Optimal. Leaf size=815 \[ \frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {x}{18 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{18\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{12\ 3^{3/4} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt {2} \sqrt [4]{3} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Rubi [A]
time = 0.52, antiderivative size = 815, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {424, 12, 482,
544, 241, 310, 225, 1893, 402} \begin {gather*} -\frac {\left (a-b x^2\right )^{2/3} x}{18 a \left (b x^2+3 a\right )}+\frac {x}{18 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\left (a-b x^2\right )^{2/3} x}{3 \left (b x^2+3 a\right )^2}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{18\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{12\ 3^{3/4} a^{2/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt {2} \sqrt [4]{3} a^{2/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 225
Rule 241
Rule 310
Rule 402
Rule 424
Rule 482
Rule 544
Rule 1893
Rubi steps
\begin {align*} \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}+\frac {\int \frac {16 a b^2 x^2}{3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{12 a b}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}+\frac {1}{9} (4 b) \int \frac {x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {\int \frac {a-\frac {b x^2}{3}}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{18 a}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {1}{9} \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx-\frac {\int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{54 a}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{18\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{36 a b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{18\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{36 a b x}+\frac {\left (\sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{18 a^{2/3} b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {x}{18 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{18\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{12\ 3^{3/4} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt {2} \sqrt [4]{3} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.19, size = 252, normalized size = 0.31 \begin {gather*} \frac {-\frac {b x^3 \sqrt [3]{1-\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{a^2}+\frac {27 x \left (3 a-4 b x^2+\frac {b^2 x^4}{a}+\frac {9 a \left (3 a+b x^2\right ) F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{\left (3 a+b x^2\right )^2}}{486 \sqrt [3]{a-b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {5}{3}}}{\left (b \,x^{2}+3 a \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a - b x^{2}\right )^{\frac {5}{3}}}{\left (3 a + b x^{2}\right )^{3}}\, dx \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*} \text {could not integrate} \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 {{\left (a-b\,x^2\right )}^{5/3}}{{\left (b\,x^2+3\,a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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