Optimal. Leaf size=131 \[ -\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{210 a^4 b^6}{x}-\frac{135 a^2 b^8}{\sqrt [3]{x}}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{a^{10}}{3 x^3}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]
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Rubi [A] time = 0.0714533, antiderivative size = 131, normalized size of antiderivative = 1., 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} \[ -\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{210 a^4 b^6}{x}-\frac{135 a^2 b^8}{\sqrt [3]{x}}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{a^{10}}{3 x^3}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^{10}+\frac{a^{10}}{x^{10}}+\frac{10 a^9 b}{x^9}+\frac{45 a^8 b^2}{x^8}+\frac{120 a^7 b^3}{x^7}+\frac{210 a^6 b^4}{x^6}+\frac{252 a^5 b^5}{x^5}+\frac{210 a^4 b^6}{x^4}+\frac{120 a^3 b^7}{x^3}+\frac{45 a^2 b^8}{x^2}+\frac{10 a b^9}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{3 x^3}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{210 a^4 b^6}{x}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{135 a^2 b^8}{\sqrt [3]{x}}+3 b^{10} \sqrt [3]{x}+10 a b^9 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0741759, size = 131, normalized size = 1. \[ -\frac{135 a^8 b^2}{7 x^{7/3}}-\frac{60 a^7 b^3}{x^2}-\frac{126 a^6 b^4}{x^{5/3}}-\frac{189 a^5 b^5}{x^{4/3}}-\frac{180 a^3 b^7}{x^{2/3}}-\frac{210 a^4 b^6}{x}-\frac{135 a^2 b^8}{\sqrt [3]{x}}-\frac{15 a^9 b}{4 x^{8/3}}-\frac{a^{10}}{3 x^3}+10 a b^9 \log (x)+3 b^{10} \sqrt [3]{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 112, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{3\,{x}^{3}}}-{\frac{15\,{a}^{9}b}{4}{x}^{-{\frac{8}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{7}{x}^{-{\frac{7}{3}}}}-60\,{\frac{{a}^{7}{b}^{3}}{{x}^{2}}}-126\,{\frac{{a}^{6}{b}^{4}}{{x}^{5/3}}}-189\,{\frac{{a}^{5}{b}^{5}}{{x}^{4/3}}}-210\,{\frac{{a}^{4}{b}^{6}}{x}}-180\,{\frac{{a}^{3}{b}^{7}}{{x}^{2/3}}}-135\,{\frac{{a}^{2}{b}^{8}}{\sqrt [3]{x}}}+3\,{b}^{10}\sqrt [3]{x}+10\,a{b}^{9}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95164, size = 151, normalized size = 1.15 \begin{align*} 10 \, a b^{9} \log \left (x\right ) + 3 \, b^{10} x^{\frac{1}{3}} - \frac{11340 \, a^{2} b^{8} x^{\frac{8}{3}} + 15120 \, a^{3} b^{7} x^{\frac{7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac{5}{3}} + 10584 \, a^{6} b^{4} x^{\frac{4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac{2}{3}} + 315 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52597, size = 290, normalized size = 2.21 \begin{align*} \frac{2520 \, a b^{9} x^{3} \log \left (x^{\frac{1}{3}}\right ) - 17640 \, a^{4} b^{6} x^{2} - 5040 \, a^{7} b^{3} x - 28 \, a^{10} - 324 \,{\left (35 \, a^{2} b^{8} x^{2} + 49 \, a^{5} b^{5} x + 5 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 63 \,{\left (4 \, b^{10} x^{3} - 240 \, a^{3} b^{7} x^{2} - 168 \, a^{6} b^{4} x - 5 \, a^{9} b\right )} x^{\frac{1}{3}}}{84 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.21492, size = 133, normalized size = 1.02 \begin{align*} - \frac{a^{10}}{3 x^{3}} - \frac{15 a^{9} b}{4 x^{\frac{8}{3}}} - \frac{135 a^{8} b^{2}}{7 x^{\frac{7}{3}}} - \frac{60 a^{7} b^{3}}{x^{2}} - \frac{126 a^{6} b^{4}}{x^{\frac{5}{3}}} - \frac{189 a^{5} b^{5}}{x^{\frac{4}{3}}} - \frac{210 a^{4} b^{6}}{x} - \frac{180 a^{3} b^{7}}{x^{\frac{2}{3}}} - \frac{135 a^{2} b^{8}}{\sqrt [3]{x}} + 10 a b^{9} \log{\left (x \right )} + 3 b^{10} \sqrt [3]{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20182, size = 153, normalized size = 1.17 \begin{align*} 10 \, a b^{9} \log \left ({\left | x \right |}\right ) + 3 \, b^{10} x^{\frac{1}{3}} - \frac{11340 \, a^{2} b^{8} x^{\frac{8}{3}} + 15120 \, a^{3} b^{7} x^{\frac{7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac{5}{3}} + 10584 \, a^{6} b^{4} x^{\frac{4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac{2}{3}} + 315 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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