Optimal. Leaf size=171 \[ \frac {3 b^{11}}{2 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {33 b^{10}}{a^{12} \left (b+a \sqrt [3]{x}\right )}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}} \]
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Rubi [A]
time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 45}
\begin {gather*} \frac {3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac {33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}-\frac {165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx &=\int \frac {x^3}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \text {Subst}\left (\int \frac {x^{11}}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {45 b^8}{a^{11}}-\frac {36 b^7 x}{a^{10}}+\frac {28 b^6 x^2}{a^9}-\frac {21 b^5 x^3}{a^8}+\frac {15 b^4 x^4}{a^7}-\frac {10 b^3 x^5}{a^6}+\frac {6 b^2 x^6}{a^5}-\frac {3 b x^7}{a^4}+\frac {x^8}{a^3}-\frac {b^{11}}{a^{11} (b+a x)^3}+\frac {11 b^{10}}{a^{11} (b+a x)^2}-\frac {55 b^9}{a^{11} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^{11}}{2 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {33 b^{10}}{a^{12} \left (b+a \sqrt [3]{x}\right )}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 171, normalized size = 1.00 \begin {gather*} \frac {-5292 b^{11}+17136 a b^{10} \sqrt [3]{x}+36288 a^2 b^9 x^{2/3}+9240 a^3 b^8 x-2310 a^4 b^7 x^{4/3}+924 a^5 b^6 x^{5/3}-462 a^6 b^5 x^2+264 a^7 b^4 x^{7/3}-165 a^8 b^3 x^{8/3}+110 a^9 b^2 x^3-77 a^{10} b x^{10/3}+56 a^{11} x^{11/3}}{168 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 145, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {9 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {18 b^{2} x^{\frac {7}{3}} a^{6}}{7}-5 a^{5} b^{3} x^{2}+9 a^{4} x^{\frac {5}{3}} b^{4}-\frac {63 b^{5} x^{\frac {4}{3}} a^{3}}{4}+28 a^{2} b^{6} x -54 a \,b^{7} x^{\frac {2}{3}}+135 b^{8} x^{\frac {1}{3}}}{a^{11}}+\frac {3 b^{11}}{2 a^{12} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {33 b^{10}}{a^{12} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {165 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{12}}\) | \(145\) |
default | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {9 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {18 b^{2} x^{\frac {7}{3}} a^{6}}{7}-5 a^{5} b^{3} x^{2}+9 a^{4} x^{\frac {5}{3}} b^{4}-\frac {63 b^{5} x^{\frac {4}{3}} a^{3}}{4}+28 a^{2} b^{6} x -54 a \,b^{7} x^{\frac {2}{3}}+135 b^{8} x^{\frac {1}{3}}}{a^{11}}+\frac {3 b^{11}}{2 a^{12} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {33 b^{10}}{a^{12} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {165 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{12}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 167, normalized size = 0.98 \begin {gather*} \frac {56 \, a^{10} - \frac {77 \, a^{9} b}{x^{\frac {1}{3}}} + \frac {110 \, a^{8} b^{2}}{x^{\frac {2}{3}}} - \frac {165 \, a^{7} b^{3}}{x} + \frac {264 \, a^{6} b^{4}}{x^{\frac {4}{3}}} - \frac {462 \, a^{5} b^{5}}{x^{\frac {5}{3}}} + \frac {924 \, a^{4} b^{6}}{x^{2}} - \frac {2310 \, a^{3} b^{7}}{x^{\frac {7}{3}}} + \frac {9240 \, a^{2} b^{8}}{x^{\frac {8}{3}}} + \frac {41580 \, a b^{9}}{x^{3}} + \frac {27720 \, b^{10}}{x^{\frac {10}{3}}}}{168 \, {\left (\frac {a^{13}}{x^{3}} + \frac {2 \, a^{12} b}{x^{\frac {10}{3}}} + \frac {a^{11} b^{2}}{x^{\frac {11}{3}}}\right )}} - \frac {165 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\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 = 225, normalized size = 1.32 \begin {gather*} \frac {56 \, a^{15} x^{5} - 728 \, a^{12} b^{3} x^{4} + 3080 \, a^{9} b^{6} x^{3} + 8568 \, a^{6} b^{9} x^{2} - 1344 \, a^{3} b^{12} x - 5292 \, b^{15} - 27720 \, {\left (a^{6} b^{9} x^{2} + 2 \, a^{3} b^{12} x + b^{15}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (3 \, a^{14} b x^{4} - 18 \, a^{11} b^{4} x^{3} + 99 \, a^{8} b^{7} x^{2} + 352 \, a^{5} b^{10} x + 220 \, a^{2} b^{13}\right )} x^{\frac {2}{3}} + 18 \, {\left (24 \, a^{13} b^{2} x^{4} - 99 \, a^{10} b^{5} x^{3} + 990 \, a^{7} b^{8} x^{2} + 2695 \, a^{4} b^{11} x + 1540 \, a b^{14}\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{18} x^{2} + 2 \, a^{15} b^{3} x + a^{12} b^{6}\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. 624 vs.
\(2 (170) = 340\).
time = 1.14, size = 624, normalized size = 3.65 \begin {gather*} \begin {cases} \frac {56 a^{11} x^{\frac {11}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {77 a^{10} b x^{\frac {10}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {110 a^{9} b^{2} x^{3}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {165 a^{8} b^{3} x^{\frac {8}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {264 a^{7} b^{4} x^{\frac {7}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {462 a^{6} b^{5} x^{2}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {924 a^{5} b^{6} x^{\frac {5}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {2310 a^{4} b^{7} x^{\frac {4}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {9240 a^{3} b^{8} x}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 a^{2} b^{9} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 b^{11} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {41580 b^{11}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4 b^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 145, normalized size = 0.85 \begin {gather*} -\frac {165 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{12}} - \frac {3 \, {\left (22 \, a b^{10} x^{\frac {1}{3}} + 21 \, b^{11}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{12}} + \frac {56 \, a^{24} x^{3} - 189 \, a^{23} b x^{\frac {8}{3}} + 432 \, a^{22} b^{2} x^{\frac {7}{3}} - 840 \, a^{21} b^{3} x^{2} + 1512 \, a^{20} b^{4} x^{\frac {5}{3}} - 2646 \, a^{19} b^{5} x^{\frac {4}{3}} + 4704 \, a^{18} b^{6} x - 9072 \, a^{17} b^{7} x^{\frac {2}{3}} + 22680 \, a^{16} b^{8} x^{\frac {1}{3}}}{168 \, a^{27}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 154, normalized size = 0.90 \begin {gather*} \frac {x^3}{3\,a^3}-\frac {\frac {63\,b^{11}}{2\,a}+33\,b^{10}\,x^{1/3}}{a^{11}\,b^2+a^{13}\,x^{2/3}+2\,a^{12}\,b\,x^{1/3}}-\frac {9\,b\,x^{8/3}}{8\,a^4}+\frac {28\,b^6\,x}{a^9}-\frac {165\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{12}}-\frac {5\,b^3\,x^2}{a^6}+\frac {18\,b^2\,x^{7/3}}{7\,a^5}+\frac {9\,b^4\,x^{5/3}}{a^7}-\frac {63\,b^5\,x^{4/3}}{4\,a^8}-\frac {54\,b^7\,x^{2/3}}{a^{10}}+\frac {135\,b^8\,x^{1/3}}{a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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