Optimal. Leaf size=91 \[ \frac {2 a^4 (d x)^{7/2}}{7 d}+\frac {8 a^3 b (d x)^{11/2}}{11 d^3}+\frac {4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac {8 a b^3 (d x)^{19/2}}{19 d^7}+\frac {2 b^4 (d x)^{23/2}}{23 d^9} \]
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Rubi [A] time = 0.04, antiderivative size = 91, 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 {4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac {8 a^3 b (d x)^{11/2}}{11 d^3}+\frac {2 a^4 (d x)^{7/2}}{7 d}+\frac {8 a b^3 (d x)^{19/2}}{19 d^7}+\frac {2 b^4 (d x)^{23/2}}{23 d^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int (d x)^{5/2} \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\int \left (a^4 b^4 (d x)^{5/2}+\frac {4 a^3 b^5 (d x)^{9/2}}{d^2}+\frac {6 a^2 b^6 (d x)^{13/2}}{d^4}+\frac {4 a b^7 (d x)^{17/2}}{d^6}+\frac {b^8 (d x)^{21/2}}{d^8}\right ) \, dx}{b^4}\\ &=\frac {2 a^4 (d x)^{7/2}}{7 d}+\frac {8 a^3 b (d x)^{11/2}}{11 d^3}+\frac {4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac {8 a b^3 (d x)^{19/2}}{19 d^7}+\frac {2 b^4 (d x)^{23/2}}{23 d^9}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.60 \begin {gather*} \frac {2 x (d x)^{5/2} \left (24035 a^4+61180 a^3 b x^2+67298 a^2 b^2 x^4+35420 a b^3 x^6+7315 b^4 x^8\right )}{168245} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 85, normalized size = 0.93 \begin {gather*} \frac {2 \left (24035 a^4 d^8 (d x)^{7/2}+61180 a^3 b d^6 (d x)^{11/2}+67298 a^2 b^2 d^4 (d x)^{15/2}+35420 a b^3 d^2 (d x)^{19/2}+7315 b^4 (d x)^{23/2}\right )}{168245 d^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.49, size = 68, normalized size = 0.75 \begin {gather*} \frac {2}{168245} \, {\left (7315 \, b^{4} d^{2} x^{11} + 35420 \, a b^{3} d^{2} x^{9} + 67298 \, a^{2} b^{2} d^{2} x^{7} + 61180 \, a^{3} b d^{2} x^{5} + 24035 \, a^{4} d^{2} x^{3}\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 = 86, normalized size = 0.95 \begin {gather*} \frac {2}{23} \, \sqrt {d x} b^{4} d^{2} x^{11} + \frac {8}{19} \, \sqrt {d x} a b^{3} d^{2} x^{9} + \frac {4}{5} \, \sqrt {d x} a^{2} b^{2} d^{2} x^{7} + \frac {8}{11} \, \sqrt {d x} a^{3} b d^{2} x^{5} + \frac {2}{7} \, \sqrt {d x} a^{4} d^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.57 \begin {gather*} \frac {2 \left (7315 b^{4} x^{8}+35420 a \,b^{3} x^{6}+67298 a^{2} b^{2} x^{4}+61180 a^{3} b \,x^{2}+24035 a^{4}\right ) \left (d x \right )^{\frac {5}{2}} x}{168245} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 73, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (7315 \, \left (d x\right )^{\frac {23}{2}} b^{4} + 35420 \, \left (d x\right )^{\frac {19}{2}} a b^{3} d^{2} + 67298 \, \left (d x\right )^{\frac {15}{2}} a^{2} b^{2} d^{4} + 61180 \, \left (d x\right )^{\frac {11}{2}} a^{3} b d^{6} + 24035 \, \left (d x\right )^{\frac {7}{2}} a^{4} d^{8}\right )}}{168245 \, d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 71, normalized size = 0.78 \begin {gather*} \frac {2\,a^4\,{\left (d\,x\right )}^{7/2}}{7\,d}+\frac {2\,b^4\,{\left (d\,x\right )}^{23/2}}{23\,d^9}+\frac {4\,a^2\,b^2\,{\left (d\,x\right )}^{15/2}}{5\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{11/2}}{11\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{19/2}}{19\,d^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.68, size = 90, normalized size = 0.99 \begin {gather*} \frac {2 a^{4} d^{\frac {5}{2}} x^{\frac {7}{2}}}{7} + \frac {8 a^{3} b d^{\frac {5}{2}} x^{\frac {11}{2}}}{11} + \frac {4 a^{2} b^{2} d^{\frac {5}{2}} x^{\frac {15}{2}}}{5} + \frac {8 a b^{3} d^{\frac {5}{2}} x^{\frac {19}{2}}}{19} + \frac {2 b^{4} d^{\frac {5}{2}} x^{\frac {23}{2}}}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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