Optimal. Leaf size=30 \[ -\frac {3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 30, 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 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (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)^{4/3} (c+d x)^{2/3}} \, dx &=-\frac {3 \sqrt [3]{c+d x}}{(b c-a d) \sqrt [3]{a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 30, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 42, normalized size = 1.40 \begin {gather*} -\frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \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 {4}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.90 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {1}{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 {4}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 26, normalized size = 0.87 \begin {gather*} \frac {3\,{\left (c+d\,x\right )}^{1/3}}{\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^{1/3}} \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 {4}{3}} \left (c + d x\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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