Optimal. Leaf size=143 \[ -\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {457, 105, 156,
157, 162, 65, 214, 212} \begin {gather*} \frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}}-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 157
Rule 162
Rule 212
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\text {Subst}\left (\int \frac {10 c d-\frac {5 d^2 x}{2}}{x (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{24 c^2}\\ &=\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {-90 c^2 d^2+15 c d^3 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{1728 c^4 d}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {-405 c^3 d^3+\frac {105}{2} c^2 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{7776 c^6 d^2}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {(5 d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{768 c^4}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{20736 c^4}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {5 \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{384 c^4}+\frac {(5 d) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{10368 c^4}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 109, normalized size = 0.76 \begin {gather*} \frac {-\frac {12 \sqrt {c} \left (108 c^2+265 c d x^3-35 d^2 x^6\right )}{x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+405 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{31104 c^{9/2}} \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 = 0.50, size = 1020, normalized size = 7.13
method | result | size |
risch | \(\text {Expression too large to display}\) | \(913\) |
default | \(\text {Expression too large to display}\) | \(1020\) |
elliptic | \(\text {Expression too large to display}\) | \(1569\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.33, size = 368, normalized size = 2.57 \begin {gather*} \left [\frac {5 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} 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 ) + 405 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} 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 (35 \, c d^{2} x^{6} - 265 \, c^{2} d x^{3} - 108 \, c^{3}\right )} \sqrt {d x^{3} + c}}{62208 \, {\left (c^{5} d^{2} x^{9} - 7 \, c^{6} d x^{6} - 8 \, c^{7} x^{3}\right )}}, -\frac {405 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + 5 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (35 \, c d^{2} x^{6} - 265 \, c^{2} d x^{3} - 108 \, c^{3}\right )} \sqrt {d x^{3} + c}}{31104 \, {\left (c^{5} d^{2} x^{9} - 7 \, c^{6} d x^{6} - 8 \, c^{7} x^{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 {1}{x^{4} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 129, normalized size = 0.90 \begin {gather*} -\frac {5 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{384 \, \sqrt {-c} c^{4}} - \frac {5 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{31104 \, \sqrt {-c} c^{4}} - \frac {35 \, {\left (d x^{3} + c\right )}^{2} d - 335 \, {\left (d x^{3} + c\right )} c d + 192 \, c^{2} d}{2592 \, {\left ({\left (d x^{3} + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 9 \, \sqrt {d x^{3} + c} c^{2}\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.56, size = 133, normalized size = 0.93 \begin {gather*} -\frac {\frac {2\,d}{9\,c^2}+\frac {35\,d\,{\left (d\,x^3+c\right )}^2}{864\,c^4}-\frac {335\,d\,\left (d\,x^3+c\right )}{864\,c^3}}{3\,{\left (d\,x^3+c\right )}^{5/2}-30\,c\,{\left (d\,x^3+c\right )}^{3/2}+27\,c^2\,\sqrt {d\,x^3+c}}-\frac {d\,\left (\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )\,1{}\mathrm {i}+\frac {\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )\,1{}\mathrm {i}}{81}\right )\,5{}\mathrm {i}}{384\,\sqrt {c^9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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