Optimal. Leaf size=34 \[ \frac {2 (a+b x)^{7/2}}{7 b^2}-\frac {2 a (a+b x)^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {2 (a+b x)^{7/2}}{7 b^2}-\frac {2 a (a+b x)^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x (a+b x)^{3/2} \, dx &=\int \left (-\frac {a (a+b x)^{3/2}}{b}+\frac {(a+b x)^{5/2}}{b}\right ) \, dx\\ &=-\frac {2 a (a+b x)^{5/2}}{5 b^2}+\frac {2 (a+b x)^{7/2}}{7 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.71 \[ \frac {2 (a+b x)^{5/2} (5 b x-2 a)}{35 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 41, normalized size = 1.21 \[ \frac {2 \, {\left (5 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x + a}}{35 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 119, normalized size = 3.50 \[ \frac {2 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{2}}{b} + \frac {14 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a}{b} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )}}{b}\right )}}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.62 \[ -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-5 b x +2 a \right )}{35 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 26, normalized size = 0.76 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.74 \[ -\frac {14\,a\,{\left (a+b\,x\right )}^{5/2}-10\,{\left (a+b\,x\right )}^{7/2}}{35\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 80, normalized size = 2.35 \[ \begin {cases} - \frac {4 a^{3} \sqrt {a + b x}}{35 b^{2}} + \frac {2 a^{2} x \sqrt {a + b x}}{35 b} + \frac {16 a x^{2} \sqrt {a + b x}}{35} + \frac {2 b x^{3} \sqrt {a + b x}}{7} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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