Optimal. Leaf size=242 \[ \frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}}-\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (\left (1-\frac {3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {3 h x}{g}+1}\right )}{2\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}}+\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h x}{g}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 h x}{g}+1}}\right )}{2^{2/3} \sqrt {3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}} \]
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Rubi [A] time = 0.09, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1009, 1008} \[ \frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}}-\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (\left (1-\frac {3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {3 h x}{g}+1}\right )}{2\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}}+\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h x}{g}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 h x}{g}+1}}\right )}{2^{2/3} \sqrt {3} h \sqrt [3]{9 c x^2-\frac {c g^2}{h^2}}} \]
Antiderivative was successfully verified.
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Rule 1008
Rule 1009
Rubi steps
\begin {align*} \int \frac {g+h x}{\sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2} \left (g^2+3 h^2 x^2\right )} \, dx &=\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \int \frac {g+h x}{\left (g^2+3 h^2 x^2\right ) \sqrt [3]{1-\frac {9 h^2 x^2}{g^2}}} \, dx}{\sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2}}\\ &=\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h x}{g}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1+\frac {3 h x}{g}}}\right )}{2^{2/3} \sqrt {3} h \sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2}}+\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2}}-\frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \log \left (\left (1-\frac {3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\frac {3 h x}{g}}\right )}{2\ 2^{2/3} h \sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.56, size = 268, normalized size = 1.11 \[ \frac {h^2 x \left (-h x \sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} F_1\left (1;\frac {1}{3},1;2;\frac {9 h^2 x^2}{g^2},-\frac {3 h^2 x^2}{g^2}\right )-\frac {2 g^5 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {9 h^2 x^2}{g^2},-\frac {3 h^2 x^2}{g^2}\right )}{\left (g^2+3 h^2 x^2\right ) \left (g^2 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {9 h^2 x^2}{g^2},-\frac {3 h^2 x^2}{g^2}\right )+2 h^2 x^2 \left (F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {9 h^2 x^2}{g^2},-\frac {3 h^2 x^2}{g^2}\right )-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {9 h^2 x^2}{g^2},-\frac {3 h^2 x^2}{g^2}\right )\right )\right )}\right ) \left (c \left (9 x^2-\frac {g^2}{h^2}\right )\right )^{2/3}}{2 c g^2 \left (g^2-9 h^2 x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {h x + g}{{\left (3 \, h^{2} x^{2} + g^{2}\right )} {\left (9 \, c x^{2} - \frac {c g^{2}}{h^{2}}\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {h x +g}{\left (9 c \,x^{2}-\frac {c \,g^{2}}{h^{2}}\right )^{\frac {1}{3}} \left (3 h^{2} x^{2}+g^{2}\right )}\, dx \]
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 \[ \int \frac {h x + g}{{\left (3 \, h^{2} x^{2} + g^{2}\right )} {\left (9 \, c x^{2} - \frac {c g^{2}}{h^{2}}\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {g+h\,x}{\left (g^2+3\,h^2\,x^2\right )\,{\left (9\,c\,x^2-\frac {c\,g^2}{h^2}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {g + h x}{\sqrt [3]{c \left (- \frac {g}{h} + 3 x\right ) \left (\frac {g}{h} + 3 x\right )} \left (g^{2} + 3 h^{2} x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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