Optimal. Leaf size=164 \[ -\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {457, 105, 156,
162, 65, 214, 212} \begin {gather*} \frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 162
Rule 212
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}-\frac {\text {Subst}\left (\int \frac {9 c d-\frac {5 d^2 x}{2}}{x^2 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {\text {Subst}\left (\int \frac {38 c^2 d^2-\frac {27}{2} c d^3 x}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{384 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac {\text {Subst}\left (\int \frac {-342 c^3 d^3+35 c^2 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27648 c^6 d}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {\left (19 d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{12288 c^4}+\frac {\left (31 d^3\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{110592 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {(19 d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{6144 c^4}+\frac {\left (31 d^2\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{55296 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 112, normalized size = 0.68 \begin {gather*} \frac {\frac {12 \sqrt {c} \sqrt {c+d x^3} \left (288 c^2-324 c d x^3+35 d^2 x^6\right )}{-8 c x^6+d x^9}+31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-513 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{165888 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 = 0.51, size = 990, normalized size = 6.04
method | result | size |
risch | \(\text {Expression too large to display}\) | \(912\) |
default | \(\text {Expression too large to display}\) | \(990\) |
elliptic | \(\text {Expression too large to display}\) | \(1580\) |
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 = 3.97, size = 310, normalized size = 1.89 \begin {gather*} \left [\frac {31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\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 ) + 513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\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} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{331776 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}, \frac {513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{165888 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\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^{7} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 128, normalized size = 0.78 \begin {gather*} \frac {19 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{6144 \, \sqrt {-c} c^{4}} - \frac {31 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{165888 \, \sqrt {-c} c^{4}} - \frac {\sqrt {d x^{3} + c} d^{2}}{13824 \, {\left (d x^{3} - 8 \, c\right )} c^{4}} + \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 2 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{4} d^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.37, size = 155, normalized size = 0.95 \begin {gather*} -\frac {\frac {647\,d^2\,\sqrt {d\,x^3+c}}{4608\,c^2}-\frac {197\,d^2\,{\left (d\,x^3+c\right )}^{3/2}}{2304\,c^3}+\frac {35\,d^2\,{\left (d\,x^3+c\right )}^{5/2}}{4608\,c^4}}{33\,c\,{\left (d\,x^3+c\right )}^2-57\,c^2\,\left (d\,x^3+c\right )-3\,{\left (d\,x^3+c\right )}^3+27\,c^3}+\frac {d^2\,\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 )\,31{}\mathrm {i}}{513}\right )\,19{}\mathrm {i}}{6144\,\sqrt {c^9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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