Optimal. Leaf size=147 \[ -\frac {3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 660, 45}
\begin {gather*} -\frac {3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx &=3 \text {Subst}\left (\int \frac {x^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+b^2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac {\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (-\frac {a}{b^3}+\frac {x}{b^2}+\frac {a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac {3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 65, normalized size = 0.44 \begin {gather*} \frac {3 \left (a+b \sqrt [3]{x}\right ) \left (b \left (-2 a+b \sqrt [3]{x}\right ) \sqrt [3]{x}+2 a^2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 103, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {3 \left (a +b \,x^{\frac {1}{3}}\right ) \left (b^{2} x^{\frac {2}{3}}+2 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )-2 a b \,x^{\frac {1}{3}}\right )}{2 \sqrt {\left (a +b \,x^{\frac {1}{3}}\right )^{2}}\, b^{3}}\) | \(52\) |
default | \(\frac {\sqrt {a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}}\, \left (3 b^{2} x^{\frac {2}{3}}-6 a b \,x^{\frac {1}{3}}+2 a^{2} \ln \left (b^{3} x +a^{3}\right )+4 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )-2 a^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )\right )}{2 \left (a +b \,x^{\frac {1}{3}}\right ) b^{3}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 36, normalized size = 0.24 \begin {gather*} \frac {3 \, a^{2} \log \left (x^{\frac {1}{3}} + \frac {a}{b}\right )}{b^{3}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {3 \, a x^{\frac {1}{3}}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 0.22 \begin {gather*} \frac {3 \, {\left (2 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right ) + b^{2} x^{\frac {2}{3}} - 2 \, a b x^{\frac {1}{3}}\right )}}{2 \, b^{3}} \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 {1}{\sqrt {a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.89, size = 61, normalized size = 0.41 \begin {gather*} \frac {3 \, {\left (b x^{\frac {2}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) - 2 \, a x^{\frac {1}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )\right )}}{2 \, b^{2}} + \frac {3 \, a^{2} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \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 {1}{\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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