Optimal. Leaf size=131 \[ \frac {2 a^6 (d x)^{3/2}}{3 d}+\frac {12 a^5 b (d x)^{7/2}}{7 d^3}+\frac {30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac {8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac {30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac {12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac {2 b^6 (d x)^{27/2}}{27 d^{13}} \]
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Rubi [A] time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {28, 270} \begin {gather*} \frac {30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac {8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac {30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac {12 a^5 b (d x)^{7/2}}{7 d^3}+\frac {2 a^6 (d x)^{3/2}}{3 d}+\frac {12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac {2 b^6 (d x)^{27/2}}{27 d^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int \sqrt {d x} \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 \sqrt {d x}+\frac {6 a^5 b^7 (d x)^{5/2}}{d^2}+\frac {15 a^4 b^8 (d x)^{9/2}}{d^4}+\frac {20 a^3 b^9 (d x)^{13/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{17/2}}{d^8}+\frac {6 a b^{11} (d x)^{21/2}}{d^{10}}+\frac {b^{12} (d x)^{25/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac {2 a^6 (d x)^{3/2}}{3 d}+\frac {12 a^5 b (d x)^{7/2}}{7 d^3}+\frac {30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac {8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac {30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac {12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac {2 b^6 (d x)^{27/2}}{27 d^{13}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.59 \begin {gather*} \frac {2 x \sqrt {d x} \left (302841 a^6+778734 a^5 b x^2+1238895 a^4 b^2 x^4+1211364 a^3 b^3 x^6+717255 a^2 b^4 x^8+237006 a b^5 x^{10}+33649 b^6 x^{12}\right )}{908523} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 121, normalized size = 0.92 \begin {gather*} \frac {2 \left (302841 a^6 d^{12} (d x)^{3/2}+778734 a^5 b d^{10} (d x)^{7/2}+1238895 a^4 b^2 d^8 (d x)^{11/2}+1211364 a^3 b^3 d^6 (d x)^{15/2}+717255 a^2 b^4 d^4 (d x)^{19/2}+237006 a b^5 d^2 (d x)^{23/2}+33649 b^6 (d x)^{27/2}\right )}{908523 d^{13}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.07, size = 73, normalized size = 0.56 \begin {gather*} \frac {2}{908523} \, {\left (33649 \, b^{6} x^{13} + 237006 \, a b^{5} x^{11} + 717255 \, a^{2} b^{4} x^{9} + 1211364 \, a^{3} b^{3} x^{7} + 1238895 \, a^{4} b^{2} x^{5} + 778734 \, a^{5} b x^{3} + 302841 \, a^{6} x\right )} \sqrt {d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 101, normalized size = 0.77 \begin {gather*} \frac {2}{27} \, \sqrt {d x} b^{6} x^{13} + \frac {12}{23} \, \sqrt {d x} a b^{5} x^{11} + \frac {30}{19} \, \sqrt {d x} a^{2} b^{4} x^{9} + \frac {8}{3} \, \sqrt {d x} a^{3} b^{3} x^{7} + \frac {30}{11} \, \sqrt {d x} a^{4} b^{2} x^{5} + \frac {12}{7} \, \sqrt {d x} a^{5} b x^{3} + \frac {2}{3} \, \sqrt {d x} a^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.56 \begin {gather*} \frac {2 \left (33649 b^{6} x^{12}+237006 a \,b^{5} x^{10}+717255 a^{2} b^{4} x^{8}+1211364 a^{3} b^{3} x^{6}+1238895 a^{4} b^{2} x^{4}+778734 a^{5} b \,x^{2}+302841 a^{6}\right ) \sqrt {d x}\, x}{908523} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 105, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (33649 \, \left (d x\right )^{\frac {27}{2}} b^{6} + 237006 \, \left (d x\right )^{\frac {23}{2}} a b^{5} d^{2} + 717255 \, \left (d x\right )^{\frac {19}{2}} a^{2} b^{4} d^{4} + 1211364 \, \left (d x\right )^{\frac {15}{2}} a^{3} b^{3} d^{6} + 1238895 \, \left (d x\right )^{\frac {11}{2}} a^{4} b^{2} d^{8} + 778734 \, \left (d x\right )^{\frac {7}{2}} a^{5} b d^{10} + 302841 \, \left (d x\right )^{\frac {3}{2}} a^{6} d^{12}\right )}}{908523 \, d^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 103, normalized size = 0.79 \begin {gather*} \frac {2\,a^6\,{\left (d\,x\right )}^{3/2}}{3\,d}+\frac {2\,b^6\,{\left (d\,x\right )}^{27/2}}{27\,d^{13}}+\frac {30\,a^4\,b^2\,{\left (d\,x\right )}^{11/2}}{11\,d^5}+\frac {8\,a^3\,b^3\,{\left (d\,x\right )}^{15/2}}{3\,d^7}+\frac {30\,a^2\,b^4\,{\left (d\,x\right )}^{19/2}}{19\,d^9}+\frac {12\,a^5\,b\,{\left (d\,x\right )}^{7/2}}{7\,d^3}+\frac {12\,a\,b^5\,{\left (d\,x\right )}^{23/2}}{23\,d^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.00, size = 131, normalized size = 1.00 \begin {gather*} \frac {2 a^{6} \sqrt {d} x^{\frac {3}{2}}}{3} + \frac {12 a^{5} b \sqrt {d} x^{\frac {7}{2}}}{7} + \frac {30 a^{4} b^{2} \sqrt {d} x^{\frac {11}{2}}}{11} + \frac {8 a^{3} b^{3} \sqrt {d} x^{\frac {15}{2}}}{3} + \frac {30 a^{2} b^{4} \sqrt {d} x^{\frac {19}{2}}}{19} + \frac {12 a b^{5} \sqrt {d} x^{\frac {23}{2}}}{23} + \frac {2 b^{6} \sqrt {d} x^{\frac {27}{2}}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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