Optimal. Leaf size=280 \[ \frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.14, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1355, 288, 290, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 288
Rule 290
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1355
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {x^4}{\left (a b+b^2 x^3\right )^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^2} \, dx}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{27 a^{4/3} b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{27 a^{4/3} b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{54 a^{4/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{18 a b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{4/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {x^2}{9 a b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 235, normalized size = 0.84 \begin {gather*} \frac {-3 a^{4/3} b^{2/3} x^2+2 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+6 \sqrt [3]{a} b^{5/3} x^5-2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{54 a^{4/3} b^{5/3} \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.03, size = 169, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^3\right ) \left (\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3}}+\frac {2 b x^5-a x^2}{18 a b \left (a+b x^3\right )^2}\right )}{\sqrt {\left (a+b x^3\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 512, normalized size = 1.83 \begin {gather*} \left [\frac {6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, \frac {6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 299, normalized size = 1.07 \begin {gather*} \frac {\left (-2 \sqrt {3}\, b^{2} x^{6} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-2 b^{2} x^{6} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )+b^{2} x^{6} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )+6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2} x^{5}-4 \sqrt {3}\, a b \,x^{3} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-4 a b \,x^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )+2 a b \,x^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )-3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b \,x^{2}-2 \sqrt {3}\, a^{2} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-2 a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )+a^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} a \,b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.81, size = 149, normalized size = 0.53 \begin {gather*} \frac {2 \, b x^{5} - a x^{2}}{18 \, {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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