Optimal. Leaf size=88 \[ -\frac{48 c^2 \sqrt{c+d x^3}}{d^2}+\frac{144 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2} \]
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Rubi [A] time = 0.0716315, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {446, 80, 50, 63, 206} \[ -\frac{48 c^2 \sqrt{c+d x^3}}{d^2}+\frac{144 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac{(8 c) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac{\left (24 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d}\\ &=-\frac{48 c^2 \sqrt{c+d x^3}}{d^2}-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac{\left (216 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{d}\\ &=-\frac{48 c^2 \sqrt{c+d x^3}}{d^2}-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac{\left (432 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d^2}\\ &=-\frac{48 c^2 \sqrt{c+d x^3}}{d^2}-\frac{16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac{144 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0359215, size = 70, normalized size = 0.8 \[ \frac{6480 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (1123 c^2+46 c d x^3+3 d^2 x^6\right )}{45 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 462, normalized size = 5.3 \begin{align*} -{\frac{2}{15\,{d}^{2}} \left ( d{x}^{3}+c \right ) ^{{\frac{5}{2}}}}-8\,{\frac{c}{d} \left ( 2/9\,{x}^{3}\sqrt{d{x}^{3}+c}+{\frac{56\,c\sqrt{d{x}^{3}+c}}{9\,d}}+{\frac{3\,ic\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-{d}^{2}c} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c}} \left ( x-{\frac{\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-{d}^{2}c}} \left ( x+1/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) }},-1/18\,{\frac{2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \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 = 1.35718, size = 363, normalized size = 4.12 \begin{align*} \left [\frac{2 \,{\left (1620 \, c^{\frac{5}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (3 \, d^{2} x^{6} + 46 \, c d x^{3} + 1123 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{2}}, -\frac{2 \,{\left (3240 \, \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) +{\left (3 \, d^{2} x^{6} + 46 \, c d x^{3} + 1123 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 78.0689, size = 90, normalized size = 1.02 \begin{align*} - \frac{144 c^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{- c}} \right )}}{d^{2} \sqrt{- c}} - \frac{48 c^{2} \sqrt{c + d x^{3}}}{d^{2}} - \frac{16 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{2}} - \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13788, size = 112, normalized size = 1.27 \begin{align*} -\frac{144 \, c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{2}} - \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{8} + 40 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{8} + 1080 \, \sqrt{d x^{3} + c} c^{2} d^{8}\right )}}{45 \, d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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