Optimal. Leaf size=246 \[ \frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} \sqrt{c} d^2} \]
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Rubi [A] time = 0.270559, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2139, 218, 2137, 203} \[ \frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} \sqrt{c} d^2} \]
Antiderivative was successfully verified.
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Rule 2139
Rule 218
Rule 2137
Rule 203
Rubi steps
\begin{align*} \int \frac{x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx &=\frac{\int \frac{1}{\sqrt{c^3+4 d^3 x^3}} \, dx}{3 d}-\frac{\int \frac{c-2 d x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx}{3 d}\\ &=\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{1+3 c^3 x^2} \, dx,x,\frac{1+\frac{2 d x}{c}}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 d^2}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} \sqrt{c} d^2}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}\\ \end{align*}
Mathematica [C] time = 0.970301, size = 372, normalized size = 1.51 \[ \frac{\sqrt [6]{2} \sqrt{\frac{\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (-\sqrt{\frac{\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )+\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) c \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt [3]{2} d x}{c}+2^{2/3}} \Pi \left (\frac{i \sqrt [3]{2} \sqrt{3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}\right )}{\left (2+\sqrt [3]{-2}\right ) d^2 \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3+4 d^3 x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.009, size = 892, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{4 \, d^{3} x^{3} + c^{3}} x}{4 \, d^{4} x^{4} + 4 \, c d^{3} x^{3} + c^{3} d x + c^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (c + d x\right ) \sqrt{c^{3} + 4 d^{3} x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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