Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3} \]
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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)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^2 (a+b x)^{3/2} \, dx &=\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(213\) vs. \(2(53)=106\).
time = 7.72, size = 197, normalized size = 3.72 \begin {gather*} \frac {2 \sqrt {a} \left (8 a^7 \left (-1+\sqrt {\frac {a+b x}{a}}\right )+4 a^6 b x \left (-6+5 \sqrt {\frac {a+b x}{a}}\right )+3 a^5 b^2 x^2 \left (-8+5 \sqrt {\frac {a+b x}{a}}\right )+5 a b^3 x^3 \left (11 a^3+38 a^2 b x+31 b^3 x^3\right ) \sqrt {\frac {a+b x}{a}}-8 a^4 b^3 x^3+258 a^2 b^5 x^5 \sqrt {\frac {a+b x}{a}}+35 b^7 x^7 \sqrt {\frac {a+b x}{a}}\right )}{315 b^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 38, normalized size = 0.72
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (35 x^{2} b^{2}-20 a b x +8 a^{2}\right )}{315 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {4 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {4 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (35 b^{4} x^{4}+50 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}-4 a^{3} b x +8 a^{4}\right ) \sqrt {b x +a}}{315 b^{3}}\) | \(54\) |
risch | \(\frac {2 \left (35 b^{4} x^{4}+50 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}-4 a^{3} b x +8 a^{4}\right ) \sqrt {b x +a}}{315 b^{3}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}}}{9 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {7}{2}} a}{7 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 53, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x + a}}{315 \, 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. 733 vs.
\(2 (49) = 98\).
time = 0.96, size = 733, normalized size = 13.83 \begin {gather*} \frac {16 a^{\frac {25}{2}} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {25}{2}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {23}{2}} b x \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {23}{2}} b x}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {21}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b^{2} x^{2}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {110 a^{\frac {19}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {19}{2}} b^{3} x^{3}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {380 a^{\frac {17}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {516 a^{\frac {15}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {310 a^{\frac {13}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {70 a^{\frac {11}{2}} b^{7} x^{7} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 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. 156 vs.
\(2 (41) = 82\).
time = 0.00, size = 257, normalized size = 4.85 \begin {gather*} \frac {\frac {2 b^{2} \left (\frac {1}{9} \sqrt {a+b x} \left (a+b x\right )^{4}-\frac {4}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a+\frac {6}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{2}-\frac {4}{3} \sqrt {a+b x} \left (a+b x\right ) a^{3}+\sqrt {a+b x} a^{4}\right )}{b^{4}}+\frac {4 a b \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}+\frac {2 a^{2} \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {70\,{\left (a+b\,x\right )}^{9/2}-180\,a\,{\left (a+b\,x\right )}^{7/2}+126\,a^2\,{\left (a+b\,x\right )}^{5/2}}{315\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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