Optimal. Leaf size=84 \[ \frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 266, 45, 37} \begin {gather*} \frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{22}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{22}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^8} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^7} \, dx,x,x^3\right )}{21 a b^3 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}}+\frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (6 a^5+35 a^4 b x^3+84 a^3 b^2 x^6+105 a^2 b^3 x^9+70 a b^4 x^{12}+21 b^5 x^{15}\right )}{126 x^{21} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.67, size = 488, normalized size = 5.81 \begin {gather*} \frac {32 b^6 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-6 a^{11} b-71 a^{10} b^2 x^3-384 a^9 b^3 x^6-1254 a^8 b^4 x^9-2750 a^7 b^5 x^{12}-4257 a^6 b^6 x^{15}-4752 a^5 b^7 x^{18}-3829 a^4 b^8 x^{21}-2184 a^3 b^9 x^{24}-840 a^2 b^{10} x^{27}-196 a b^{11} x^{30}-21 b^{12} x^{33}\right )+32 \sqrt {b^2} b^6 \left (6 a^{12}+77 a^{11} b x^3+455 a^{10} b^2 x^6+1638 a^9 b^3 x^9+4004 a^8 b^4 x^{12}+7007 a^7 b^5 x^{15}+9009 a^6 b^6 x^{18}+8581 a^5 b^7 x^{21}+6013 a^4 b^8 x^{24}+3024 a^3 b^9 x^{27}+1036 a^2 b^{10} x^{30}+217 a b^{11} x^{33}+21 b^{12} x^{36}\right )}{63 \sqrt {b^2} x^{21} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-64 a^6 b^6-384 a^5 b^7 x^3-960 a^4 b^8 x^6-1280 a^3 b^9 x^9-960 a^2 b^{10} x^{12}-384 a b^{11} x^{15}-64 b^{12} x^{18}\right )+63 x^{21} \left (64 a^7 b^7+448 a^6 b^8 x^3+1344 a^5 b^9 x^6+2240 a^4 b^{10} x^9+2240 a^3 b^{11} x^{12}+1344 a^2 b^{12} x^{15}+448 a b^{13} x^{18}+64 b^{14} x^{21}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 59, normalized size = 0.70 \begin {gather*} -\frac {21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 107, normalized size = 1.27 \begin {gather*} -\frac {21 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 70 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 105 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 84 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 35 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{126 \, x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.95 \begin {gather*} -\frac {\left (21 b^{5} x^{15}+70 a \,b^{4} x^{12}+105 a^{2} b^{3} x^{9}+84 a^{3} b^{2} x^{6}+35 a^{4} b \,x^{3}+6 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{126 \left (b \,x^{3}+a \right )^{5} x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.16, size = 241, normalized size = 2.87 \begin {gather*} -\frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{7}}{18 \, a^{7}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{6}}{18 \, a^{6} x^{3}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{5}}{18 \, a^{7} x^{6}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{4}}{18 \, a^{6} x^{9}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{3}}{18 \, a^{5} x^{12}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{18 \, a^{4} x^{15}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{18 \, a^{3} x^{18}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{21 \, a^{2} x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 231, normalized size = 2.75 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{21\,x^{21}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{6\,x^6\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^9\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{18\,x^{18}\,\left (b\,x^3+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{6\,x^{12}\,\left (b\,x^3+a\right )}-\frac {2\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^{15}\,\left (b\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{22}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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