Optimal. Leaf size=97 \[ \frac {32 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {32 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {457, 90, 52, 65,
209} \begin {gather*} -\frac {32 c^{5/2} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^3}+\frac {32 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 209
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt {c+d x^3}}{4 c+d x^3} \, 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 \sqrt {c+d x}}{d^2}+\frac {(c+d x)^{3/2}}{d^2}+\frac {16 c^2 \sqrt {c+d x}}{d^2 (4 c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {\left (16 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{4 c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac {32 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {\left (16 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )}{d^2}\\ &=\frac {32 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {\left (32 c^3\right ) \text {Subst}\left (\int \frac {1}{3 c+x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^3}\\ &=\frac {32 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {32 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 78, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {c+d x^3} \left (218 c^2-19 c d x^3+3 d^2 x^6\right )}{45 d^3}-\frac {32 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} 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 = 1.22, size = 503, normalized size = 5.19 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 69, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (240 \, \sqrt {3} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - 3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} + 25 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c - 240 \, \sqrt {d x^{3} + c} c^{2}\right )}}{45 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.24, size = 156, normalized size = 1.61 \begin {gather*} \left [\frac {2 \, {\left (120 \, \sqrt {3} \sqrt {-c} c^{2} \log \left (\frac {d x^{3} - 2 \, \sqrt {3} \sqrt {d x^{3} + c} \sqrt {-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + {\left (3 \, d^{2} x^{6} - 19 \, c d x^{3} + 218 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}, -\frac {2 \, {\left (240 \, \sqrt {3} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - {\left (3 \, d^{2} x^{6} - 19 \, c d x^{3} + 218 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 11.85, size = 85, normalized size = 0.88 \begin {gather*} \frac {2 \left (- \frac {16 \sqrt {3} c^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {c + d x^{3}}}{3 \sqrt {c}} \right )}}{3} + \frac {16 c^{2} \sqrt {c + d x^{3}}}{3} - \frac {5 c \left (c + d x^{3}\right )^{\frac {3}{2}}}{9} + \frac {\left (c + d x^{3}\right )^{\frac {5}{2}}}{15}\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 82, normalized size = 0.85 \begin {gather*} -\frac {32 \, \sqrt {3} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right )}{3 \, d^{3}} + \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{12} - 25 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{12} + 240 \, \sqrt {d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.54, size = 109, normalized size = 1.12 \begin {gather*} \frac {436\,c^2\,\sqrt {d\,x^3+c}}{45\,d^3}+\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15\,d}-\frac {38\,c\,x^3\,\sqrt {d\,x^3+c}}{45\,d^2}+\frac {\sqrt {3}\,c^{5/2}\,\ln \left (\frac {2\,\sqrt {3}\,c-\sqrt {3}\,d\,x^3+\sqrt {c}\,\sqrt {d\,x^3+c}\,6{}\mathrm {i}}{d\,x^3+4\,c}\right )\,16{}\mathrm {i}}{3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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