Optimal. Leaf size=122 \[ -\frac {2 a^3 (A b-a B) (a+b x)^{5/2}}{5 b^5}+\frac {2 a^2 (3 A b-4 a B) (a+b x)^{7/2}}{7 b^5}-\frac {2 a (A b-2 a B) (a+b x)^{9/2}}{3 b^5}+\frac {2 (A b-4 a B) (a+b x)^{11/2}}{11 b^5}+\frac {2 B (a+b x)^{13/2}}{13 b^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78}
\begin {gather*} -\frac {2 a^3 (a+b x)^{5/2} (A b-a B)}{5 b^5}+\frac {2 a^2 (a+b x)^{7/2} (3 A b-4 a B)}{7 b^5}+\frac {2 (a+b x)^{11/2} (A b-4 a B)}{11 b^5}-\frac {2 a (a+b x)^{9/2} (A b-2 a B)}{3 b^5}+\frac {2 B (a+b x)^{13/2}}{13 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int x^3 (a+b x)^{3/2} (A+B x) \, dx &=\int \left (\frac {a^3 (-A b+a B) (a+b x)^{3/2}}{b^4}-\frac {a^2 (-3 A b+4 a B) (a+b x)^{5/2}}{b^4}+\frac {3 a (-A b+2 a B) (a+b x)^{7/2}}{b^4}+\frac {(A b-4 a B) (a+b x)^{9/2}}{b^4}+\frac {B (a+b x)^{11/2}}{b^4}\right ) \, dx\\ &=-\frac {2 a^3 (A b-a B) (a+b x)^{5/2}}{5 b^5}+\frac {2 a^2 (3 A b-4 a B) (a+b x)^{7/2}}{7 b^5}-\frac {2 a (A b-2 a B) (a+b x)^{9/2}}{3 b^5}+\frac {2 (A b-4 a B) (a+b x)^{11/2}}{11 b^5}+\frac {2 B (a+b x)^{13/2}}{13 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 87, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (128 a^4 B+105 b^4 x^3 (13 A+11 B x)-70 a b^3 x^2 (13 A+12 B x)+40 a^2 b^2 x (13 A+14 B x)-16 a^3 b (13 A+20 B x)\right )}{15015 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 110, normalized size = 0.90
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-1155 B \,x^{4} b^{4}-1365 A \,b^{4} x^{3}+840 B a \,b^{3} x^{3}+910 A a \,b^{3} x^{2}-560 B \,a^{2} b^{2} x^{2}-520 A \,a^{2} b^{2} x +320 B \,a^{3} b x +208 A \,a^{3} b -128 a^{4} B \right )}{15015 b^{5}}\) | \(95\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -4 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} B -3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-a^{3} B +3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{5}}\) | \(110\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -4 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} B -3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-a^{3} B +3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{5}}\) | \(110\) |
trager | \(-\frac {2 \left (-1155 B \,b^{6} x^{6}-1365 A \,b^{6} x^{5}-1470 B a \,b^{5} x^{5}-1820 A a \,b^{5} x^{4}-35 B \,a^{2} b^{4} x^{4}-65 A \,a^{2} b^{4} x^{3}+40 B \,a^{3} b^{3} x^{3}+78 A \,a^{3} b^{3} x^{2}-48 B \,a^{4} b^{2} x^{2}-104 A \,a^{4} b^{2} x +64 B \,a^{5} b x +208 A \,a^{5} b -128 B \,a^{6}\right ) \sqrt {b x +a}}{15015 b^{5}}\) | \(143\) |
risch | \(-\frac {2 \left (-1155 B \,b^{6} x^{6}-1365 A \,b^{6} x^{5}-1470 B a \,b^{5} x^{5}-1820 A a \,b^{5} x^{4}-35 B \,a^{2} b^{4} x^{4}-65 A \,a^{2} b^{4} x^{3}+40 B \,a^{3} b^{3} x^{3}+78 A \,a^{3} b^{3} x^{2}-48 B \,a^{4} b^{2} x^{2}-104 A \,a^{4} b^{2} x +64 B \,a^{5} b x +208 A \,a^{5} b -128 B \,a^{6}\right ) \sqrt {b x +a}}{15015 b^{5}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 100, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (1155 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 1365 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 5005 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 2145 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 3003 \, {\left (B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{15015 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 143, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (1155 \, B b^{6} x^{6} + 128 \, B a^{6} - 208 \, A a^{5} b + 105 \, {\left (14 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 35 \, {\left (B a^{2} b^{4} + 52 \, A a b^{5}\right )} x^{4} - 5 \, {\left (8 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x + a}}{15015 \, b^{5}} \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. 298 vs.
\(2 (121) = 242\).
time = 5.84, size = 298, normalized size = 2.44 \begin {gather*} \frac {2 A a \left (- \frac {a^{3} \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {3 a^{2} \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {3 a \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a + b x\right )^{\frac {9}{2}}}{9}\right )}{b^{4}} + \frac {2 A \left (\frac {a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{3} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {4 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{4}} + \frac {2 B a \left (\frac {a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{3} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {4 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{5}} + \frac {2 B \left (- \frac {a^{5} \left (a + b x\right )^{\frac {3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac {5}{2}} - \frac {10 a^{3} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {10 a^{2} \left (a + b x\right )^{\frac {9}{2}}}{9} - \frac {5 a \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a + b x\right )^{\frac {13}{2}}}{13}\right )}{b^{5}} \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. 422 vs.
\(2 (104) = 208\).
time = 1.44, size = 422, normalized size = 3.46 \begin {gather*} \frac {2 \, {\left (\frac {1287 \, {\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 a^{2}}{b^{3}} + \frac {143 \, {\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 )} B a^{2}}{b^{4}} + \frac {286 \, {\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 a}{b^{3}} + \frac {130 \, {\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 a}{b^{4}} + \frac {65 \, {\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 )} A}{b^{3}} + \frac {15 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} B}{b^{4}}\right )}}{45045 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 111, normalized size = 0.91 \begin {gather*} \frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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