Optimal. Leaf size=51 \[ \frac {2 a^2 (d x)^{3/2}}{3 d}+\frac {4 a b (d x)^{7/2}}{7 d^3}+\frac {2 b^2 (d x)^{11/2}}{11 d^5} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14} \begin {gather*} \frac {2 a^2 (d x)^{3/2}}{3 d}+\frac {4 a b (d x)^{7/2}}{7 d^3}+\frac {2 b^2 (d x)^{11/2}}{11 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 \sqrt {d x}+\frac {2 a b (d x)^{5/2}}{d^2}+\frac {b^2 (d x)^{9/2}}{d^4}\right ) \, dx\\ &=\frac {2 a^2 (d x)^{3/2}}{3 d}+\frac {4 a b (d x)^{7/2}}{7 d^3}+\frac {2 b^2 (d x)^{11/2}}{11 d^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.65 \begin {gather*} \frac {2}{231} x \sqrt {d x} \left (77 a^2+66 a b x^2+21 b^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 44, normalized size = 0.86 \begin {gather*} \frac {2 (d x)^{3/2} \left (77 a^2 d^4+66 a b d^4 x^2+21 b^2 d^4 x^4\right )}{231 d^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 29, normalized size = 0.57 \begin {gather*} \frac {2}{231} \, {\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt {d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.73 \begin {gather*} \frac {2}{11} \, \sqrt {d x} b^{2} x^{5} + \frac {4}{7} \, \sqrt {d x} a b x^{3} + \frac {2}{3} \, \sqrt {d x} a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.59 \begin {gather*} \frac {2 \left (21 b^{2} x^{4}+66 a b \,x^{2}+77 a^{2}\right ) \sqrt {d x}\, x}{231} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (21 \, \left (d x\right )^{\frac {11}{2}} b^{2} + 66 \, \left (d x\right )^{\frac {7}{2}} a b d^{2} + 77 \, \left (d x\right )^{\frac {3}{2}} a^{2} d^{4}\right )}}{231 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 41, normalized size = 0.80 \begin {gather*} \frac {42\,b^2\,{\left (d\,x\right )}^{11/2}+154\,a^2\,d^4\,{\left (d\,x\right )}^{3/2}+132\,a\,b\,d^2\,{\left (d\,x\right )}^{7/2}}{231\,d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 a^{2} \sqrt {d} x^{\frac {3}{2}}}{3} + \frac {4 a b \sqrt {d} x^{\frac {7}{2}}}{7} + \frac {2 b^{2} \sqrt {d} x^{\frac {11}{2}}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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