Optimal. Leaf size=71 \[ \frac{128 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]
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Rubi [A] time = 0.0689688, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {446, 88, 63, 206} \[ \frac{128 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^8}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{7 c}{d^2 \sqrt{c+d x}}+\frac{64 c^2}{d^2 (8 c-d x) \sqrt{c+d x}}-\frac{\sqrt{c+d x}}{d^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{\left (64 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{\left (128 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{3 d^3}\\ &=-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{128 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}\\ \end{align*}
Mathematica [A] time = 0.0493628, size = 58, normalized size = 0.82 \[ \frac{128 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (22 c+d x^3\right )}{9 d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 468, normalized size = 6.6 \begin{align*} -{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,{x}^{3}}{9\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,c}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{16\,c}{3\,d}\sqrt{d{x}^{3}+c}} \right ) }-{\frac{{\frac{64\,i}{27}}c\sqrt{2}}{{d}^{5}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,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}{18\,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 = 1.31304, size = 296, normalized size = 4.17 \begin{align*} \left [\frac{2 \,{\left (32 \, 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} + 22 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{3}}, -\frac{2 \,{\left (64 \, \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) +{\left (d x^{3} + 22 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{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{x^{8}}{- 8 c \sqrt{c + d x^{3}} + d x^{3} \sqrt{c + d x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10726, size = 88, normalized size = 1.24 \begin{align*} -\frac{128 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d^{3}} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{6} + 21 \, \sqrt{d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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