Optimal. Leaf size=30 \[ \frac {3 \sqrt [3]{a+b x}}{(b c-a d) \sqrt [3]{c+d x}} \]
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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]{a+b x}}{\sqrt [3]{c+d 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)^{2/3} (c+d x)^{4/3}} \, dx &=\frac {3 \sqrt [3]{a+b x}}{(b c-a d) \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{a+b x}}{(b c-a d) \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 27, normalized size = 0.90
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (a d -b c \right )}\) | \(27\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.13, size = 42, normalized size = 1.40 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} 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 {2}{3}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \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*} \text {could not integrate} \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 )}^{2/3}\,{\left (c+d\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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