Optimal. Leaf size=93 \[ -\frac {3}{5 b x^{5/3}}+\frac {3 a}{4 b^2 x^{4/3}}-\frac {a^2}{b^3 x}+\frac {3 a^3}{2 b^4 x^{2/3}}-\frac {3 a^4}{b^5 \sqrt [3]{x}}+\frac {3 a^5 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac {a^5 \log (x)}{b^6} \]
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Rubi [A]
time = 0.03, antiderivative size = 93, 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, 46}
\begin {gather*} \frac {3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {a^5 \log (x)}{b^6}-\frac {3 a^4}{b^5 \sqrt [3]{x}}+\frac {3 a^3}{2 b^4 x^{2/3}}-\frac {a^2}{b^3 x}+\frac {3 a}{4 b^2 x^{4/3}}-\frac {3}{5 b x^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right ) x^{8/3}} \, dx\\ &=3 \text {Subst}\left (\int \frac {1}{x^6 (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {1}{b x^6}-\frac {a}{b^2 x^5}+\frac {a^2}{b^3 x^4}-\frac {a^3}{b^4 x^3}+\frac {a^4}{b^5 x^2}-\frac {a^5}{b^6 x}+\frac {a^6}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3}{5 b x^{5/3}}+\frac {3 a}{4 b^2 x^{4/3}}-\frac {a^2}{b^3 x}+\frac {3 a^3}{2 b^4 x^{2/3}}-\frac {3 a^4}{b^5 \sqrt [3]{x}}+\frac {3 a^5 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac {a^5 \log (x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 84, normalized size = 0.90 \begin {gather*} \frac {\frac {b \left (-12 b^4+15 a b^3 \sqrt [3]{x}-20 a^2 b^2 x^{2/3}+30 a^3 b x-60 a^4 x^{4/3}\right )}{x^{5/3}}+60 a^5 \log \left (b+a \sqrt [3]{x}\right )-20 a^5 \log (x)}{20 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 78, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {3}{5 b \,x^{\frac {5}{3}}}+\frac {3 a}{4 b^{2} x^{\frac {4}{3}}}-\frac {a^{2}}{b^{3} x}+\frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}}-\frac {3 a^{4}}{b^{5} x^{\frac {1}{3}}}+\frac {3 a^{5} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {a^{5} \ln \left (x \right )}{b^{6}}\) | \(78\) |
default | \(-\frac {3}{5 b \,x^{\frac {5}{3}}}+\frac {3 a}{4 b^{2} x^{\frac {4}{3}}}-\frac {a^{2}}{b^{3} x}+\frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}}-\frac {3 a^{4}}{b^{5} x^{\frac {1}{3}}}+\frac {3 a^{5} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {a^{5} \ln \left (x \right )}{b^{6}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 95, normalized size = 1.02 \begin {gather*} \frac {3 \, a^{5} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{6}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5}}{5 \, b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a}{4 \, b^{6}} - \frac {10 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{2}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{3}}{b^{6}} - \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{4}}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 85, normalized size = 0.91 \begin {gather*} \frac {60 \, a^{5} x^{2} \log \left (a x^{\frac {1}{3}} + b\right ) - 60 \, a^{5} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 20 \, a^{2} b^{3} x - 15 \, {\left (4 \, a^{4} b x - a b^{4}\right )} x^{\frac {2}{3}} + 6 \, {\left (5 \, a^{3} b^{2} x - 2 \, b^{5}\right )} x^{\frac {1}{3}}}{20 \, b^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.94, size = 116, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {3}{5 b x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {a^{5} \log {\left (x \right )}}{b^{6}} + \frac {3 a^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{6}} - \frac {3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}} - \frac {a^{2}}{b^{3} x} + \frac {3 a}{4 b^{2} x^{\frac {4}{3}}} - \frac {3}{5 b x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 81, normalized size = 0.87 \begin {gather*} \frac {3 \, a^{5} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{6}} - \frac {a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{\frac {4}{3}} - 30 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac {2}{3}} - 15 \, a b^{4} x^{\frac {1}{3}} + 12 \, b^{5}}{20 \, b^{6} x^{\frac {5}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 71, normalized size = 0.76 \begin {gather*} \frac {6\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^6}-\frac {\frac {3}{5\,b}-\frac {3\,a\,x^{1/3}}{4\,b^2}-\frac {3\,a^3\,x}{2\,b^4}+\frac {a^2\,x^{2/3}}{b^3}+\frac {3\,a^4\,x^{4/3}}{b^5}}{x^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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