Optimal. Leaf size=122 \[ -\frac {3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac {24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}+\frac {21 a^6 \sqrt [3]{x}}{b^8}-\frac {9 a^5 x^{2/3}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {3 a^3 x^{4/3}}{b^5}+\frac {9 a^2 x^{5/3}}{5 b^4}-\frac {a x^2}{b^3}+\frac {3 x^{7/3}}{7 b^2} \]
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Rubi [A] time = 0.09, antiderivative size = 122, 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 = {266, 43} \[ -\frac {9 a^5 x^{2/3}}{b^7}-\frac {3 a^3 x^{4/3}}{b^5}+\frac {9 a^2 x^{5/3}}{5 b^4}-\frac {3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac {21 a^6 \sqrt [3]{x}}{b^8}+\frac {5 a^4 x}{b^6}-\frac {24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac {a x^2}{b^3}+\frac {3 x^{7/3}}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^2} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^8}{(a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {7 a^6}{b^8}-\frac {6 a^5 x}{b^7}+\frac {5 a^4 x^2}{b^6}-\frac {4 a^3 x^3}{b^5}+\frac {3 a^2 x^4}{b^4}-\frac {2 a x^5}{b^3}+\frac {x^6}{b^2}+\frac {a^8}{b^8 (a+b x)^2}-\frac {8 a^7}{b^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac {21 a^6 \sqrt [3]{x}}{b^8}-\frac {9 a^5 x^{2/3}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {3 a^3 x^{4/3}}{b^5}+\frac {9 a^2 x^{5/3}}{5 b^4}-\frac {a x^2}{b^3}+\frac {3 x^{7/3}}{7 b^2}-\frac {24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 122, normalized size = 1.00 \[ -\frac {3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac {24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}+\frac {21 a^6 \sqrt [3]{x}}{b^8}-\frac {9 a^5 x^{2/3}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {3 a^3 x^{4/3}}{b^5}+\frac {9 a^2 x^{5/3}}{5 b^4}-\frac {a x^2}{b^3}+\frac {3 x^{7/3}}{7 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 147, normalized size = 1.20 \[ -\frac {35 \, a b^{9} x^{3} - 140 \, a^{4} b^{6} x^{2} - 175 \, a^{7} b^{3} x + 105 \, a^{10} + 840 \, {\left (a^{7} b^{3} x + a^{10}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 21 \, {\left (3 \, a^{2} b^{8} x^{2} - 12 \, a^{5} b^{5} x - 20 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} - 15 \, {\left (b^{10} x^{3} - 6 \, a^{3} b^{7} x^{2} + 42 \, a^{6} b^{4} x + 56 \, a^{9} b\right )} x^{\frac {1}{3}}}{35 \, {\left (b^{12} x + a^{3} b^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 111, normalized size = 0.91 \[ -\frac {24 \, a^{7} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{9}} - \frac {3 \, a^{8}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{9}} + \frac {15 \, b^{12} x^{\frac {7}{3}} - 35 \, a b^{11} x^{2} + 63 \, a^{2} b^{10} x^{\frac {5}{3}} - 105 \, a^{3} b^{9} x^{\frac {4}{3}} + 175 \, a^{4} b^{8} x - 315 \, a^{5} b^{7} x^{\frac {2}{3}} + 735 \, a^{6} b^{6} x^{\frac {1}{3}}}{35 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 0.86 \[ \frac {3 x^{\frac {7}{3}}}{7 b^{2}}-\frac {a \,x^{2}}{b^{3}}+\frac {9 a^{2} x^{\frac {5}{3}}}{5 b^{4}}-\frac {3 a^{3} x^{\frac {4}{3}}}{b^{5}}-\frac {3 a^{8}}{\left (b \,x^{\frac {1}{3}}+a \right ) b^{9}}-\frac {24 a^{7} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{9}}+\frac {5 a^{4} x}{b^{6}}-\frac {9 a^{5} x^{\frac {2}{3}}}{b^{7}}+\frac {21 a^{6} x^{\frac {1}{3}}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 146, normalized size = 1.20 \[ -\frac {24 \, a^{7} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{9}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7}}{7 \, b^{9}} - \frac {4 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a}{b^{9}} + \frac {84 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{2}}{5 \, b^{9}} - \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{3}}{b^{9}} + \frac {70 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{4}}{b^{9}} - \frac {84 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{5}}{b^{9}} + \frac {84 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{6}}{b^{9}} - \frac {3 \, a^{8}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 110, normalized size = 0.90 \[ \frac {3\,x^{7/3}}{7\,b^2}-\frac {3\,a^8}{b\,\left (a\,b^8+b^9\,x^{1/3}\right )}-\frac {a\,x^2}{b^3}+\frac {5\,a^4\,x}{b^6}-\frac {24\,a^7\,\ln \left (a+b\,x^{1/3}\right )}{b^9}+\frac {9\,a^2\,x^{5/3}}{5\,b^4}-\frac {3\,a^3\,x^{4/3}}{b^5}-\frac {9\,a^5\,x^{2/3}}{b^7}+\frac {21\,a^6\,x^{1/3}}{b^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 56.23, size = 343, normalized size = 2.81 \[ - \frac {840 a^{8} x^{\frac {176}{3}} \log {\left (1 + \frac {b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {840 a^{7} b x^{59} \log {\left (1 + \frac {b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {840 a^{7} b x^{59}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {420 a^{6} b^{2} x^{\frac {178}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {140 a^{5} b^{3} x^{\frac {179}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {70 a^{4} b^{4} x^{60}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {42 a^{3} b^{5} x^{\frac {181}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {28 a^{2} b^{6} x^{\frac {182}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {20 a b^{7} x^{61}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {15 b^{8} x^{\frac {184}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} \]
Verification of antiderivative is not currently implemented for this CAS.
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