Optimal. Leaf size=216 \[ -\frac{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 d^2}-\frac{5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b} d^{8/3}}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d} \]
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Rubi [A] time = 0.080201, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {50, 59} \[ -\frac{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 d^2}-\frac{5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b} d^{8/3}}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/3}}{(c+d x)^{2/3}} \, dx &=\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d}-\frac{(5 (b c-a d)) \int \frac{(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx}{6 d}\\ &=-\frac{5 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 d^2}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d}+\frac{\left (5 (b c-a d)^2\right ) \int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{9 d^2}\\ &=-\frac{5 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 d^2}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d}-\frac{5 (b c-a d)^2 \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 )}{3 \sqrt{3} \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log \left (-1+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 \sqrt [3]{b} d^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.0345999, size = 73, normalized size = 0.34 \[ \frac{3 (a+b x)^{8/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{8}{3};\frac{11}{3};\frac{d (a+b x)}{a d-b c}\right )}{8 b (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09284, size = 1835, normalized size = 8.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{5}{3}}}{\left (c + d x\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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