Optimal. Leaf size=113 \[ -\frac {4 b}{a^2 \sqrt [3]{a+b x}}-\frac {1}{a x \sqrt [3]{a+b x}}-\frac {4 b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {44, 53, 57, 631,
210, 31} \begin {gather*} -\frac {4 b \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac {4 b}{a^2 \sqrt [3]{a+b x}}-\frac {1}{a x \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 53
Rule 57
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx &=\frac {3}{a x \sqrt [3]{a+b x}}+\frac {4 \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx}{a}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}-\frac {(4 b) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{3 a^2}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}+\frac {2 b \log (x)}{3 a^{7/3}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{a^2}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{7/3}}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}-\frac {4 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 123, normalized size = 1.09 \begin {gather*} \frac {-\frac {3 \sqrt [3]{a} (a+4 b x)}{x \sqrt [3]{a+b x}}-4 \sqrt {3} b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+2 b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 a^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 112, normalized size = 0.99
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a^{2} x}-\frac {3 b}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}-\frac {4 b \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {7}{3}}}+\frac {2 b \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{3 a^{\frac {7}{3}}}-\frac {4 b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {7}{3}}}\) | \(108\) |
derivativedivides | \(3 b \left (-\frac {1}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{2}}\right )\) | \(112\) |
default | \(3 b \left (-\frac {1}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{2}}\right )\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 122, normalized size = 1.08 \begin {gather*} -\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left (b x + a\right )}^{\frac {4}{3}} a^{2} - {\left (b x + a\right )}^{\frac {1}{3}} a^{3}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {7}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs.
\(2 (88) = 176\).
time = 1.26, size = 407, normalized size = 3.60 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.38, size = 857, normalized size = 7.58 \begin {gather*} - \frac {9 a^{\frac {4}{3}} b^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} + \frac {12 \sqrt [3]{a} b^{\frac {5}{3}} \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {4 a b \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {4 a b \sqrt [3]{\frac {a}{b} + x} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {4 a b \sqrt [3]{\frac {a}{b} + x} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} + \frac {4 b^{2} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} + \frac {4 b^{2} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} + \frac {4 b^{2} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{- 9 a^{\frac {10}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) + 9 a^{\frac {7}{3}} b \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 120, normalized size = 1.06 \begin {gather*} -\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left ({\left (b x + a\right )}^{\frac {4}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 173, normalized size = 1.53 \begin {gather*} -\frac {\frac {3\,b}{a}-\frac {4\,b\,\left (a+b\,x\right )}{a^2}}{a\,{\left (a+b\,x\right )}^{1/3}-{\left (a+b\,x\right )}^{4/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}-\frac {4\,b\,\ln \left (16\,a^{7/3}\,b^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )}{3\,a^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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