Optimal. Leaf size=61 \[ a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \]
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Rubi [A] time = 0.17, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6741, 6742} \begin {gather*} a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \left (a+c \sqrt {x}+b x^{2/3}\right )^2 \, dx &=6 \operatorname {Subst}\left (\int x^5 \left (a+x^3 (c+b x)\right )^2 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int x^5 \left (a+c x^3+b x^4\right )^2 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (a^2 x^5+2 a c x^8+2 a b x^9+c^2 x^{11}+2 b c x^{12}+b^2 x^{13}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=a^2 x+\frac {4}{3} a c x^{3/2}+\frac {6}{5} a b x^{5/3}+\frac {c^2 x^2}{2}+\frac {12}{13} b c x^{13/6}+\frac {3}{7} b^2 x^{7/3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 1.00 \begin {gather*} a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 56, normalized size = 0.92 \begin {gather*} \frac {2730 a^2 x+3276 a b x^{5/3}+3640 a c x^{3/2}+1170 b^2 x^{7/3}+2520 b c x^{13/6}+1365 c^2 x^2}{2730} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 43, normalized size = 0.70 \begin {gather*} \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {4}{3} \, a c x^{\frac {3}{2}} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 43, normalized size = 0.70 \begin {gather*} \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {4}{3} \, a c x^{\frac {3}{2}} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.75 \begin {gather*} \frac {3 b^{2} x^{\frac {7}{3}}}{7}+\frac {c^{2} x^{2}}{2}+\frac {6 a b \,x^{\frac {5}{3}}}{5}+a^{2} x +2 \left (\frac {6 b \,x^{\frac {13}{6}}}{13}+\frac {2 a \,x^{\frac {3}{2}}}{3}\right ) c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 45, normalized size = 0.74 \begin {gather*} \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + a^{2} x + \frac {2}{15} \, {\left (9 \, b x^{\frac {5}{3}} + 10 \, c x^{\frac {3}{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 43, normalized size = 0.70 \begin {gather*} a^2\,x+\frac {3\,b^2\,x^{7/3}}{7}+\frac {c^2\,x^2}{2}+\frac {6\,a\,b\,x^{5/3}}{5}+\frac {4\,a\,c\,x^{3/2}}{3}+\frac {12\,b\,c\,x^{13/6}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.42, size = 60, normalized size = 0.98 \begin {gather*} a^{2} x + \frac {6 a b x^{\frac {5}{3}}}{5} + \frac {4 a c x^{\frac {3}{2}}}{3} + \frac {3 b^{2} x^{\frac {7}{3}}}{7} + \frac {12 b c x^{\frac {13}{6}}}{13} + \frac {c^{2} x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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