Optimal. Leaf size=120 \[ -\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \left (a+b \sqrt [3]{x}\right )^{10} x \, dx &=3 \text {Subst}\left (\int x^5 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (-\frac {a^5 (a+b x)^{10}}{b^5}+\frac {5 a^4 (a+b x)^{11}}{b^5}-\frac {10 a^3 (a+b x)^{12}}{b^5}+\frac {10 a^2 (a+b x)^{13}}{b^5}-\frac {5 a (a+b x)^{14}}{b^5}+\frac {(a+b x)^{15}}{b^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 130, normalized size = 1.08 \begin {gather*} \frac {8008 a^{10} x^2+68640 a^9 b x^{7/3}+270270 a^8 b^2 x^{8/3}+640640 a^7 b^3 x^3+1009008 a^6 b^4 x^{10/3}+1100736 a^5 b^5 x^{11/3}+840840 a^4 b^6 x^4+443520 a^3 b^7 x^{13/3}+154440 a^2 b^8 x^{14/3}+32032 a b^9 x^5+3003 b^{10} x^{16/3}}{16016} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 113, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {3 b^{10} x^{\frac {16}{3}}}{16}+2 a \,b^{9} x^{5}+\frac {135 a^{2} b^{8} x^{\frac {14}{3}}}{14}+\frac {360 a^{3} b^{7} x^{\frac {13}{3}}}{13}+\frac {105 a^{4} b^{6} x^{4}}{2}+\frac {756 a^{5} b^{5} x^{\frac {11}{3}}}{11}+63 a^{6} b^{4} x^{\frac {10}{3}}+40 a^{7} b^{3} x^{3}+\frac {135 a^{8} b^{2} x^{\frac {8}{3}}}{8}+\frac {30 a^{9} b \,x^{\frac {7}{3}}}{7}+\frac {a^{10} x^{2}}{2}\) | \(113\) |
default | \(\frac {3 b^{10} x^{\frac {16}{3}}}{16}+2 a \,b^{9} x^{5}+\frac {135 a^{2} b^{8} x^{\frac {14}{3}}}{14}+\frac {360 a^{3} b^{7} x^{\frac {13}{3}}}{13}+\frac {105 a^{4} b^{6} x^{4}}{2}+\frac {756 a^{5} b^{5} x^{\frac {11}{3}}}{11}+63 a^{6} b^{4} x^{\frac {10}{3}}+40 a^{7} b^{3} x^{3}+\frac {135 a^{8} b^{2} x^{\frac {8}{3}}}{8}+\frac {30 a^{9} b \,x^{\frac {7}{3}}}{7}+\frac {a^{10} x^{2}}{2}\) | \(113\) |
trager | \(\frac {a \left (4 b^{9} x^{4}+105 a^{3} b^{6} x^{3}+4 b^{9} x^{3}+80 a^{6} b^{3} x^{2}+105 a^{3} b^{6} x^{2}+4 b^{9} x^{2}+a^{9} x +80 x \,a^{6} b^{3}+105 a^{3} b^{6} x +4 x \,b^{9}+a^{9}+80 a^{6} b^{3}+105 a^{3} b^{6}+4 b^{9}\right ) \left (x -1\right )}{2}+\frac {3 b \,x^{\frac {7}{3}} \left (91 b^{9} x^{3}+13440 a^{3} b^{6} x^{2}+30576 x \,a^{6} b^{3}+2080 a^{9}\right )}{1456}+\frac {27 a^{2} b^{2} x^{\frac {8}{3}} \left (220 b^{6} x^{2}+1568 a^{3} b^{3} x +385 a^{6}\right )}{616}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 98, normalized size = 0.82 \begin {gather*} \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16}}{16 \, b^{6}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{15} a}{b^{6}} + \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{2}}{7 \, b^{6}} - \frac {30 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{3}}{13 \, b^{6}} + \frac {5 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{4}}{4 \, b^{6}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{5}}{11 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 124, normalized size = 1.03 \begin {gather*} 2 \, a b^{9} x^{5} + \frac {105}{2} \, a^{4} b^{6} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac {1}{2} \, a^{10} x^{2} + \frac {27}{616} \, {\left (220 \, a^{2} b^{8} x^{4} + 1568 \, a^{5} b^{5} x^{3} + 385 \, a^{8} b^{2} x^{2}\right )} x^{\frac {2}{3}} + \frac {3}{1456} \, {\left (91 \, b^{10} x^{5} + 13440 \, a^{3} b^{7} x^{4} + 30576 \, a^{6} b^{4} x^{3} + 2080 \, a^{9} b x^{2}\right )} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.74, size = 143, normalized size = 1.19 \begin {gather*} \frac {a^{10} x^{2}}{2} + \frac {30 a^{9} b x^{\frac {7}{3}}}{7} + \frac {135 a^{8} b^{2} x^{\frac {8}{3}}}{8} + 40 a^{7} b^{3} x^{3} + 63 a^{6} b^{4} x^{\frac {10}{3}} + \frac {756 a^{5} b^{5} x^{\frac {11}{3}}}{11} + \frac {105 a^{4} b^{6} x^{4}}{2} + \frac {360 a^{3} b^{7} x^{\frac {13}{3}}}{13} + \frac {135 a^{2} b^{8} x^{\frac {14}{3}}}{14} + 2 a b^{9} x^{5} + \frac {3 b^{10} x^{\frac {16}{3}}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 112, normalized size = 0.93 \begin {gather*} \frac {3}{16} \, b^{10} x^{\frac {16}{3}} + 2 \, a b^{9} x^{5} + \frac {135}{14} \, a^{2} b^{8} x^{\frac {14}{3}} + \frac {360}{13} \, a^{3} b^{7} x^{\frac {13}{3}} + \frac {105}{2} \, a^{4} b^{6} x^{4} + \frac {756}{11} \, a^{5} b^{5} x^{\frac {11}{3}} + 63 \, a^{6} b^{4} x^{\frac {10}{3}} + 40 \, a^{7} b^{3} x^{3} + \frac {135}{8} \, a^{8} b^{2} x^{\frac {8}{3}} + \frac {30}{7} \, a^{9} b x^{\frac {7}{3}} + \frac {1}{2} \, a^{10} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 112, normalized size = 0.93 \begin {gather*} \frac {a^{10}\,x^2}{2}+\frac {3\,b^{10}\,x^{16/3}}{16}+2\,a\,b^9\,x^5+\frac {30\,a^9\,b\,x^{7/3}}{7}+40\,a^7\,b^3\,x^3+\frac {105\,a^4\,b^6\,x^4}{2}+\frac {135\,a^8\,b^2\,x^{8/3}}{8}+63\,a^6\,b^4\,x^{10/3}+\frac {756\,a^5\,b^5\,x^{11/3}}{11}+\frac {360\,a^3\,b^7\,x^{13/3}}{13}+\frac {135\,a^2\,b^8\,x^{14/3}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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