Optimal. Leaf size=99 \[ \frac {(b c-a d) x}{a b \sqrt [3]{a+b x^3}}+\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{4/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 99, 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 = {393, 245}
\begin {gather*} \frac {d \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3}}-\frac {d \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{4/3}}+\frac {x (b c-a d)}{a b \sqrt [3]{a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 393
Rubi steps
\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx &=\frac {(b c-a d) x}{a b \sqrt [3]{a+b x^3}}+\frac {d \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{b}\\ &=\frac {(b c-a d) x}{a b \sqrt [3]{a+b x^3}}+\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 150, normalized size = 1.52 \begin {gather*} \frac {\frac {6 \sqrt [3]{b} (b c-a d) x}{a \sqrt [3]{a+b x^3}}+2 \sqrt {3} d \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+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 )}{6 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{3}+c}{\left (b \,x^{3}+a \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 134, normalized size = 1.35 \begin {gather*} -\frac {1}{6} \, d {\left (\frac {2 \, \sqrt {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 {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\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 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {c x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} a} \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. 220 vs.
\(2 (81) = 162\).
time = 3.93, size = 488, normalized size = 4.93 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x - 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x + 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}\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 = 3.08, size = 71, normalized size = 0.72 \begin {gather*} \frac {c x \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {4}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {4}{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 \frac {d\,x^3+c}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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