Optimal. Leaf size=149 \[ -\frac {3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac {\sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{b^{4/3}}-\frac {\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}}-\frac {3 \sqrt [3]{d} \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {49, 61}
\begin {gather*} -\frac {\sqrt {3} \sqrt [3]{d} \text {ArcTan}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{b^{4/3}}-\frac {3 \sqrt [3]{d} \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3}}-\frac {3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac {\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 61
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{4/3}} \, dx &=-\frac {3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}+\frac {d \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{b}\\ &=-\frac {3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac {\sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{b^{4/3}}-\frac {\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}}-\frac {3 \sqrt [3]{d} \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 191, normalized size = 1.28 \begin {gather*} \frac {-\frac {6 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+2 \sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )-2 \sqrt [3]{d} \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+\sqrt [3]{d} \log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{2 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {4}{3}}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs.
\(2 (109) = 218\).
time = 0.88, size = 233, normalized size = 1.56 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (b x + a\right )} \left (-\frac {d}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b \left (-\frac {d}{b}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + a d\right )}}{3 \, {\left (b d x + a d\right )}}\right ) + {\left (b x + a\right )} \left (-\frac {d}{b}\right )^{\frac {1}{3}} \log \left (\frac {{\left (b x + a\right )} \left (-\frac {d}{b}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \left (-\frac {d}{b}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{b x + a}\right ) - 2 \, {\left (b x + a\right )} \left (-\frac {d}{b}\right )^{\frac {1}{3}} \log \left (\frac {{\left (b x + a\right )} \left (-\frac {d}{b}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) + 6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{2 \, {\left (b^{2} x + a b\right )}} \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 {\sqrt [3]{c + d x}}{\left (a + b 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 = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{1/3}}{{\left (a+b\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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