Optimal. Leaf size=57 \[ \frac{2 \sqrt{c+d x^3}}{3 d}-\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d} \]
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Rubi [A] time = 0.045898, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {444, 50, 63, 203} \[ \frac{2 \sqrt{c+d x^3}}{3 d}-\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d} \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{c+d x^3}}{4 c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{4 c+d x} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{c+d x^3}}{3 d}-c \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} (4 c+d x)} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{c+d x^3}}{3 d}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{3 c+x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d}\\ &=\frac{2 \sqrt{c+d x^3}}{3 d}-\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d}\\ \end{align*}
Mathematica [A] time = 0.0142963, size = 54, normalized size = 0.95 \[ \frac{2 \left (\sqrt{c+d x^3}-\sqrt{3} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 425, normalized size = 7.5 \begin{align*}{\frac{2}{3\,d}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d+4\,c \right ) }{\sqrt [3]{-{d}^{2}c}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-{d}^{2}c}} \right ) \left ( -3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i\sqrt{3} \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-{d}^{2}c}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}},{\frac{1}{6\,cd} \left ( 2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c} \left ( -{\frac{3}{2\,d}\sqrt [3]{-{d}^{2}c}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0047, size = 281, normalized size = 4.93 \begin{align*} \left [\frac{\sqrt{3} \sqrt{-c} \log \left (\frac{d x^{3} - 2 \, \sqrt{3} \sqrt{d x^{3} + c} \sqrt{-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + 2 \, \sqrt{d x^{3} + c}}{3 \, d}, -\frac{2 \,{\left (\sqrt{3} \sqrt{c} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right ) - \sqrt{d x^{3} + c}\right )}}{3 \, d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.40482, size = 51, normalized size = 0.89 \begin{align*} \frac{2 \left (- \frac{\sqrt{3} \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{3} + \frac{\sqrt{c + d x^{3}}}{3}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12275, size = 59, normalized size = 1.04 \begin{align*} -\frac{2 \, \sqrt{3} \sqrt{c} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{3 \, d} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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