Optimal. Leaf size=88 \[ -\frac{\sqrt{c+d x^3}}{12 c^2 x^3}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}} \]
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Rubi [A] time = 0.0769034, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {446, 103, 156, 63, 208, 203} \[ -\frac{\sqrt{c+d x^3}}{12 c^2 x^3}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 156
Rule 63
Rule 208
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c+d x} (4 c+d x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}-\frac{\operatorname{Subst}\left (\int \frac{3 c d+\frac{d^2 x}{2}}{x \sqrt{c+d x} (4 c+d x)} \, dx,x,x^3\right )}{12 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{16 c^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} (4 c+d x)} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{8 c^2}+\frac{d \operatorname{Subst}\left (\int \frac{1}{3 c+x^2} \, dx,x,\sqrt{c+d x^3}\right )}{24 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0319086, size = 88, normalized size = 1. \[ -\frac{\sqrt{c+d x^3}}{12 c^2 x^3}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 477, normalized size = 5.4 \begin{align*}{\frac{-{\frac{i}{144}}\sqrt{2}}{d{c}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+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}}}}}+{\frac{1}{4\,c} \left ( -{\frac{1}{3\,c{x}^{3}}\sqrt{d{x}^{3}+c}}+{\frac{d}{3}{\it Artanh} \left ({\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }+{\frac{d}{24}{\it Artanh} \left ({\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}} \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{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60369, size = 505, normalized size = 5.74 \begin{align*} \left [\frac{2 \, \sqrt{3} \sqrt{c} d x^{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right ) + 9 \, \sqrt{c} d x^{3} \log \left (\frac{d x^{3} + 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) - 12 \, \sqrt{d x^{3} + c} c}{144 \, c^{3} x^{3}}, -\frac{\sqrt{3} \sqrt{-c} d x^{3} \log \left (\frac{d x^{3} - 2 \, \sqrt{3} \sqrt{d x^{3} + c} \sqrt{-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + 18 \, \sqrt{-c} d x^{3} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) + 12 \, \sqrt{d x^{3} + c} c}{144 \, c^{3} x^{3}}\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{1}{x^{4} \sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12235, size = 101, normalized size = 1.15 \begin{align*} \frac{1}{72} \, d{\left (\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{c^{\frac{5}{2}}} - \frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{6 \, \sqrt{d x^{3} + c}}{c^{2} d x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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