Optimal. Leaf size=72 \[ -\frac {2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac {6 a^2 (a+b x)^{7/2}}{7 b^4}-\frac {2 a (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{11/2}}{11 b^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, 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^3 (a+b x)^{5/2}}{5 b^4}+\frac {6 a^2 (a+b x)^{7/2}}{7 b^4}+\frac {2 (a+b x)^{11/2}}{11 b^4}-\frac {2 a (a+b x)^{9/2}}{3 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^3 (a+b x)^{3/2} \, dx &=\int \left (-\frac {a^3 (a+b x)^{3/2}}{b^3}+\frac {3 a^2 (a+b x)^{5/2}}{b^3}-\frac {3 a (a+b x)^{7/2}}{b^3}+\frac {(a+b x)^{9/2}}{b^3}\right ) \, dx\\ &=-\frac {2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac {6 a^2 (a+b x)^{7/2}}{7 b^4}-\frac {2 a (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{11/2}}{11 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.64 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-16 a^3+40 a^2 b x-70 a b^2 x^2+105 b^3 x^3\right )}{1155 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 50, normalized size = 0.69
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-105 b^{3} x^{3}+70 a \,b^{2} x^{2}-40 a^{2} b x +16 a^{3}\right )}{1155 b^{4}}\) | \(43\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {2 a \left (b x +a \right )^{\frac {9}{2}}}{3}+\frac {6 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{4}}\) | \(50\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {2 a \left (b x +a \right )^{\frac {9}{2}}}{3}+\frac {6 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{4}}\) | \(50\) |
trager | \(-\frac {2 \left (-105 b^{5} x^{5}-140 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}+6 a^{3} b^{2} x^{2}-8 a^{4} b x +16 a^{5}\right ) \sqrt {b x +a}}{1155 b^{4}}\) | \(65\) |
risch | \(-\frac {2 \left (-105 b^{5} x^{5}-140 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}+6 a^{3} b^{2} x^{2}-8 a^{4} b x +16 a^{5}\right ) \sqrt {b x +a}}{1155 b^{4}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 56, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}}}{11 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a}{3 \, b^{4}} + \frac {6 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{5 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 64, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x + a}}{1155 \, b^{4}} \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. 1742 vs.
\(2 (68) = 136\).
time = 1.27, size = 1742, normalized size = 24.19 \begin {gather*} - \frac {32 a^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {32 a^{\frac {51}{2}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} - \frac {176 a^{\frac {49}{2}} b x \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {192 a^{\frac {49}{2}} b x}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} - \frac {396 a^{\frac {47}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {480 a^{\frac {47}{2}} b^{2} x^{2}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} - \frac {462 a^{\frac {45}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {640 a^{\frac {45}{2}} b^{3} x^{3}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {480 a^{\frac {43}{2}} b^{4} x^{4}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {1848 a^{\frac {41}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {192 a^{\frac {41}{2}} b^{5} x^{5}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {5544 a^{\frac {39}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {32 a^{\frac {39}{2}} b^{6} x^{6}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {8844 a^{\frac {37}{2}} b^{7} x^{7} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {8448 a^{\frac {35}{2}} b^{8} x^{8} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {4840 a^{\frac {33}{2}} b^{9} x^{9} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {1540 a^{\frac {31}{2}} b^{10} x^{10} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} + \frac {210 a^{\frac {29}{2}} b^{11} x^{11} \sqrt {1 + \frac {b x}{a}}}{1155 a^{20} b^{4} + 6930 a^{19} b^{5} x + 17325 a^{18} b^{6} x^{2} + 23100 a^{17} b^{7} x^{3} + 17325 a^{16} b^{8} x^{4} + 6930 a^{15} b^{9} x^{5} + 1155 a^{14} b^{10} x^{6}} \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. 193 vs.
\(2 (56) = 112\).
time = 0.69, size = 193, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left (\frac {99 \, {\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 )} a^{2}}{b^{3}} + \frac {22 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a}{b^{3}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )}}{b^{3}}\right )}}{3465 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {6\,a^2\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}-\frac {2\,a\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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