Optimal. Leaf size=74 \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 74, 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 = {1369, 14}
\begin {gather*} \frac {b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1369
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^3} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (b^2+\frac {a b}{x^3}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.50 \begin {gather*} -\frac {\left (a-2 b x^3\right ) \sqrt {\left (a+b x^3\right )^2}}{2 x^2 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 34, normalized size = 0.46
method | result | size |
gosper | \(-\frac {\left (-2 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 \left (b \,x^{3}+a \right ) x^{2}}\) | \(34\) |
default | \(-\frac {\left (-2 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 \left (b \,x^{3}+a \right ) x^{2}}\) | \(34\) |
risch | \(-\frac {a \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x^{2} \left (b \,x^{3}+a \right )}+\frac {b x \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.20 \begin {gather*} \frac {2 \, b x^{3} - a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 15, normalized size = 0.20 \begin {gather*} \frac {2 \, b x^{3} - a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 8, normalized size = 0.11 \begin {gather*} - \frac {a}{2 x^{2}} + b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.77, size = 26, normalized size = 0.35 \begin {gather*} b x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a \mathrm {sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {{\left (b\,x^3+a\right )}^2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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