3.789 \(\int \sqrt{x+x^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt{x}}-\frac{16 \left (x^{3/2}+x\right )^{3/2}}{35 x}+\frac{32 \left (x^{3/2}+x\right )^{3/2}}{105 x^{3/2}} \]

[Out]

(32*(x + x^(3/2))^(3/2))/(105*x^(3/2)) - (16*(x + x^(3/2))^(3/2))/(35*x) + (4*(x
+ x^(3/2))^(3/2))/(7*Sqrt[x])

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Rubi [A]  time = 0.0867029, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt{x}}-\frac{16 \left (x^{3/2}+x\right )^{3/2}}{35 x}+\frac{32 \left (x^{3/2}+x\right )^{3/2}}{105 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x + x^(3/2)],x]

[Out]

(32*(x + x^(3/2))^(3/2))/(105*x^(3/2)) - (16*(x + x^(3/2))^(3/2))/(35*x) + (4*(x
+ x^(3/2))^(3/2))/(7*Sqrt[x])

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Rubi in Sympy [A]  time = 4.63909, size = 51, normalized size = 0.86 \[ - \frac{16 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{35 x} + \frac{4 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{7 \sqrt{x}} + \frac{32 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{105 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+x**(3/2))**(1/2),x)

[Out]

-16*(x**(3/2) + x)**(3/2)/(35*x) + 4*(x**(3/2) + x)**(3/2)/(7*sqrt(x)) + 32*(x**
(3/2) + x)**(3/2)/(105*x**(3/2))

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Mathematica [A]  time = 0.0208424, size = 41, normalized size = 0.69 \[ \left (\frac{4 x}{7}+\frac{4 \sqrt{x}}{35}+\frac{32}{105 \sqrt{x}}-\frac{16}{105}\right ) \sqrt{\left (\sqrt{x}+1\right ) x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x + x^(3/2)],x]

[Out]

(-16/105 + 32/(105*Sqrt[x]) + (4*Sqrt[x])/35 + (4*x)/7)*Sqrt[(1 + Sqrt[x])*x]

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Maple [A]  time = 0.012, size = 28, normalized size = 0.5 \[{\frac{4}{105}\sqrt{x+{x}^{{\frac{3}{2}}}} \left ( 1+\sqrt{x} \right ) \left ( 15\,x-12\,\sqrt{x}+8 \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+x^(3/2))^(1/2),x)

[Out]

4/105*(x+x^(3/2))^(1/2)*(1+x^(1/2))*(15*x-12*x^(1/2)+8)/x^(1/2)

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Maxima [A]  time = 0.721673, size = 38, normalized size = 0.64 \[ \frac{4}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{8}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(3/2) + x),x, algorithm="maxima")

[Out]

4/7*(sqrt(x) + 1)^(7/2) - 8/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2)

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Fricas [A]  time = 0.300985, size = 41, normalized size = 0.69 \[ \frac{4 \,{\left (15 \, x^{2} +{\left (3 \, x + 8\right )} \sqrt{x} - 4 \, x\right )} \sqrt{x^{\frac{3}{2}} + x}}{105 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(3/2) + x),x, algorithm="fricas")

[Out]

4/105*(15*x^2 + (3*x + 8)*sqrt(x) - 4*x)*sqrt(x^(3/2) + x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{\frac{3}{2}} + x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x+x**(3/2))**(1/2),x)

[Out]

Integral(sqrt(x**(3/2) + x), x)

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GIAC/XCAS [A]  time = 0.281018, size = 39, normalized size = 0.66 \[ \frac{4}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{8}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - \frac{32}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(3/2) + x),x, algorithm="giac")

[Out]

4/7*(sqrt(x) + 1)^(7/2) - 8/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2) - 32
/105