Optimal. Leaf size=165 \[ -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {3 a b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 165, 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, 276}
\begin {gather*} \frac {3 a b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1369
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^3}{x^2} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^3 b^3}{x^2}+3 a^2 b^4 x+3 a b^5 x^4+b^6 x^7\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {3 a b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 61, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-40 a^3+60 a^2 b x^3+24 a b^2 x^6+5 b^3 x^9\right )}{40 x \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 58, normalized size = 0.35
method | result | size |
gosper | \(-\frac {\left (-5 b^{3} x^{9}-24 a \,b^{2} x^{6}-60 a^{2} b \,x^{3}+40 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{40 x \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(-\frac {\left (-5 b^{3} x^{9}-24 a \,b^{2} x^{6}-60 a^{2} b \,x^{3}+40 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{40 x \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b \left (\frac {1}{8} b^{2} x^{8}+\frac {3}{5} a b \,x^{5}+\frac {3}{2} a^{2} x^{2}\right )}{b \,x^{3}+a}-\frac {a^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{x \left (b \,x^{3}+a \right )}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.22 \begin {gather*} \frac {5 \, b^{3} x^{9} + 24 \, a b^{2} x^{6} + 60 \, a^{2} b x^{3} - 40 \, a^{3}}{40 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 37, normalized size = 0.22 \begin {gather*} \frac {5 \, b^{3} x^{9} + 24 \, a b^{2} x^{6} + 60 \, a^{2} b x^{3} - 40 \, a^{3}}{40 \, x} \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 {3}{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.67, size = 67, normalized size = 0.41 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{5} \, a b^{2} x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{2} \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{x} \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 {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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