Optimal. Leaf size=109 \[ -\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{1152 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3} \]
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Rubi [A] time = 0.102259, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {446, 88, 50, 63, 206} \[ -\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{1152 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 (c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{7 c (c+d x)^{3/2}}{d^2}+\frac{64 c^2 (c+d x)^{3/2}}{d^2 (8 c-d x)}-\frac{(c+d x)^{5/2}}{d^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac{\left (64 c^2\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac{\left (192 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d^2}\\ &=-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac{\left (1728 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{d^2}\\ &=-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac{\left (3456 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d^3}\\ &=-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac{1152 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0699023, size = 81, normalized size = 0.74 \[ \frac{362880 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (2579 c^2 d x^3+62882 c^3+192 c d^2 x^6+15 d^3 x^9\right )}{315 d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 541, normalized size = 5. \begin{align*} -{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,d{x}^{9}}{21}\sqrt{d{x}^{3}+c}}+{\frac{16\,c{x}^{6}}{105}\sqrt{d{x}^{3}+c}}+{\frac{2\,{c}^{2}{x}^{3}}{105\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,{c}^{3}}{105\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{16\,c}{15\,d} \left ( d{x}^{3}+c \right ) ^{{\frac{5}{2}}}} \right ) }-64\,{\frac{{c}^{2}}{{d}^{2}} \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 = 1.38414, size = 429, normalized size = 3.94 \begin{align*} \left [\frac{2 \,{\left (90720 \, c^{\frac{7}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, d^{3}}, -\frac{2 \,{\left (181440 \, \sqrt{-c} c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) +{\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, d^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12501, size = 135, normalized size = 1.24 \begin{align*} -\frac{1152 \, c^{4} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{3}} - \frac{2 \,{\left (15 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} d^{18} + 147 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} c d^{18} + 2240 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c^{2} d^{18} + 60480 \, \sqrt{d x^{3} + c} c^{3} d^{18}\right )}}{315 \, d^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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