Optimal. Leaf size=88 \[ \frac {a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x}{a+b \sqrt [3]{x}}+\frac {3 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x^{4/3}}{4 \left (a+b \sqrt [3]{x}\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 88, 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 b x^{4/3} \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac {a x \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rule 1355
Rubi steps
\begin {align*} \int \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \, dx &=3 \text {Subst}\left (\int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \text {Subst}\left (\int x^2 \left (a b+b^2 x\right ) \, dx,x,\sqrt [3]{x}\right )}{b \left (a+b \sqrt [3]{x}\right )}\\ &=\frac {\left (3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \text {Subst}\left (\int \left (a b x^2+b^2 x^3\right ) \, dx,x,\sqrt [3]{x}\right )}{b \left (a+b \sqrt [3]{x}\right )}\\ &=\frac {a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x}{a+b \sqrt [3]{x}}+\frac {3 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x^{4/3}}{4 \left (a+b \sqrt [3]{x}\right )}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b \sqrt [3]{x}\right )^2} \left (4 a x+3 b x^{4/3}\right )}{4 \left (a+b \sqrt [3]{x}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 43, normalized size = 0.49
method | result | size |
derivativedivides | \(\frac {\mathrm {csgn}\left (a +b \,x^{\frac {1}{3}}\right ) \left (a +b \,x^{\frac {1}{3}}\right )^{2} \left (3 b^{2} x^{\frac {2}{3}}-2 a b \,x^{\frac {1}{3}}+a^{2}\right )}{4 b^{3}}\) | \(42\) |
default | \(\frac {\sqrt {a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}}\, \left (3 b \,x^{\frac {4}{3}}+4 a x \right )}{4 a +4 b \,x^{\frac {1}{3}}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 114, normalized size = 1.30 \begin {gather*} \frac {3 \, \sqrt {b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}} a^{2} x^{\frac {1}{3}}}{2 \, b^{2}} + \frac {3 \, \sqrt {b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}} a^{3}}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {3}{2}} x^{\frac {1}{3}}}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 10, normalized size = 0.11 \begin {gather*} \frac {3}{4} \, b x^{\frac {4}{3}} + a 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 \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 = 4.42, size = 26, normalized size = 0.30 \begin {gather*} \frac {3}{4} \, b x^{\frac {4}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + a x \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 [B]
time = 1.56, size = 71, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^3-4\,a^2\,b\,x^{1/3}-5\,a\,b^2\,x^{2/3}+3\,b\,x^{1/3}\,\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )\right )}{4\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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