Optimal. Leaf size=125 \[ -\frac {2 a^6}{d \sqrt {d x}}+\frac {4 a^5 b (d x)^{3/2}}{d^3}+\frac {30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac {40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac {2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac {12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac {2 b^6 (d x)^{23/2}}{23 d^{13}} \]
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Rubi [A] time = 0.06, antiderivative size = 125, 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 {2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac {40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac {30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac {4 a^5 b (d x)^{3/2}}{d^3}-\frac {2 a^6}{d \sqrt {d x}}+\frac {12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac {2 b^6 (d x)^{23/2}}{23 d^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{(d x)^{3/2}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{(d x)^{3/2}}+\frac {6 a^5 b^7 \sqrt {d x}}{d^2}+\frac {15 a^4 b^8 (d x)^{5/2}}{d^4}+\frac {20 a^3 b^9 (d x)^{9/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{13/2}}{d^8}+\frac {6 a b^{11} (d x)^{17/2}}{d^{10}}+\frac {b^{12} (d x)^{21/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=-\frac {2 a^6}{d \sqrt {d x}}+\frac {4 a^5 b (d x)^{3/2}}{d^3}+\frac {30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac {40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac {2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac {12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac {2 b^6 (d x)^{23/2}}{23 d^{13}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.62 \begin {gather*} \frac {2 x \left (-33649 a^6+67298 a^5 b x^2+72105 a^4 b^2 x^4+61180 a^3 b^3 x^6+33649 a^2 b^4 x^8+10626 a b^5 x^{10}+1463 b^6 x^{12}\right )}{33649 (d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 100, normalized size = 0.80 \begin {gather*} \frac {2 \left (-33649 a^6 d^{12}+67298 a^5 b d^{12} x^2+72105 a^4 b^2 d^{12} x^4+61180 a^3 b^3 d^{12} x^6+33649 a^2 b^4 d^{12} x^8+10626 a b^5 d^{12} x^{10}+1463 b^6 d^{12} x^{12}\right )}{33649 d^{13} \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 78, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (1463 \, b^{6} x^{12} + 10626 \, a b^{5} x^{10} + 33649 \, a^{2} b^{4} x^{8} + 61180 \, a^{3} b^{3} x^{6} + 72105 \, a^{4} b^{2} x^{4} + 67298 \, a^{5} b x^{2} - 33649 \, a^{6}\right )} \sqrt {d x}}{33649 \, d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 127, normalized size = 1.02 \begin {gather*} -\frac {2 \, {\left (\frac {33649 \, a^{6}}{\sqrt {d x}} - \frac {1463 \, \sqrt {d x} b^{6} d^{275} x^{11} + 10626 \, \sqrt {d x} a b^{5} d^{275} x^{9} + 33649 \, \sqrt {d x} a^{2} b^{4} d^{275} x^{7} + 61180 \, \sqrt {d x} a^{3} b^{3} d^{275} x^{5} + 72105 \, \sqrt {d x} a^{4} b^{2} d^{275} x^{3} + 67298 \, \sqrt {d x} a^{5} b d^{275} x}{d^{276}}\right )}}{33649 \, d} \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 = 0.59 \begin {gather*} -\frac {2 \left (-1463 b^{6} x^{12}-10626 a \,b^{5} x^{10}-33649 a^{2} b^{4} x^{8}-61180 a^{3} b^{3} x^{6}-72105 a^{4} b^{2} x^{4}-67298 a^{5} b \,x^{2}+33649 a^{6}\right ) x}{33649 \left (d x \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 108, normalized size = 0.86 \begin {gather*} -\frac {2 \, {\left (\frac {33649 \, a^{6}}{\sqrt {d x}} - \frac {1463 \, \left (d x\right )^{\frac {23}{2}} b^{6} + 10626 \, \left (d x\right )^{\frac {19}{2}} a b^{5} d^{2} + 33649 \, \left (d x\right )^{\frac {15}{2}} a^{2} b^{4} d^{4} + 61180 \, \left (d x\right )^{\frac {11}{2}} a^{3} b^{3} d^{6} + 72105 \, \left (d x\right )^{\frac {7}{2}} a^{4} b^{2} d^{8} + 67298 \, \left (d x\right )^{\frac {3}{2}} a^{5} b d^{10}}{d^{12}}\right )}}{33649 \, d} \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 = 0.82 \begin {gather*} \frac {2\,b^6\,{\left (d\,x\right )}^{23/2}}{23\,d^{13}}-\frac {2\,a^6}{d\,\sqrt {d\,x}}+\frac {30\,a^4\,b^2\,{\left (d\,x\right )}^{7/2}}{7\,d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{11/2}}{11\,d^7}+\frac {2\,a^2\,b^4\,{\left (d\,x\right )}^{15/2}}{d^9}+\frac {4\,a^5\,b\,{\left (d\,x\right )}^{3/2}}{d^3}+\frac {12\,a\,b^5\,{\left (d\,x\right )}^{19/2}}{19\,d^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.01, size = 126, normalized size = 1.01 \begin {gather*} - \frac {2 a^{6}}{d^{\frac {3}{2}} \sqrt {x}} + \frac {4 a^{5} b x^{\frac {3}{2}}}{d^{\frac {3}{2}}} + \frac {30 a^{4} b^{2} x^{\frac {7}{2}}}{7 d^{\frac {3}{2}}} + \frac {40 a^{3} b^{3} x^{\frac {11}{2}}}{11 d^{\frac {3}{2}}} + \frac {2 a^{2} b^{4} x^{\frac {15}{2}}}{d^{\frac {3}{2}}} + \frac {12 a b^{5} x^{\frac {19}{2}}}{19 d^{\frac {3}{2}}} + \frac {2 b^{6} x^{\frac {23}{2}}}{23 d^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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