Optimal. Leaf size=111 \[ \frac {1024 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {1024 c^3 \sqrt {c+d x^3}}{3 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4} \]
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Rubi [A] time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {1024 c^3 \sqrt {c+d x^3}}{3 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}+\frac {1024 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{11} \sqrt {c+d x^3}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {57 c^2 \sqrt {c+d x}}{d^3}+\frac {512 c^3 \sqrt {c+d x}}{d^3 (8 c-d x)}-\frac {6 c (c+d x)^{3/2}}{d^3}-\frac {(c+d x)^{5/2}}{d^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4}+\frac {\left (512 c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{3 d^3}\\ &=-\frac {1024 c^3 \sqrt {c+d x^3}}{3 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4}+\frac {\left (1536 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^3}\\ &=-\frac {1024 c^3 \sqrt {c+d x^3}}{3 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4}+\frac {\left (3072 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^4}\\ &=-\frac {1024 c^3 \sqrt {c+d x^3}}{3 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac {4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^4}+\frac {1024 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 81, normalized size = 0.73 \[ \frac {107520 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (18632 c^3+764 c^2 d x^3+57 c d^2 x^6+5 d^3 x^9\right )}{105 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 169, normalized size = 1.52 \[ \left [\frac {2 \, {\left (26880 \, c^{\frac {7}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (5 \, d^{3} x^{9} + 57 \, c d^{2} x^{6} + 764 \, c^{2} d x^{3} + 18632 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{105 \, d^{4}}, -\frac {2 \, {\left (53760 \, \sqrt {-c} c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (5 \, d^{3} x^{9} + 57 \, c d^{2} x^{6} + 764 \, c^{2} d x^{3} + 18632 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{105 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 100, normalized size = 0.90 \[ -\frac {1024 \, c^{4} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{4}} - \frac {2 \, {\left (5 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} d^{24} + 42 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c d^{24} + 665 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} d^{24} + 17920 \, \sqrt {d x^{3} + c} c^{3} d^{24}\right )}}{105 \, d^{28}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 582, normalized size = 5.24 \[ -\frac {512 \left (\frac {2 \sqrt {d \,x^{3}+c}}{3 d}+\frac {i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) d}}\right )}{3 d^{3} \sqrt {d \,x^{3}+c}}\right ) c^{3}}{d^{3}}-\frac {8 \left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{6}}{15}+\frac {2 \sqrt {d \,x^{3}+c}\, c \,x^{3}}{45 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c^{2}}{45 d^{2}}\right ) c}{d^{2}}-\frac {\frac {2 \sqrt {d \,x^{3}+c}\, x^{9}}{21}+\frac {2 \sqrt {d \,x^{3}+c}\, c \,x^{6}}{105 d}-\frac {8 \sqrt {d \,x^{3}+c}\, c^{2} x^{3}}{315 d^{2}}+\frac {16 \sqrt {d \,x^{3}+c}\, c^{3}}{315 d^{3}}}{d}-\frac {128 \left (d \,x^{3}+c \right )^{\frac {3}{2}} c^{2}}{9 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 96, normalized size = 0.86 \[ -\frac {2 \, {\left (26880 \, c^{\frac {7}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 5 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} + 42 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c + 665 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} + 17920 \, \sqrt {d x^{3} + c} c^{3}\right )}}{105 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 118, normalized size = 1.06 \[ \frac {512\,c^{7/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^4}-\frac {37264\,c^3\,\sqrt {d\,x^3+c}}{105\,d^4}-\frac {2\,x^9\,\sqrt {d\,x^3+c}}{21\,d}-\frac {38\,c\,x^6\,\sqrt {d\,x^3+c}}{35\,d^2}-\frac {1528\,c^2\,x^3\,\sqrt {d\,x^3+c}}{105\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.27, size = 99, normalized size = 0.89 \[ \frac {2 \left (- \frac {512 c^{4} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {512 c^{3} \sqrt {c + d x^{3}}}{3} - \frac {19 c^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x^{3}\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x^{3}\right )^{\frac {7}{2}}}{21}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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