Optimal. Leaf size=141 \[ \frac {(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac {d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac {a (6 b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{4/3}}-\frac {a (6 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {396, 201, 245}
\begin {gather*} \frac {a \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (6 b c-a d)}{9 \sqrt {3} b^{4/3}}-\frac {a (6 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac {x \left (a+b x^3\right )^{2/3} (6 b c-a d)}{18 b}+\frac {d x \left (a+b x^3\right )^{5/3}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 245
Rule 396
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right ) \, dx &=\frac {d x \left (a+b x^3\right )^{5/3}}{6 b}-\frac {(-6 b c+a d) \int \left (a+b x^3\right )^{2/3} \, dx}{6 b}\\ &=\frac {(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac {d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac {(a (6 b c-a d)) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{9 b}\\ &=\frac {(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac {d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac {a (6 b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{4/3}}-\frac {a (6 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 180, normalized size = 1.28 \begin {gather*} \frac {3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3} \left (6 b c+2 a d+3 b d x^3\right )-2 \sqrt {3} a (-6 b c+a d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )+2 a (-6 b c+a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-a (-6 b c+a d) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 322 vs.
\(2 (114) = 228\).
time = 0.51, size = 322, normalized size = 2.28 \begin {gather*} -\frac {1}{9} \, {\left (\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {1}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{2}}\right )} c + \frac {1}{54} \, {\left (\frac {2 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} - \frac {a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} + \frac {3 \, {\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.69, size = 424, normalized size = 3.01 \begin {gather*} \left [-\frac {3 \, \sqrt {\frac {1}{3}} {\left (6 \, a b^{2} c - a^{2} b d\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 2 \, {\left (6 \, a b c - a^{2} d\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (6 \, a b c - a^{2} d\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (3 \, b^{2} d x^{4} + 2 \, {\left (3 \, b^{2} c + a b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{2}}, -\frac {2 \, {\left (6 \, a b c - a^{2} d\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (6 \, a b c - a^{2} d\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {6 \, \sqrt {\frac {1}{3}} {\left (6 \, a b^{2} c - a^{2} b d\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}} - 3 \, {\left (3 \, b^{2} d x^{4} + 2 \, {\left (3 \, b^{2} c + a b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.96, size = 82, normalized size = 0.58 \begin {gather*} \frac {a^{\frac {2}{3}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {2}{3}} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \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 {\left (b\,x^3+a\right )}^{2/3}\,\left (d\,x^3+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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