Optimal. Leaf size=27 \[ \frac {a+b x}{2 b \left (\frac {c}{(a+b x)^{3/2}}\right )^{2/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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} \[ \frac {a+b x}{2 b \left (\frac {c}{(a+b x)^{3/2}}\right )^{2/3}} \]
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)^{3/2}}\right )^{2/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (\frac {c}{x^{3/2}}\right )^{2/3}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}(\int x \, dx,x,a+b x)}{b \left (\frac {c}{(a+b x)^{3/2}}\right )^{2/3} (a+b x)}\\ &=\frac {a+b x}{2 b \left (\frac {c}{(a+b x)^{3/2}}\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 1.26 \[ \frac {x (2 a+b x)}{2 (a+b x) \left (\frac {c}{(a+b x)^{3/2}}\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 51, normalized size = 1.89 \[ \frac {{\left (b x^{2} + 2 \, a x\right )} \sqrt {b x + a} \left (\frac {\sqrt {b x + a} c}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{\frac {1}{3}}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {c}{{\left (b x + a\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.07 \[ \frac {\left (b x +2 a \right ) x}{2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 21, normalized size = 0.78 \[ \frac {b x + a}{2 \, b \left (\frac {c}{{\left (b x + a\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 31, normalized size = 1.15 \[ \frac {x\,{\left (\frac {c}{{\left (a+b\,x\right )}^{3/2}}\right )}^{1/3}\,\left (2\,a+b\,x\right )\,\sqrt {a+b\,x}}{2\,c} \]
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 \[ \int \frac {1}{\left (\frac {c}{\left (a + b x\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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