Optimal. Leaf size=157 \[ -\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}-\frac {8 b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{11/3}}-\frac {8 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{11/3}}+\frac {4 b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{11/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {269, 294, 308,
298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {8 b^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{11/3}}+\frac {4 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{11/3}}-\frac {8 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{11/3}}-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (a x^3+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 269
Rule 294
Rule 298
Rule 308
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+\frac {b}{x^3}\right )^2} \, dx &=\int \frac {x^{10}}{\left (b+a x^3\right )^2} \, dx\\ &=-\frac {x^8}{3 a \left (b+a x^3\right )}+\frac {8 \int \frac {x^7}{b+a x^3} \, dx}{3 a}\\ &=-\frac {x^8}{3 a \left (b+a x^3\right )}+\frac {8 \int \left (-\frac {b x}{a^2}+\frac {x^4}{a}+\frac {b^2 x}{a^2 \left (b+a x^3\right )}\right ) \, dx}{3 a}\\ &=-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}+\frac {\left (8 b^2\right ) \int \frac {x}{b+a x^3} \, dx}{3 a^3}\\ &=-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}-\frac {\left (8 b^{5/3}\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 a^{10/3}}+\frac {\left (8 b^{5/3}\right ) \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 a^{10/3}}\\ &=-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}-\frac {8 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{11/3}}+\frac {\left (4 b^{5/3}\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 a^{11/3}}+\frac {\left (4 b^2\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^{10/3}}\\ &=-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}-\frac {8 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{11/3}}+\frac {4 b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{11/3}}+\frac {\left (8 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{11/3}}\\ &=-\frac {4 b x^2}{3 a^3}+\frac {8 x^5}{15 a^2}-\frac {x^8}{3 a \left (b+a x^3\right )}-\frac {8 b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{11/3}}-\frac {8 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{11/3}}+\frac {4 b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{11/3}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 144, normalized size = 0.92 \begin {gather*} \frac {-45 a^{2/3} b x^2+9 a^{5/3} x^5-\frac {15 a^{2/3} b^2 x^2}{b+a x^3}-40 \sqrt {3} b^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-40 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+20 b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{45 a^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 130, normalized size = 0.83
method | result | size |
risch | \(\frac {x^{5}}{5 a^{2}}-\frac {b \,x^{2}}{a^{3}}-\frac {b^{2} x^{2}}{3 a^{3} \left (a \,x^{3}+b \right )}+\frac {8 b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{9 a^{4}}\) | \(68\) |
default | \(\frac {b^{2} \left (-\frac {x^{2}}{3 \left (a \,x^{3}+b \right )}-\frac {8 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {4 \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {8 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{3}}+\frac {\frac {1}{5} a \,x^{5}-b \,x^{2}}{a^{3}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 147, normalized size = 0.94 \begin {gather*} -\frac {b^{2} x^{2}}{3 \, {\left (a^{4} x^{3} + a^{3} b\right )}} + \frac {8 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {4 \, b^{2} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {8 \, b^{2} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {a x^{5} - 5 \, b x^{2}}{5 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 190, normalized size = 1.21 \begin {gather*} \frac {9 \, a^{2} x^{8} - 36 \, a b x^{5} - 60 \, b^{2} x^{2} + 40 \, \sqrt {3} {\left (a b x^{3} + b^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + \sqrt {3} b}{3 \, b}\right ) - 20 \, {\left (a b x^{3} + b^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - b \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 40 \, {\left (a b x^{3} + b^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right )}{45 \, {\left (a^{4} x^{3} + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 70, normalized size = 0.45 \begin {gather*} - \frac {b^{2} x^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \operatorname {RootSum} {\left (729 t^{3} a^{11} + 512 b^{5}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{7}}{64 b^{3}} + x \right )} \right )\right )} + \frac {x^{5}}{5 a^{2}} - \frac {b x^{2}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.75, size = 151, normalized size = 0.96 \begin {gather*} -\frac {b^{2} x^{2}}{3 \, {\left (a x^{3} + b\right )} a^{3}} - \frac {8 \, b \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} - \frac {8 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5}} + \frac {4 \, \left (-a^{2} b\right )^{\frac {2}{3}} b \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{5}} + \frac {a^{8} x^{5} - 5 \, a^{7} b x^{2}}{5 \, a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.29, size = 172, normalized size = 1.10 \begin {gather*} \frac {x^5}{5\,a^2}-\frac {b^2\,x^2}{3\,\left (a^4\,x^3+b\,a^3\right )}-\frac {b\,x^2}{a^3}+\frac {8\,{\left (-b\right )}^{5/3}\,\ln \left (\frac {64\,b^4\,x}{9\,a^5}-\frac {64\,{\left (-b\right )}^{13/3}}{9\,a^{16/3}}\right )}{9\,a^{11/3}}+\frac {8\,{\left (-b\right )}^{5/3}\,\ln \left (\frac {64\,b^4\,x}{9\,a^5}-\frac {64\,{\left (-b\right )}^{13/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{16/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{11/3}}-\frac {8\,{\left (-b\right )}^{5/3}\,\ln \left (\frac {64\,b^4\,x}{9\,a^5}-\frac {64\,{\left (-b\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{16/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{11/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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