Optimal. Leaf size=150 \[ \frac {\sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-1146880 x^4-168960 x^3+201344 x^2-264264 x+495495\right )}{5160960 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {1146880 x^4+1387520 x^3-18304 x^2+37752 x-165165}{5160960}-\frac {1573 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{16384 \sqrt {2} x}\right ) \]
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Rubi [F] time = 1.76, 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^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {x^{7/2} \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^8 \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 = 0.57, size = 156, normalized size = 1.04 \begin {gather*} \frac {\left (x+\sqrt {x (x+1)}\right )^4 \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right ) \left (11 \, _2F_1\left (-\frac {9}{2},3;-\frac {7}{2};1+\frac {1}{2 \left (x+\sqrt {x (x+1)}\right )}\right )+72 x \left (32 x^3+4 \left (8 \sqrt {x (x+1)}+13\right ) x^2+\left (36 \sqrt {x (x+1)}+19\right ) x+5 \sqrt {x (x+1)}\right )\right )}{1152 \sqrt {x (x+1)} \left (2 x+2 \sqrt {x (x+1)}+1\right )^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.12, size = 150, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-1146880 x^4-168960 x^3+201344 x^2-264264 x+495495\right )}{5160960 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {1146880 x^4+1387520 x^3-18304 x^2+37752 x-165165}{5160960}-\frac {1573 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{16384 \sqrt {2} x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 138, normalized size = 0.92 \begin {gather*} \frac {495495 \, \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 (1146880 \, x^{5} + 1387520 \, x^{4} - 18304 \, x^{3} + 37752 \, x^{2} - {\left (1146880 \, x^{4} + 168960 \, x^{3} - 201344 \, x^{2} + 264264 \, x - 495495\right )} \sqrt {x^{2} + x} - 165165 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{20643840 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x} x^{3}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x} x^{3}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\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.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \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.
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