Optimal. Leaf size=128 \[ -\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac {b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}} \]
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Rubi [A] time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 266, 45, 37} \[ -\frac {b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}} \]
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^{25}} \, 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^{25}} \, 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^9} \, 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}}{24 a x^{24}}-\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 )}{12 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}}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{84 a^2 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 )}{84 a^2 b^2 \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}}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{84 a^2 x^{21}}-\frac {b^2 \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{504 a^3 x^{18}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.65 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (21 a^5+120 a^4 b x^3+280 a^3 b^2 x^6+336 a^2 b^3 x^9+210 a b^4 x^{12}+56 b^5 x^{15}\right )}{504 x^{24} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 59, normalized size = 0.46 \[ -\frac {56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 107, normalized size = 0.84 \[ -\frac {56 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 210 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 336 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 280 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 120 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 21 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{504 \, x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.62 \[ -\frac {\left (56 b^{5} x^{15}+210 a \,b^{4} x^{12}+336 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+120 a^{4} b \,x^{3}+21 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b \,x^{3}+a \right )^{5} x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 272, normalized size = 2.12 \[ \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{8}}{18 \, a^{8}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{7}}{18 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{6}}{18 \, a^{8} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{5}}{18 \, a^{7} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{4}}{18 \, a^{6} x^{12}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{3}}{18 \, a^{5} x^{15}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{18 \, a^{4} x^{18}} + \frac {3 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{56 \, a^{3} x^{21}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{24 \, a^{2} x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 231, normalized size = 1.80 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{24\,x^{24}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^9\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{12\,x^{12}\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{21\,x^{21}\,\left (b\,x^3+a\right )}-\frac {2\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^{15}\,\left (b\,x^3+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^{18}\,\left (b\,x^3+a\right )} \]
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 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{25}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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