Optimal. Leaf size=149 \[ \frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}+\frac {3 b^7}{2 a^8 x^{2/3}}-\frac {b^6}{a^7 x}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^2}{7 a^3 x^{7/3}}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {1}{3 a x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 149, 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, 44} \[ \frac {3 b^7}{2 a^8 x^{2/3}}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^2}{7 a^3 x^{7/3}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}-\frac {b^6}{a^7 x}+\frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^{10} (a+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{a x^{10}}-\frac {b}{a^2 x^9}+\frac {b^2}{a^3 x^8}-\frac {b^3}{a^4 x^7}+\frac {b^4}{a^5 x^6}-\frac {b^5}{a^6 x^5}+\frac {b^6}{a^7 x^4}-\frac {b^7}{a^8 x^3}+\frac {b^8}{a^9 x^2}-\frac {b^9}{a^{10} x}+\frac {b^{10}}{a^{10} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {1}{3 a x^3}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {3 b^2}{7 a^3 x^{7/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {b^6}{a^7 x}+\frac {3 b^7}{2 a^8 x^{2/3}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}+\frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 138, normalized size = 0.93 \[ -\frac {280 a^9-315 a^8 b \sqrt [3]{x}+360 a^7 b^2 x^{2/3}-420 a^6 b^3 x+504 a^5 b^4 x^{4/3}-630 a^4 b^5 x^{5/3}+840 a^3 b^6 x^2-1260 a^2 b^7 x^{7/3}-2520 b^9 x^3 \log \left (a+b \sqrt [3]{x}\right )+2520 a b^8 x^{8/3}+840 b^9 x^3 \log (x)}{840 a^{10} x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 126, normalized size = 0.85 \[ \frac {2520 \, b^{9} x^{3} \log \left (b x^{\frac {1}{3}} + a\right ) - 2520 \, b^{9} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 840 \, a^{3} b^{6} x^{2} + 420 \, a^{6} b^{3} x - 280 \, a^{9} - 90 \, {\left (28 \, a b^{8} x^{2} - 7 \, a^{4} b^{5} x + 4 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} + 63 \, {\left (20 \, a^{2} b^{7} x^{2} - 8 \, a^{5} b^{4} x + 5 \, a^{8} b\right )} x^{\frac {1}{3}}}{840 \, a^{10} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 125, normalized size = 0.84 \[ \frac {3 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{10}} - \frac {b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{\frac {8}{3}} - 1260 \, a^{2} b^{7} x^{\frac {7}{3}} + 840 \, a^{3} b^{6} x^{2} - 630 \, a^{4} b^{5} x^{\frac {5}{3}} + 504 \, a^{5} b^{4} x^{\frac {4}{3}} - 420 \, a^{6} b^{3} x + 360 \, a^{7} b^{2} x^{\frac {2}{3}} - 315 \, a^{8} b x^{\frac {1}{3}} + 280 \, a^{9}}{840 \, a^{10} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 122, normalized size = 0.82 \[ -\frac {b^{9} \ln \relax (x )}{a^{10}}+\frac {3 b^{9} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{10}}-\frac {3 b^{8}}{a^{9} x^{\frac {1}{3}}}+\frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}}-\frac {b^{6}}{a^{7} x}+\frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}}-\frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}}+\frac {b^{3}}{2 a^{4} x^{2}}-\frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}}+\frac {3 b}{8 a^{2} x^{\frac {8}{3}}}-\frac {1}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 120, normalized size = 0.81 \[ \frac {3 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{10}} - \frac {b^{9} \log \relax (x)}{a^{10}} - \frac {2520 \, b^{8} x^{\frac {8}{3}} - 1260 \, a b^{7} x^{\frac {7}{3}} + 840 \, a^{2} b^{6} x^{2} - 630 \, a^{3} b^{5} x^{\frac {5}{3}} + 504 \, a^{4} b^{4} x^{\frac {4}{3}} - 420 \, a^{5} b^{3} x + 360 \, a^{6} b^{2} x^{\frac {2}{3}} - 315 \, a^{7} b x^{\frac {1}{3}} + 280 \, a^{8}}{840 \, a^{9} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 116, normalized size = 0.78 \[ -\frac {280\,a^9-5040\,b^9\,x^3\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )-420\,a^6\,b^3\,x-315\,a^8\,b\,x^{1/3}+2520\,a\,b^8\,x^{8/3}+840\,a^3\,b^6\,x^2+360\,a^7\,b^2\,x^{2/3}+504\,a^5\,b^4\,x^{4/3}-630\,a^4\,b^5\,x^{5/3}-1260\,a^2\,b^7\,x^{7/3}}{840\,a^{10}\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.37, size = 172, normalized size = 1.15 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {10}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{10 b x^{\frac {10}{3}}} & \text {for}\: a = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {1}{3 a x^{3}} + \frac {3 b}{8 a^{2} x^{\frac {8}{3}}} - \frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}} + \frac {b^{3}}{2 a^{4} x^{2}} - \frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}} + \frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}} - \frac {b^{6}}{a^{7} x} + \frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}} - \frac {3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac {b^{9} \log {\relax (x )}}{a^{10}} + \frac {3 b^{9} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{10}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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