Optimal. Leaf size=32 \[ -\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx &=-\frac {3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} -\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 32, normalized size = 1.00 \begin {gather*} -\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 42, normalized size = 1.31 \begin {gather*} -\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {2}{3}}}{2 \left (b x +a \right )^{\frac {2}{3}} \left (a d -b c \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d 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 = -0.03 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{5/3}\,{\left (c+d\,x\right )}^{1/3}} \,d 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 {1}{\left (a + b x\right )^{\frac {5}{3}} \sqrt [3]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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