Optimal. Leaf size=117 \[ \frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {2 c \sqrt {c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}-\frac {3968 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {457, 99, 158,
152, 65, 212} \begin {gather*} -\frac {3968 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^4}+\frac {2 c \sqrt {c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}+\frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{11} \sqrt {c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3 \sqrt {c+d x}}{(8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 c+\frac {7 d x}{2}\right )}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 d}\\ &=\frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {2 \text {Subst}\left (\int \frac {x \left (-56 c^2 d-\frac {141}{2} c d^2 x\right )}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{15 d^3}\\ &=\frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {2 c \sqrt {c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}-\frac {\left (1984 c^3\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 d^3}\\ &=\frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {2 c \sqrt {c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}-\frac {\left (3968 c^3\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^4}\\ &=\frac {7 x^6 \sqrt {c+d x^3}}{15 d^2}+\frac {x^9 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {2 c \sqrt {c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}-\frac {3968 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 91, normalized size = 0.78 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {c+d x^3} \left (-9168 c^3+770 c^2 d x^3+19 c d^2 x^6+d^3 x^9\right )}{-8 c+d x^3}-9920 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{45 d^4} \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 = 0.42, size = 953, normalized size = 8.15
method | result | size |
elliptic | \(\frac {512 c^{3} \sqrt {d \,x^{3}+c}}{3 d^{4} \left (-d \,x^{3}+8 c \right )}+\frac {2 x^{6} \sqrt {d \,x^{3}+c}}{15 d^{2}}+\frac {18 c \,x^{3} \sqrt {d \,x^{3}+c}}{5 d^{3}}+\frac {1972 c^{2} \sqrt {d \,x^{3}+c}}{15 d^{4}}+\frac {1984 i c^{2} \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \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 )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \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 )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\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}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \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 )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{27 d^{6}}\) | \(495\) |
risch | \(\text {Expression too large to display}\) | \(901\) |
default | \(\text {Expression too large to display}\) | \(953\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 107, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (4960 \, c^{\frac {5}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} + 75 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 2880 \, \sqrt {d x^{3} + c} c^{2} - \frac {3840 \, \sqrt {d x^{3} + c} c^{3}}{d x^{3} - 8 \, c}\right )}}{45 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.11, size = 219, normalized size = 1.87 \begin {gather*} \left [\frac {2 \, {\left (4960 \, {\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \, {\left (d^{3} x^{9} + 19 \, c d^{2} x^{6} + 770 \, c^{2} d x^{3} - 9168 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, {\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}, \frac {2 \, {\left (9920 \, {\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, {\left (d^{3} x^{9} + 19 \, c d^{2} x^{6} + 770 \, c^{2} d x^{3} - 9168 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, {\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}\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^{11} \sqrt {c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 110, normalized size = 0.94 \begin {gather*} \frac {3968 \, c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{9 \, \sqrt {-c} d^{4}} - \frac {512 \, \sqrt {d x^{3} + c} c^{3}}{3 \, {\left (d x^{3} - 8 \, c\right )} d^{4}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{16} + 25 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{16} + 960 \, \sqrt {d x^{3} + c} c^{2} d^{16}\right )}}{15 \, d^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.09, size = 127, normalized size = 1.09 \begin {gather*} \frac {1984\,c^{5/2}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{9\,d^4}+\frac {1972\,c^2\,\sqrt {d\,x^3+c}}{15\,d^4}+\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15\,d^2}+\frac {18\,c\,x^3\,\sqrt {d\,x^3+c}}{5\,d^3}+\frac {512\,c^3\,\sqrt {d\,x^3+c}}{3\,d^4\,\left (8\,c-d\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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