Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b x} \, dx &=\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{3/2}}{3 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (8 a^2-12 a b x+15 b^2 x^2\right )}{105 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 38, normalized size = 0.72
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (15 x^{2} b^{2}-12 a b x +8 a^{2}\right )}{105 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (15 b^{3} x^{3}+3 a \,b^{2} x^{2}-4 a^{2} b x +8 a^{3}\right ) \sqrt {b x +a}}{105 b^{3}}\) | \(43\) |
risch | \(\frac {2 \left (15 b^{3} x^{3}+3 a \,b^{2} x^{2}-4 a^{2} b x +8 a^{3}\right ) \sqrt {b x +a}}{105 b^{3}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 42, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{105 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs.
\(2 (49) = 98\).
time = 0.81, size = 666, normalized size = 12.57 \begin {gather*} \frac {16 a^{\frac {23}{2}} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {23}{2}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {21}{2}} b x \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b x}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {19}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b^{2} x^{2}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {17}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {17}{2}} b^{3} x^{3}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {100 a^{\frac {15}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {96 a^{\frac {13}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {11}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (41) = 82\).
time = 0.69, size = 93, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (\frac {7 \, {\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^{2}} + \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^{2}}\right )}}{105 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 37, normalized size = 0.70 \begin {gather*} \frac {30\,{\left (a+b\,x\right )}^{7/2}-84\,a\,{\left (a+b\,x\right )}^{5/2}+70\,a^2\,{\left (a+b\,x\right )}^{3/2}}{105\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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