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

Optimal. Leaf size=94 \[ \frac{4}{11} \sqrt{x} \left (x^{3/2}+x\right )^{3/2}+\frac{64 \left (x^{3/2}+x\right )^{3/2}}{231 \sqrt{x}}-\frac{256 \left (x^{3/2}+x\right )^{3/2}}{1155 x}+\frac{512 \left (x^{3/2}+x\right )^{3/2}}{3465 x^{3/2}}-\frac{32}{99} \left (x^{3/2}+x\right )^{3/2} \]

[Out]

(-32*(x + x^(3/2))^(3/2))/99 + (512*(x + x^(3/2))^(3/2))/(3465*x^(3/2)) - (256*(
x + x^(3/2))^(3/2))/(1155*x) + (64*(x + x^(3/2))^(3/2))/(231*Sqrt[x]) + (4*Sqrt[
x]*(x + x^(3/2))^(3/2))/11

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Rubi [A]  time = 0.149267, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{4}{11} \sqrt{x} \left (x^{3/2}+x\right )^{3/2}+\frac{64 \left (x^{3/2}+x\right )^{3/2}}{231 \sqrt{x}}-\frac{256 \left (x^{3/2}+x\right )^{3/2}}{1155 x}+\frac{512 \left (x^{3/2}+x\right )^{3/2}}{3465 x^{3/2}}-\frac{32}{99} \left (x^{3/2}+x\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[x + x^(3/2)],x]

[Out]

(-32*(x + x^(3/2))^(3/2))/99 + (512*(x + x^(3/2))^(3/2))/(3465*x^(3/2)) - (256*(
x + x^(3/2))^(3/2))/(1155*x) + (64*(x + x^(3/2))^(3/2))/(231*Sqrt[x]) + (4*Sqrt[
x]*(x + x^(3/2))^(3/2))/11

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Rubi in Sympy [A]  time = 7.87215, size = 83, normalized size = 0.88 \[ \frac{4 \sqrt{x} \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{11} - \frac{32 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{99} - \frac{256 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{1155 x} + \frac{64 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{231 \sqrt{x}} + \frac{512 \left (x^{\frac{3}{2}} + x\right )^{\frac{3}{2}}}{3465 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(x)*(x**(3/2) + x)**(3/2)/11 - 32*(x**(3/2) + x)**(3/2)/99 - 256*(x**(3/2)
+ x)**(3/2)/(1155*x) + 64*(x**(3/2) + x)**(3/2)/(231*sqrt(x)) + 512*(x**(3/2) +
x)**(3/2)/(3465*x**(3/2))

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Mathematica [A]  time = 0.021794, size = 51, normalized size = 0.54 \[ \frac{4 \sqrt{x^{3/2}+x} \left (315 x^{5/2}-40 x^{3/2}+35 x^2+48 x-64 \sqrt{x}+128\right )}{3465 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[x + x^(3/2)],x]

[Out]

(4*Sqrt[x + x^(3/2)]*(128 - 64*Sqrt[x] + 48*x - 40*x^(3/2) + 35*x^2 + 315*x^(5/2
)))/(3465*Sqrt[x])

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Maple [A]  time = 0.005, size = 38, normalized size = 0.4 \[{\frac{4}{3465}\sqrt{x+{x}^{{\frac{3}{2}}}} \left ( 1+\sqrt{x} \right ) \left ( 315\,{x}^{2}-280\,{x}^{3/2}+240\,x-192\,\sqrt{x}+128 \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x+x^(3/2))^(1/2),x)

[Out]

4/3465*(x+x^(3/2))^(1/2)*(1+x^(1/2))*(315*x^2-280*x^(3/2)+240*x-192*x^(1/2)+128)
/x^(1/2)

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Maxima [A]  time = 0.711294, size = 62, normalized size = 0.66 \[ \frac{4}{11} \,{\left (\sqrt{x} + 1\right )}^{\frac{11}{2}} - \frac{16}{9} \,{\left (\sqrt{x} + 1\right )}^{\frac{9}{2}} + \frac{24}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{16}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/11*(sqrt(x) + 1)^(11/2) - 16/9*(sqrt(x) + 1)^(9/2) + 24/7*(sqrt(x) + 1)^(7/2)
- 16/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2)

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Fricas [A]  time = 0.30126, size = 54, normalized size = 0.57 \[ \frac{4 \,{\left (315 \, x^{3} - 40 \, x^{2} +{\left (35 \, x^{2} + 48 \, x + 128\right )} \sqrt{x} - 64 \, x\right )} \sqrt{x^{\frac{3}{2}} + x}}{3465 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3465*(315*x^3 - 40*x^2 + (35*x^2 + 48*x + 128)*sqrt(x) - 64*x)*sqrt(x^(3/2) +
x)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(x**(3/2) + x), x)

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GIAC/XCAS [A]  time = 0.268615, size = 63, normalized size = 0.67 \[ \frac{4}{11} \,{\left (\sqrt{x} + 1\right )}^{\frac{11}{2}} - \frac{16}{9} \,{\left (\sqrt{x} + 1\right )}^{\frac{9}{2}} + \frac{24}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{16}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - \frac{512}{3465} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/11*(sqrt(x) + 1)^(11/2) - 16/9*(sqrt(x) + 1)^(9/2) + 24/7*(sqrt(x) + 1)^(7/2)
- 16/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2) - 512/3465