Optimal. Leaf size=34 \[ \frac {(a+b x)^{5/3}}{2 b c \sqrt {\frac {c}{(a+b x)^{2/3}}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {253, 15, 30}
\begin {gather*} \frac {(a+b x)^{5/3}}{2 b c \sqrt {\frac {c}{(a+b x)^{2/3}}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {c}{(a+b x)^{2/3}}\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (\frac {c}{x^{2/3}}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}(\int x \, dx,x,a+b x)}{b c \sqrt {\frac {c}{(a+b x)^{2/3}}} \sqrt [3]{a+b x}}\\ &=\frac {(a+b x)^{5/3}}{2 b c \sqrt {\frac {c}{(a+b x)^{2/3}}}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {x (2 a+b x)}{2 \left (\frac {c}{(a+b x)^{2/3}}\right )^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 29, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {b x +a}{2 b \left (\frac {c}{\left (b x +a \right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}}\) | \(22\) |
gosper | \(\frac {x \left (b x +2 a \right )}{2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}}\) | \(29\) |
default | \(\frac {x \left (b x +2 a \right )}{2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.44 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.44 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (26) = 52\).
time = 0.80, size = 80, normalized size = 2.35 \begin {gather*} \frac {2 a x}{2 a \left (\frac {c}{\left (a + b x\right )^{\frac {2}{3}}}\right )^{\frac {3}{2}} + 2 b x \left (\frac {c}{\left (a + b x\right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}} + \frac {b x^{2}}{2 a \left (\frac {c}{\left (a + b x\right )^{\frac {2}{3}}}\right )^{\frac {3}{2}} + 2 b x \left (\frac {c}{\left (a + b x\right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 15, normalized size = 0.44 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 42, normalized size = 1.24 \begin {gather*} \sqrt {\frac {c}{{\left (a+b\,x\right )}^{2/3}}}\,\left (\frac {a\,x\,{\left (a+b\,x\right )}^{1/3}}{c^2}+\frac {b\,x^2\,{\left (a+b\,x\right )}^{1/3}}{2\,c^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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