Optimal. Leaf size=174 \[ \frac {5 a (9 b c-a d) x \left (a+b x^3\right )^{2/3}}{162 b}+\frac {(9 b c-a d) x \left (a+b x^3\right )^{5/3}}{54 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}+\frac {5 a^2 (9 b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} b^{4/3}}-\frac {5 a^2 (9 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{162 b^{4/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {5 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (9 b c-a d)}{81 \sqrt {3} b^{4/3}}-\frac {5 a^2 (9 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{162 b^{4/3}}+\frac {x \left (a+b x^3\right )^{5/3} (9 b c-a d)}{54 b}+\frac {5 a x \left (a+b x^3\right )^{2/3} (9 b c-a d)}{162 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 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 )^{5/3} \left (c+d x^3\right ) \, dx &=\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}-\frac {(-9 b c+a d) \int \left (a+b x^3\right )^{5/3} \, dx}{9 b}\\ &=\frac {(9 b c-a d) x \left (a+b x^3\right )^{5/3}}{54 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}+\frac {(5 a (9 b c-a d)) \int \left (a+b x^3\right )^{2/3} \, dx}{54 b}\\ &=\frac {5 a (9 b c-a d) x \left (a+b x^3\right )^{2/3}}{162 b}+\frac {(9 b c-a d) x \left (a+b x^3\right )^{5/3}}{54 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}+\frac {\left (5 a^2 (9 b c-a d)\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{81 b}\\ &=\frac {5 a (9 b c-a d) x \left (a+b x^3\right )^{2/3}}{162 b}+\frac {(9 b c-a d) x \left (a+b x^3\right )^{5/3}}{54 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}+\frac {5 a^2 (9 b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} b^{4/3}}-\frac {5 a^2 (9 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{162 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 208, normalized size = 1.20 \begin {gather*} \frac {3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3} \left (10 a^2 d+9 b^2 x^3 \left (3 c+2 d x^3\right )+a b \left (72 c+33 d x^3\right )\right )-10 \sqrt {3} a^2 (-9 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 )+10 a^2 (-9 b c+a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-5 a^2 (-9 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 )}{486 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (d \,x^{3}+c \right )\, dx\]
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. 406 vs.
\(2 (143) = 286\).
time = 0.49, size = 406, normalized size = 2.33 \begin {gather*} -\frac {1}{54} \, {\left (\frac {10 \, \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 {1}{3}}} - \frac {5 \, 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 {1}{3}}} + \frac {10 \, a^{2} \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 (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} - \frac {8 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{2} - \frac {2 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2}}{x^{6}}}\right )} c + \frac {1}{486} \, {\left (\frac {10 \, \sqrt {3} a^{3} \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 {5 \, a^{3} \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 {10 \, a^{3} \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 {5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3} b^{2}}{x^{2}} - \frac {13 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3} b}{x^{5}} - \frac {10 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}\right )}}{b^{4} - \frac {3 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b}{x^{9}}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.55, size = 482, normalized size = 2.77 \begin {gather*} \left [-\frac {15 \, \sqrt {\frac {1}{3}} {\left (9 \, a^{2} b^{2} c - a^{3} 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 ) + 10 \, {\left (9 \, a^{2} b c - a^{3} 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 ) - 5 \, {\left (9 \, a^{2} b c - a^{3} 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 (18 \, b^{3} d x^{7} + 3 \, {\left (9 \, b^{3} c + 11 \, a b^{2} d\right )} x^{4} + 2 \, {\left (36 \, a b^{2} c + 5 \, a^{2} b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{486 \, b^{2}}, -\frac {10 \, {\left (9 \, a^{2} b c - a^{3} 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 ) - 5 \, {\left (9 \, a^{2} b c - a^{3} 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 {30 \, \sqrt {\frac {1}{3}} {\left (9 \, a^{2} b^{2} c - a^{3} 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 (18 \, b^{3} d x^{7} + 3 \, {\left (9 \, b^{3} c + 11 \, a b^{2} d\right )} x^{4} + 2 \, {\left (36 \, a b^{2} c + 5 \, a^{2} b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{486 \, 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 = 7.07, size = 170, normalized size = 0.98 \begin {gather*} \frac {a^{\frac {5}{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 {5}{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 )} + \frac {a^{\frac {2}{3}} b c 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 )} + \frac {a^{\frac {2}{3}} b d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{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 )}^{5/3}\,\left (d\,x^3+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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