Optimal. Leaf size=80 \[ -\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]
[Out]
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Rubi [A] time = 0.131948, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]
Antiderivative was successfully verified.
[In] Int[x^7*(a + b*x^2)^(4/3),x]
[Out]
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Rubi in Sympy [A] time = 15.5822, size = 75, normalized size = 0.94 \[ - \frac{3 a^{3} \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{4}} + \frac{9 a^{2} \left (a + b x^{2}\right )^{\frac{10}{3}}}{20 b^{4}} - \frac{9 a \left (a + b x^{2}\right )^{\frac{13}{3}}}{26 b^{4}} + \frac{3 \left (a + b x^{2}\right )^{\frac{16}{3}}}{32 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(b*x**2+a)**(4/3),x)
[Out]
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Mathematica [A] time = 0.0454171, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (-81 a^3+189 a^2 b x^2-315 a b^2 x^4+455 b^3 x^6\right )}{14560 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^7*(a + b*x^2)^(4/3),x]
[Out]
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Maple [A] time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-1365\,{b}^{3}{x}^{6}+945\,a{b}^{2}{x}^{4}-567\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{14560\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(b*x^2+a)^(4/3),x)
[Out]
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Maxima [A] time = 1.35407, size = 86, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}}}{32 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a}{26 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2}}{20 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3}}{14 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)*x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206399, size = 92, normalized size = 1.15 \[ \frac{3 \,{\left (455 \, b^{5} x^{10} + 595 \, a b^{4} x^{8} + 14 \, a^{2} b^{3} x^{6} - 18 \, a^{3} b^{2} x^{4} + 27 \, a^{4} b x^{2} - 81 \, a^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{14560 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)*x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 20.8648, size = 136, normalized size = 1.7 \[ \begin{cases} - \frac{243 a^{5} \sqrt [3]{a + b x^{2}}}{14560 b^{4}} + \frac{81 a^{4} x^{2} \sqrt [3]{a + b x^{2}}}{14560 b^{3}} - \frac{27 a^{3} x^{4} \sqrt [3]{a + b x^{2}}}{7280 b^{2}} + \frac{3 a^{2} x^{6} \sqrt [3]{a + b x^{2}}}{1040 b} + \frac{51 a x^{8} \sqrt [3]{a + b x^{2}}}{416} + \frac{3 b x^{10} \sqrt [3]{a + b x^{2}}}{32} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{8}}{8} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(b*x**2+a)**(4/3),x)
[Out]
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GIAC/XCAS [A] time = 0.220508, size = 181, normalized size = 2.26 \[ \frac{3 \,{\left (\frac{4 \,{\left (140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}\right )} a}{b^{3}} + \frac{455 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}} - 2240 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a + 4368 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2} - 4160 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3} + 1820 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{4}}{b^{3}}\right )}}{14560 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(4/3)*x^7,x, algorithm="giac")
[Out]