Optimal. Leaf size=69 \[ \frac{16 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]
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Rubi [A] time = 0.0571967, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{16 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]
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 \sqrt{c+d x^3}}{8 c-d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x \sqrt{c+d x}}{8 c-d x} \, dx,x,x^3\right )\\ &=-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{(8 c) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{8 c-d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{\left (24 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{d}\\ &=-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{\left (48 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d^2}\\ &=-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{16 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0323542, size = 58, normalized size = 0.84 \[ \frac{144 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (25 c+d x^3\right )}{9 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 446, normalized size = 6.5 \begin{align*} -{\frac{2}{9\,{d}^{2}} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}}-8\,{\frac{c}{d} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\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 = 1.81016, size = 296, normalized size = 4.29 \begin{align*} \left [\frac{2 \,{\left (36 \, c^{\frac{3}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (d x^{3} + 25 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{2}}, -\frac{2 \,{\left (72 \, \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) +{\left (d x^{3} + 25 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.7207, size = 65, normalized size = 0.94 \begin{align*} \frac{2 \left (- \frac{8 c^{2} \operatorname{atan}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{- c}} \right )}}{\sqrt{- c}} - \frac{8 c \sqrt{c + d x^{3}}}{3} - \frac{\left (c + d x^{3}\right )^{\frac{3}{2}}}{9}\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1145, size = 93, normalized size = 1.35 \begin{align*} -\frac{2 \,{\left (\frac{72 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} + \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{2} + 24 \, \sqrt{d x^{3} + c} c d^{2}}{d^{3}}\right )}}{9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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