Optimal. Leaf size=78 \[ -\frac {10 c \sqrt {c+d x^3}}{3 d^3}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {32 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {457, 90, 65,
209} \begin {gather*} \frac {32 c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^3}-\frac {10 c \sqrt {c+d x^3}}{3 d^3}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 209
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {5 c}{d^2 \sqrt {c+d x}}+\frac {\sqrt {c+d x}}{d^2}+\frac {16 c^2}{d^2 \sqrt {c+d x} (4 c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {10 c \sqrt {c+d x^3}}{3 d^3}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {\left (16 c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {10 c \sqrt {c+d x^3}}{3 d^3}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {\left (32 c^2\right ) \text {Subst}\left (\int \frac {1}{3 c+x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^3}\\ &=-\frac {10 c \sqrt {c+d x^3}}{3 d^3}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {32 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 0.83 \begin {gather*} \frac {2 \left (-14 c+d x^3\right ) \sqrt {c+d x^3}+32 \sqrt {3} c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.39, size = 464, normalized size = 5.95 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 53, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (16 \, \sqrt {3} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) + {\left (d x^{3} + c\right )}^{\frac {3}{2}} - 15 \, \sqrt {d x^{3} + c} c\right )}}{9 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 129, normalized size = 1.65 \begin {gather*} \left [\frac {2 \, {\left (8 \, \sqrt {3} \sqrt {-c} c \log \left (\frac {d x^{3} + 2 \, \sqrt {3} \sqrt {d x^{3} + c} \sqrt {-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + \sqrt {d x^{3} + c} {\left (d x^{3} - 14 \, c\right )}\right )}}{9 \, d^{3}}, \frac {2 \, {\left (16 \, \sqrt {3} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) + \sqrt {d x^{3} + c} {\left (d x^{3} - 14 \, c\right )}\right )}}{9 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\sqrt {c + d x^{3}} \cdot \left (4 c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 64, normalized size = 0.82 \begin {gather*} \frac {32 \, \sqrt {3} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right )}{9 \, d^{3}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{6} - 15 \, \sqrt {d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.38, size = 88, normalized size = 1.13 \begin {gather*} \frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d^2}-\frac {28\,c\,\sqrt {d\,x^3+c}}{9\,d^3}+\frac {\sqrt {3}\,c^{3/2}\,\ln \left (\frac {\sqrt {3}\,d\,x^3-2\,\sqrt {3}\,c+\sqrt {c}\,\sqrt {d\,x^3+c}\,6{}\mathrm {i}}{d\,x^3+4\,c}\right )\,16{}\mathrm {i}}{9\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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