Optimal. Leaf size=219 \[ \frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 b^{4/3} d^{5/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 219, normalized size of antiderivative = 1.00, 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 = {52, 61}
\begin {gather*} \frac {(b c-a d)^2 \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 )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(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 b^{4/3} d^{5/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rubi steps
\begin {align*} \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx &=\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx}{6 b}\\ &=\frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}-\frac {(b c-a d)^2 \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{9 b d}\\ &=\frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(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} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(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 b^{4/3} d^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 220, normalized size = 1.00 \begin {gather*} \frac {(b c-a d)^2 \left (\frac {3 \sqrt [3]{b} d^{2/3} (a+b x)^{2/3} \sqrt [3]{c+d x} (2 a d+b (c+3 d x))}{(b c-a d)^2}-2 \sqrt {3} \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 \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )-\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 )\right )}{18 b^{4/3} d^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {2}{3}} \left (d x +c \right )^{\frac {1}{3}}\, dx\]
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 [A]
time = 1.00, size = 717, normalized size = 3.27 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} + 3 \, \left (b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right ) + 3 \, {\left (3 \, b^{2} d^{3} x + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}}{b d^{2} x + a d^{2}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right ) - 3 \, {\left (3 \, b^{2} d^{3} x + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}\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 \left (a + b x\right )^{\frac {2}{3}} \sqrt [3]{c + d x}\, 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 = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{2/3}\,{\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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