3.20.86 \(\int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx\) [1986]

Optimal. Leaf size=140 \[ \frac {\sqrt {x+x^2} \left (-1575+840 x-640 x^2-3072 x^3\right ) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{10752 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {525-120 x+3968 x^2+3072 x^3}{10752}+\frac {75 \sqrt {-x+\sqrt {x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{512 \sqrt {2} x}\right ) \]

[Out]

1/10752*(x^2+x)^(1/2)*(-3072*x^3-640*x^2+840*x-1575)*(x*(x+(x^2+x)^(1/2)))^(1/2)/x+(x*(x+(x^2+x)^(1/2)))^(1/2)
*(2/7*x^3+31/84*x^2-5/448*x+25/512+75/1024*2^(1/2)*(-x+(x^2+x)^(1/2))^(1/2)*arctanh(2^(1/2)*(-x+(x^2+x)^(1/2))
^(1/2))/x)

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Rubi [F]
time = 1.15, 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^2 \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^2*Sqrt[x + x^2])/Sqrt[x^2 + x*Sqrt[x + x^2]],x]

[Out]

(2*Sqrt[x + x^2]*Defer[Subst][Defer[Int][(x^6*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^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {x^{5/2} \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 ) \text {Subst}\left (\int \frac {x^6 \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 [A]
time = 2.93, size = 160, normalized size = 1.14 \begin {gather*} \frac {(1+x) \sqrt {x \left (x+\sqrt {x (1+x)}\right )} \left (-6144 x^5+1050 x \left (-3+\sqrt {x (1+x)}\right )-30 x^2 \left (49+8 \sqrt {x (1+x)}\right )+256 x^4 \left (-29+24 \sqrt {x (1+x)}\right )+16 x^3 \left (25+496 \sqrt {x (1+x)}\right )+1575 \sqrt {2} \sqrt {x (1+x)} \sqrt {-x+\sqrt {x (1+x)}} \tanh ^{-1}\left (\sqrt {-2 x+2 \sqrt {x (1+x)}}\right )\right )}{21504 (x (1+x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)*Sqrt[x*(x + Sqrt[x*(1 + x)])]*(-6144*x^5 + 1050*x*(-3 + Sqrt[x*(1 + x)]) - 30*x^2*(49 + 8*Sqrt[x*(1 +
x)]) + 256*x^4*(-29 + 24*Sqrt[x*(1 + x)]) + 16*x^3*(25 + 496*Sqrt[x*(1 + x)]) + 1575*Sqrt[2]*Sqrt[x*(1 + x)]*
Sqrt[-x + Sqrt[x*(1 + x)]]*ArcTanh[Sqrt[-2*x + 2*Sqrt[x*(1 + x)]]]))/(21504*(x*(1 + x))^(3/2))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]
time = 0.42, size = 128, normalized size = 0.91 \begin {gather*} \frac {1575 \, \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 (3072 \, x^{4} + 3968 \, x^{3} - 120 \, x^{2} - {\left (3072 \, x^{3} + 640 \, x^{2} - 840 \, x + 1575\right )} \sqrt {x^{2} + x} + 525 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{43008 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43008*(1575*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*sqr
t(x^2 + x)*x + x)/x) + 4*(3072*x^4 + 3968*x^3 - 120*x^2 - (3072*x^3 + 640*x^2 - 840*x + 1575)*sqrt(x^2 + x) +
525*x)*sqrt(x^2 + sqrt(x^2 + x)*x))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \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**2*(x**2+x)**(1/2)/(x**2+x*(x**2+x)**(1/2))**(1/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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