Optimal. Leaf size=38 \[ \frac {3 \left (a+b x^2\right )^{10/3}}{20 b^2}-\frac {3 a \left (a+b x^2\right )^{7/3}}{14 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 = {266, 43} \[ \frac {3 \left (a+b x^2\right )^{10/3}}{20 b^2}-\frac {3 a \left (a+b x^2\right )^{7/3}}{14 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^{4/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^{4/3}}{b}+\frac {(a+b x)^{7/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a \left (a+b x^2\right )^{7/3}}{14 b^2}+\frac {3 \left (a+b x^2\right )^{10/3}}{20 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {3 \left (a+b x^2\right )^{7/3} \left (7 b x^2-3 a\right )}{140 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 45, normalized size = 1.18 \[ \frac {3 \, {\left (7 \, b^{3} x^{6} + 11 \, a b^{2} x^{4} + a^{2} b x^{2} - 3 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{140 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 29, normalized size = 0.76 \[ \frac {3 \, {\left (7 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} - 10 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a\right )}}{140 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{3}} \left (-7 b \,x^{2}+3 a \right )}{140 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}}}{20 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a}{14 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 42, normalized size = 1.11 \[ {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {33\,a\,x^4}{140}+\frac {3\,b\,x^6}{20}-\frac {9\,a^3}{140\,b^2}+\frac {3\,a^2\,x^2}{140\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.54, size = 88, normalized size = 2.32 \[ \begin {cases} - \frac {9 a^{3} \sqrt [3]{a + b x^{2}}}{140 b^{2}} + \frac {3 a^{2} x^{2} \sqrt [3]{a + b x^{2}}}{140 b} + \frac {33 a x^{4} \sqrt [3]{a + b x^{2}}}{140} + \frac {3 b x^{6} \sqrt [3]{a + b x^{2}}}{20} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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