3.19.87 \(\int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx\)

Optimal. Leaf size=130 \[ \frac {\sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-384 x^2-136 x+255\right )}{960 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {1}{960} \left (384 x^2+568 x-85\right )-\frac {17 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{64 \sqrt {2} x}\right ) \]

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Rubi [F]  time = 1.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[x + x^2]*Defer[Subst][Defer[Int][(x^4*Sqrt[1 + x^2])/Sqrt[x^4 + x^2*Sqrt[x^2 + x^4]], x], x, Sqrt[x]])
/(Sqrt[x]*Sqrt[1 + x])

Rubi steps

\begin {align*} \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {x^{3/2} \sqrt {1+x}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x} \sqrt {1+x}}\\ &=\frac {\left (2 \sqrt {x+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^2}}{\sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.43, size = 137, normalized size = 1.05 \begin {gather*} \frac {\left (x+\sqrt {x (x+1)}\right )^2 \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right ) \left (17 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1+\frac {1}{2 \left (x+\sqrt {x (x+1)}\right )}\right )+10 \left (8 x^2+\left (8 \sqrt {x (x+1)}+11\right ) x+7 \sqrt {x (x+1)}\right )\right )}{80 \sqrt {x (x+1)} \left (2 x+2 \sqrt {x (x+1)}+1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x + Sqrt[x*(1 + x)])^2*Sqrt[x*(x + Sqrt[x*(1 + x)])]*(1 + x + Sqrt[x*(1 + x)])*(10*(8*x^2 + 7*Sqrt[x*(1 + x)
] + x*(11 + 8*Sqrt[x*(1 + x)])) + 17*Hypergeometric2F1[-5/2, 1, -3/2, 1 + 1/(2*(x + Sqrt[x*(1 + x)]))]))/(80*S
qrt[x*(1 + x)]*(1 + 2*x + 2*Sqrt[x*(1 + x)])^3)

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IntegrateAlgebraic [A]  time = 4.52, size = 130, normalized size = 1.00 \begin {gather*} \frac {\left (255-136 x-384 x^2\right ) \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{960 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {1}{960} \left (-85+568 x+384 x^2\right )-\frac {17 \sqrt {-x+\sqrt {x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{64 \sqrt {2} x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*Sqrt[x + x^2])/Sqrt[x^2 + x*Sqrt[x + x^2]],x]

[Out]

((255 - 136*x - 384*x^2)*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/(960*x) + Sqrt[x*(x + Sqrt[x + x^2])]*((-8
5 + 568*x + 384*x^2)/960 - (17*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/(64*Sqrt[2]
*x))

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fricas [A]  time = 0.46, size = 118, normalized size = 0.91 \begin {gather*} \frac {255 \, \sqrt {2} x \log \left (\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, {\left (384 \, x^{3} + 568 \, x^{2} - {\left (384 \, x^{2} + 136 \, x - 255\right )} \sqrt {x^{2} + x} - 85 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{3840 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(255*sqrt(2)*x*log((4*x^2 - 2*sqrt(x^2 + sqrt(x^2 + x)*x)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 + x)) + 4*sqrt(
x^2 + x)*x + x)/x) + 4*(384*x^3 + 568*x^2 - (384*x^2 + 136*x - 255)*sqrt(x^2 + x) - 85*x)*sqrt(x^2 + sqrt(x^2
+ x)*x))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x} x}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x} x}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\sqrt {x^2+x}}{\sqrt {x^2+x\,\sqrt {x^2+x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {x \left (x + 1\right )}}{\sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(x*(x + 1))/sqrt(x*(x + sqrt(x**2 + x))), x)

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