Optimal. Leaf size=119 \[ \frac {160 c^2 \sqrt {c+d x^3}}{d^3}+\frac {160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {480 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {457, 91, 81, 52,
65, 212} \begin {gather*} -\frac {480 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}+\frac {160 c^2 \sqrt {c+d x^3}}{d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac {160 c \left (c+d x^3\right )^{3/2}}{27 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 81
Rule 91
Rule 212
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 (c+d x)^{3/2}}{(8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\text {Subst}\left (\int \frac {(c+d x)^{3/2} \left (168 c^2 d+9 c d^2 x\right )}{8 c-d x} \, dx,x,x^3\right )}{27 c d^3}\\ &=\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {(80 c) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{9 d^2}\\ &=\frac {160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\left (80 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d^2}\\ &=\frac {160 c^2 \sqrt {c+d x^3}}{d^3}+\frac {160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\left (720 c^3\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^2}\\ &=\frac {160 c^2 \sqrt {c+d x^3}}{d^3}+\frac {160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\left (1440 c^3\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^3}\\ &=\frac {160 c^2 \sqrt {c+d x^3}}{d^3}+\frac {160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}-\frac {480 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 93, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {c+d x^3} \left (-29944 c^3+2515 c^2 d x^3+62 c d^2 x^6+3 d^3 x^9\right )}{45 d^3 \left (-8 c+d x^3\right )}-\frac {480 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{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 = 0.41, size = 921, normalized size = 7.74
method | result | size |
elliptic | \(\frac {192 c^{3} \sqrt {d \,x^{3}+c}}{d^{3} \left (-d \,x^{3}+8 c \right )}+\frac {2 x^{6} \sqrt {d \,x^{3}+c}}{15 d}+\frac {172 c \,x^{3} \sqrt {d \,x^{3}+c}}{45 d^{2}}+\frac {6406 c^{2} \sqrt {d \,x^{3}+c}}{45 d^{3}}+\frac {80 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 )}{d^{5}}\) | \(495\) |
risch | \(\text {Expression too large to display}\) | \(902\) |
default | \(\text {Expression too large to display}\) | \(921\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 107, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (5400 \, 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}} + 80 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 3120 \, \sqrt {d x^{3} + c} c^{2} - \frac {4320 \, \sqrt {d x^{3} + c} c^{3}}{d x^{3} - 8 \, c}\right )}}{45 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 219, normalized size = 1.84 \begin {gather*} \left [\frac {2 \, {\left (5400 \, {\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 ) + {\left (3 \, d^{3} x^{9} + 62 \, c d^{2} x^{6} + 2515 \, c^{2} d x^{3} - 29944 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, \frac {2 \, {\left (10800 \, {\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 ) + {\left (3 \, d^{3} x^{9} + 62 \, c d^{2} x^{6} + 2515 \, c^{2} d x^{3} - 29944 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\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^{8} \left (c + d x^{3}\right )^{\frac {3}{2}}}{\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 = 0.61, size = 111, normalized size = 0.93 \begin {gather*} \frac {480 \, c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{3}} - \frac {192 \, \sqrt {d x^{3} + c} c^{3}}{{\left (d x^{3} - 8 \, c\right )} d^{3}} + \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{12} + 80 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{12} + 3120 \, \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 = 4.05, size = 127, normalized size = 1.07 \begin {gather*} \frac {240\,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 )}{d^3}+\frac {6406\,c^2\,\sqrt {d\,x^3+c}}{45\,d^3}+\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15\,d}+\frac {172\,c\,x^3\,\sqrt {d\,x^3+c}}{45\,d^2}+\frac {192\,c^3\,\sqrt {d\,x^3+c}}{d^3\,\left (8\,c-d\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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