Optimal. Leaf size=124 \[ \frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{7/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{7/2}}+\frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {446, 103, 151, 156, 63, 208, 206} \[ \frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{7/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{7/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 151
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {2 c d-\frac {3 d^2 x}{2}}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{24 c^2}\\ &=\frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {-18 c^2 d^2+5 c d^3 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{1728 c^4 d}\\ &=\frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac {d \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{768 c^3}+\frac {\left (11 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6912 c^3}\\ &=\frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{384 c^3}+\frac {(11 d) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3456 c^3}\\ &=\frac {5 d \sqrt {c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{7/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 97, normalized size = 0.78 \[ \frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+27 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )+\frac {12 \sqrt {c} \sqrt {c+d x^3} \left (36 c-5 d x^3\right )}{d x^6-8 c x^3}}{10368 c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 280, normalized size = 2.26 \[ \left [\frac {11 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 27 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 24 \, {\left (5 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt {d x^{3} + c}}{20736 \, {\left (c^{4} d x^{6} - 8 \, c^{5} x^{3}\right )}}, -\frac {27 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + 11 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (5 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt {d x^{3} + c}}{10368 \, {\left (c^{4} d x^{6} - 8 \, c^{5} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 114, normalized size = 0.92 \[ -\frac {d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{384 \, \sqrt {-c} c^{3}} - \frac {11 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{10368 \, \sqrt {-c} c^{3}} - \frac {5 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d - 41 \, \sqrt {d x^{3} + c} c d}{864 \, {\left ({\left (d x^{3} + c\right )}^{2} - 10 \, {\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 926, normalized size = 7.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.11, size = 117, normalized size = 0.94 \[ \frac {\frac {41\,d\,\sqrt {d\,x^3+c}}{288\,c^2}-\frac {5\,d\,{\left (d\,x^3+c\right )}^{3/2}}{288\,c^3}}{3\,{\left (d\,x^3+c\right )}^2-30\,c\,\left (d\,x^3+c\right )+27\,c^2}-\frac {d\,\left (\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{\sqrt {c^7}}\right )\,1{}\mathrm {i}+\frac {\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^7}}\right )\,11{}\mathrm {i}}{27}\right )\,1{}\mathrm {i}}{384\,\sqrt {c^7}} \]
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 \[ \int \frac {1}{x^{4} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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