Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 14} \begin {gather*} -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^7} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^7} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^7}+\frac {b^2}{x^4}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a+2 b x^3\right )}{6 x^6 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 118, normalized size = 1.49 \begin {gather*} \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (-a b-2 b^2 x^3\right )+\sqrt {b^2} \left (a^2+3 a b x^3+2 b^2 x^6\right )}{6 x^6 \left (a b+b^2 x^3\right )-6 \sqrt {b^2} x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 13, normalized size = 0.16 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 30, normalized size = 0.38 \begin {gather*} -\frac {2 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a \mathrm {sgn}\left (b x^{3} + a\right )}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.43 \begin {gather*} -\frac {\left (2 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{6 \left (b \,x^{3}+a \right ) x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 86, normalized size = 1.09 \begin {gather*} \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2}}{6 \, a^{2}} + \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b}{6 \, a x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{6 \, a^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 33, normalized size = 0.42 \begin {gather*} -\frac {\left (2\,b\,x^3+a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{6\,x^6\,\left (b\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 14, normalized size = 0.18 \begin {gather*} \frac {- a - 2 b x^{3}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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