Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
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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} \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
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^8} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^8} \, 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^8}+\frac {b^2}{x^5}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (4 a+7 b x^3\right )}{28 x^7 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 15, normalized size = 0.19 \[ -\frac {7 \, b x^{3} + 4 \, a}{28 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 31, normalized size = 0.39 \[ -\frac {7 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{28 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.46 \[ -\frac {\left (7 b \,x^{3}+4 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{28 \left (b \,x^{3}+a \right ) x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 15, normalized size = 0.19 \[ -\frac {7 \, b x^{3} + 4 \, a}{28 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 35, normalized size = 0.44 \[ -\frac {\left (7\,b\,x^3+4\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{28\,x^7\,\left (b\,x^3+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 15, normalized size = 0.19 \[ \frac {- 4 a - 7 b x^{3}}{28 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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