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 = {247, 15, 30} \[ \frac {(a+b x)^{5/3}}{2 b c \sqrt {\frac {c}{(a+b x)^{2/3}}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {c}{(a+b x)^{2/3}}\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (\frac {c}{x^{2/3}}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {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 = 0.03, size = 34, normalized size = 1.00 \[ \frac {x (2 a+b x)}{2 (a+b x) \left (\frac {c}{(a+b x)^{2/3}}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 15, normalized size = 0.44 \[ \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.44 \[ \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.85 \[ \frac {\left (b x +2 a \right ) x}{2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{\frac {2}{3}}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 15, normalized size = 0.44 \[ \frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}}} \]
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 \[ \sqrt {\frac {c}{{\left (a+b\,x\right )}^{2/3}}}\,\left (\frac {a\,x\,{\left (a+b\,x\right )}^{1/3}}{c^2}+\frac {b\,x^2\,{\left (a+b\,x\right )}^{1/3}}{2\,c^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.15, size = 76, normalized size = 2.24 \[ \begin {cases} \frac {2 a x}{\frac {2 a c^{\frac {3}{2}}}{a + b x} + \frac {2 b c^{\frac {3}{2}} x}{a + b x}} + \frac {b x^{2}}{\frac {2 a c^{\frac {3}{2}}}{a + b x} + \frac {2 b c^{\frac {3}{2}} x}{a + b x}} & \text {for}\: a \neq 0 \vee b \neq 0 \\\frac {x}{\left (\tilde {\infty } c\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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