Optimal. Leaf size=183 \[ -\frac {3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}-\frac {30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^3 x^3}+\frac {9 b}{8 a^4 x^{8/3}}-\frac {18 b^2}{7 a^5 x^{7/3}}+\frac {5 b^3}{a^6 x^2}-\frac {9 b^4}{a^7 x^{5/3}}+\frac {63 b^5}{4 a^8 x^{4/3}}-\frac {28 b^6}{a^9 x}+\frac {54 b^7}{a^{10} x^{2/3}}-\frac {135 b^8}{a^{11} \sqrt [3]{x}}+\frac {165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac {55 b^9 \log (x)}{a^{12}} \]
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Rubi [A]
time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46}
\begin {gather*} \frac {165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac {55 b^9 \log (x)}{a^{12}}-\frac {30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac {135 b^8}{a^{11} \sqrt [3]{x}}-\frac {3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}+\frac {54 b^7}{a^{10} x^{2/3}}-\frac {28 b^6}{a^9 x}+\frac {63 b^5}{4 a^8 x^{4/3}}-\frac {9 b^4}{a^7 x^{5/3}}+\frac {5 b^3}{a^6 x^2}-\frac {18 b^2}{7 a^5 x^{7/3}}+\frac {9 b}{8 a^4 x^{8/3}}-\frac {1}{3 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3 x^4} \, dx &=3 \text {Subst}\left (\int \frac {1}{x^{10} (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {1}{a^3 x^{10}}-\frac {3 b}{a^4 x^9}+\frac {6 b^2}{a^5 x^8}-\frac {10 b^3}{a^6 x^7}+\frac {15 b^4}{a^7 x^6}-\frac {21 b^5}{a^8 x^5}+\frac {28 b^6}{a^9 x^4}-\frac {36 b^7}{a^{10} x^3}+\frac {45 b^8}{a^{11} x^2}-\frac {55 b^9}{a^{12} x}+\frac {b^{10}}{a^{10} (a+b x)^3}+\frac {10 b^{10}}{a^{11} (a+b x)^2}+\frac {55 b^{10}}{a^{12} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}-\frac {30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^3 x^3}+\frac {9 b}{8 a^4 x^{8/3}}-\frac {18 b^2}{7 a^5 x^{7/3}}+\frac {5 b^3}{a^6 x^2}-\frac {9 b^4}{a^7 x^{5/3}}+\frac {63 b^5}{4 a^8 x^{4/3}}-\frac {28 b^6}{a^9 x}+\frac {54 b^7}{a^{10} x^{2/3}}-\frac {135 b^8}{a^{11} \sqrt [3]{x}}+\frac {165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac {55 b^9 \log (x)}{a^{12}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 167, normalized size = 0.91 \begin {gather*} -\frac {\frac {a \left (56 a^{10}-77 a^9 b \sqrt [3]{x}+110 a^8 b^2 x^{2/3}-165 a^7 b^3 x+264 a^6 b^4 x^{4/3}-462 a^5 b^5 x^{5/3}+924 a^4 b^6 x^2-2310 a^3 b^7 x^{7/3}+9240 a^2 b^8 x^{8/3}+41580 a b^9 x^3+27720 b^{10} x^{10/3}\right )}{\left (a+b \sqrt [3]{x}\right )^2 x^3}-27720 b^9 \log \left (a+b \sqrt [3]{x}\right )+9240 b^9 \log (x)}{168 a^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 156, normalized size = 0.85
method | result | size |
derivativedivides | \(-\frac {3 b^{9}}{2 a^{10} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}-\frac {30 b^{9}}{a^{11} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{3} x^{3}}+\frac {9 b}{8 a^{4} x^{\frac {8}{3}}}-\frac {18 b^{2}}{7 a^{5} x^{\frac {7}{3}}}+\frac {5 b^{3}}{a^{6} x^{2}}-\frac {9 b^{4}}{a^{7} x^{\frac {5}{3}}}+\frac {63 b^{5}}{4 a^{8} x^{\frac {4}{3}}}-\frac {28 b^{6}}{a^{9} x}+\frac {54 b^{7}}{a^{10} x^{\frac {2}{3}}}-\frac {135 b^{8}}{a^{11} x^{\frac {1}{3}}}+\frac {165 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{12}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}\) | \(156\) |
default | \(-\frac {3 b^{9}}{2 a^{10} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}-\frac {30 b^{9}}{a^{11} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{3} x^{3}}+\frac {9 b}{8 a^{4} x^{\frac {8}{3}}}-\frac {18 b^{2}}{7 a^{5} x^{\frac {7}{3}}}+\frac {5 b^{3}}{a^{6} x^{2}}-\frac {9 b^{4}}{a^{7} x^{\frac {5}{3}}}+\frac {63 b^{5}}{4 a^{8} x^{\frac {4}{3}}}-\frac {28 b^{6}}{a^{9} x}+\frac {54 b^{7}}{a^{10} x^{\frac {2}{3}}}-\frac {135 b^{8}}{a^{11} x^{\frac {1}{3}}}+\frac {165 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{12}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 165, normalized size = 0.90 \begin {gather*} -\frac {27720 \, b^{10} x^{\frac {10}{3}} + 41580 \, a b^{9} x^{3} + 9240 \, a^{2} b^{8} x^{\frac {8}{3}} - 2310 \, a^{3} b^{7} x^{\frac {7}{3}} + 924 \, a^{4} b^{6} x^{2} - 462 \, a^{5} b^{5} x^{\frac {5}{3}} + 264 \, a^{6} b^{4} x^{\frac {4}{3}} - 165 \, a^{7} b^{3} x + 110 \, a^{8} b^{2} x^{\frac {2}{3}} - 77 \, a^{9} b x^{\frac {1}{3}} + 56 \, a^{10}}{168 \, {\left (a^{11} b^{2} x^{\frac {11}{3}} + 2 \, a^{12} b x^{\frac {10}{3}} + a^{13} x^{3}\right )}} + \frac {165 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{12}} - \frac {55 \, b^{9} \log \left (x\right )}{a^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 263, normalized size = 1.44 \begin {gather*} -\frac {9240 \, a^{3} b^{12} x^{4} + 13860 \, a^{6} b^{9} x^{3} + 3080 \, a^{9} b^{6} x^{2} - 728 \, a^{12} b^{3} x + 56 \, a^{15} - 27720 \, {\left (b^{15} x^{5} + 2 \, a^{3} b^{12} x^{4} + a^{6} b^{9} x^{3}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 27720 \, {\left (b^{15} x^{5} + 2 \, a^{3} b^{12} x^{4} + a^{6} b^{9} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, {\left (1540 \, a b^{14} x^{4} + 2695 \, a^{4} b^{11} x^{3} + 990 \, a^{7} b^{8} x^{2} - 99 \, a^{10} b^{5} x + 24 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} - 63 \, {\left (220 \, a^{2} b^{13} x^{4} + 352 \, a^{5} b^{10} x^{3} + 99 \, a^{8} b^{7} x^{2} - 18 \, a^{11} b^{4} x + 3 \, a^{14} b\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{12} b^{6} x^{5} + 2 \, a^{15} b^{3} x^{4} + a^{18} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs.
\(2 (184) = 368\).
time = 7.35, size = 847, normalized size = 4.63 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{4}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 a^{3} x^{3}} & \text {for}\: b = 0 \\- \frac {1}{4 b^{3} x^{4}} & \text {for}\: a = 0 \\- \frac {56 a^{11} x^{\frac {2}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {77 a^{10} b x}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {110 a^{9} b^{2} x^{\frac {4}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {165 a^{8} b^{3} x^{\frac {5}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {264 a^{7} b^{4} x^{2}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {462 a^{6} b^{5} x^{\frac {7}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {924 a^{5} b^{6} x^{\frac {8}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {2310 a^{4} b^{7} x^{3}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {9240 a^{3} b^{8} x^{\frac {10}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {9240 a^{2} b^{9} x^{\frac {11}{3}} \log {\left (x \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {27720 a^{2} b^{9} x^{\frac {11}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {41580 a^{2} b^{9} x^{\frac {11}{3}}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {18480 a b^{10} x^{4} \log {\left (x \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {55440 a b^{10} x^{4} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {27720 a b^{10} x^{4}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} - \frac {9240 b^{11} x^{\frac {13}{3}} \log {\left (x \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} + \frac {27720 b^{11} x^{\frac {13}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{14} x^{\frac {11}{3}} + 336 a^{13} b x^{4} + 168 a^{12} b^{2} x^{\frac {13}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 156, normalized size = 0.85 \begin {gather*} \frac {165 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{12}} - \frac {55 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{12}} - \frac {27720 \, a b^{10} x^{\frac {10}{3}} + 41580 \, a^{2} b^{9} x^{3} + 9240 \, a^{3} b^{8} x^{\frac {8}{3}} - 2310 \, a^{4} b^{7} x^{\frac {7}{3}} + 924 \, a^{5} b^{6} x^{2} - 462 \, a^{6} b^{5} x^{\frac {5}{3}} + 264 \, a^{7} b^{4} x^{\frac {4}{3}} - 165 \, a^{8} b^{3} x + 110 \, a^{9} b^{2} x^{\frac {2}{3}} - 77 \, a^{10} b x^{\frac {1}{3}} + 56 \, a^{11}}{168 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{12} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.29, size = 159, normalized size = 0.87 \begin {gather*} \frac {330\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^{12}}-\frac {\frac {1}{3\,a}-\frac {11\,b\,x^{1/3}}{24\,a^2}-\frac {55\,b^3\,x}{56\,a^4}+\frac {55\,b^2\,x^{2/3}}{84\,a^3}+\frac {11\,b^6\,x^2}{2\,a^7}+\frac {11\,b^4\,x^{4/3}}{7\,a^5}-\frac {11\,b^5\,x^{5/3}}{4\,a^6}+\frac {495\,b^9\,x^3}{2\,a^{10}}-\frac {55\,b^7\,x^{7/3}}{4\,a^8}+\frac {55\,b^8\,x^{8/3}}{a^9}+\frac {165\,b^{10}\,x^{10/3}}{a^{11}}}{a^2\,x^3+b^2\,x^{11/3}+2\,a\,b\,x^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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