Optimal. Leaf size=80 \[ -\frac {3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac {9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac {9 a \left (a+b x^2\right )^{13/3}}{26 b^4}+\frac {3 \left (a+b x^2\right )^{16/3}}{32 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2\right )^{4/3} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{4/3}}{b^3}+\frac {3 a^2 (a+b x)^{7/3}}{b^3}-\frac {3 a (a+b x)^{10/3}}{b^3}+\frac {(a+b x)^{13/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\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 {9 a \left (a+b x^2\right )^{13/3}}{26 b^4}+\frac {3 \left (a+b x^2\right )^{16/3}}{32 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 50, normalized size = 0.62 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 47, normalized size = 0.59
method | result | size |
gosper | \(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{3}} \left (-455 b^{3} x^{6}+315 a \,b^{2} x^{4}-189 a^{2} b \,x^{2}+81 a^{3}\right )}{14560 b^{4}}\) | \(47\) |
trager | \(-\frac {3 \left (-455 b^{5} x^{10}-595 a \,b^{4} x^{8}-14 a^{2} b^{3} x^{6}+18 a^{3} b^{2} x^{4}-27 a^{4} b \,x^{2}+81 a^{5}\right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{14560 b^{4}}\) | \(69\) |
risch | \(-\frac {3 \left (-455 b^{5} x^{10}-595 a \,b^{4} x^{8}-14 a^{2} b^{3} x^{6}+18 a^{3} b^{2} x^{4}-27 a^{4} b \,x^{2}+81 a^{5}\right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{14560 b^{4}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 64, normalized size = 0.80 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 68, normalized size = 0.85 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 136, normalized size = 1.70 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 57, normalized size = 0.71 \begin {gather*} \frac {3 \, {\left (455 \, {\left (b x^{2} + a\right )}^{\frac {16}{3}} - 1680 \, {\left (b x^{2} + a\right )}^{\frac {13}{3}} a + 2184 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} a^{2} - 1040 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a^{3}\right )}}{14560 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.19, size = 64, normalized size = 0.80 \begin {gather*} {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {51\,a\,x^8}{416}+\frac {3\,b\,x^{10}}{32}-\frac {243\,a^5}{14560\,b^4}+\frac {3\,a^2\,x^6}{1040\,b}-\frac {27\,a^3\,x^4}{7280\,b^2}+\frac {81\,a^4\,x^2}{14560\,b^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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