Optimal. Leaf size=152 \[ -\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {44, 53, 58, 631,
210, 31} \begin {gather*} \frac {14 \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 53
Rule 58
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^{4/3} (a+b x)^3} \, dx &=\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7 \int \frac {1}{x^{4/3} (a+b x)^2} \, dx}{6 a}\\ &=\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \int \frac {1}{x^{4/3} (a+b x)} \, dx}{9 a^2}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}-\frac {(14 b) \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{9 a^3}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}-\frac {7 \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{3 a^3}+\frac {\left (7 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{3 a^{10/3}}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}-\frac {\left (14 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{10/3}}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 153, normalized size = 1.01 \begin {gather*} \frac {-\frac {3 \sqrt [3]{a} \left (18 a^2+49 a b x+28 b^2 x^2\right )}{\sqrt [3]{x} (a+b x)^2}+28 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-14 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{18 a^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 133, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {3 b \left (\frac {\frac {5 b \,x^{\frac {5}{3}}}{9}+\frac {13 a \,x^{\frac {2}{3}}}{18}}{\left (b x +a \right )^{2}}-\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}-\frac {3}{a^{3} x^{\frac {1}{3}}}\) | \(133\) |
default | \(-\frac {3 b \left (\frac {\frac {5 b \,x^{\frac {5}{3}}}{9}+\frac {13 a \,x^{\frac {2}{3}}}{18}}{\left (b x +a \right )^{2}}-\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}-\frac {3}{a^{3} x^{\frac {1}{3}}}\) | \(133\) |
risch | \(-\frac {3}{a^{3} x^{\frac {1}{3}}}-\frac {5 b^{2} x^{\frac {5}{3}}}{3 a^{3} \left (b x +a \right )^{2}}-\frac {13 b \,x^{\frac {2}{3}}}{6 a^{2} \left (b x +a \right )^{2}}+\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 154, normalized size = 1.01 \begin {gather*} -\frac {28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}}{6 \, {\left (a^{3} b^{2} x^{\frac {7}{3}} + 2 \, a^{4} b x^{\frac {4}{3}} + a^{5} x^{\frac {1}{3}}\right )}} - \frac {14 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {14 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 211, normalized size = 1.39 \begin {gather*} -\frac {28 \, \sqrt {3} {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (-a x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 28 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {1}{3}}\right ) + 3 \, {\left (28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 155, normalized size = 1.02 \begin {gather*} \frac {14 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4}} + \frac {14 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} b} - \frac {3}{a^{3} x^{\frac {1}{3}}} - \frac {7 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{4} b} - \frac {10 \, b^{2} x^{\frac {5}{3}} + 13 \, a b x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 174, normalized size = 1.14 \begin {gather*} \frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}+588\,a^3\,b^3\,x^{1/3}\right )}{9\,a^{10/3}}-\frac {\frac {3}{a}+\frac {14\,b^2\,x^2}{3\,a^3}+\frac {49\,b\,x}{6\,a^2}}{a^2\,x^{1/3}+b^2\,x^{7/3}+2\,a\,b\,x^{4/3}}+\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}}-\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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