1)/(x^6 + 3*x^
5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268933, size = 61, normalized size = 0.8 \[ \frac{4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \, \sqrt{x}{\left (2 \, \sqrt{x} + 5\right )} + 35\right )} \sqrt{x} + 65\right )} \sqrt{x} + 355\right )} \sqrt{x} + 203\right )}}{405 \,{\left (x + \sqrt{x} + 1\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/405*(2*(8*(2*(4*sqrt(x)*(2*sqrt(x) + 5) + 35)*sqrt(x) + 65)*sqrt(x) + 355)*sqr
t(x) + 203)/(x + sqrt(x) + 1)^(5/2)