Optimal. Leaf size=151 \[ \frac {2 a^4 (A b-a B) (a+b x)^{3/2}}{3 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{7/2}}{7 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {2 (A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 B (a+b x)^{13/2}}{13 b^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 151, 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^4 (a+b x)^{3/2} (A b-a B)}{3 b^6}-\frac {2 a^3 (a+b x)^{5/2} (4 A b-5 a B)}{5 b^6}+\frac {4 a^2 (a+b x)^{7/2} (3 A b-5 a B)}{7 b^6}+\frac {2 (a+b x)^{11/2} (A b-5 a B)}{11 b^6}-\frac {4 a (a+b x)^{9/2} (2 A b-5 a B)}{9 b^6}+\frac {2 B (a+b x)^{13/2}}{13 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int x^4 \sqrt {a+b x} (A+B x) \, dx &=\int \left (-\frac {a^4 (-A b+a B) \sqrt {a+b x}}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^{3/2}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{5/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{7/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{9/2}}{b^5}+\frac {B (a+b x)^{11/2}}{b^5}\right ) \, dx\\ &=\frac {2 a^4 (A b-a B) (a+b x)^{3/2}}{3 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{7/2}}{7 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {2 (A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 B (a+b x)^{13/2}}{13 b^6}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 106, normalized size = 0.70 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (-1280 a^5 B+315 b^5 x^4 (13 A+11 B x)+128 a^4 b (13 A+15 B x)-96 a^3 b^2 x (26 A+25 B x)+80 a^2 b^3 x^2 (39 A+35 B x)-70 a b^4 x^3 (52 A+45 B x)\right )}{45045 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 138, normalized size = 0.91
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (3465 b^{5} B \,x^{5}+4095 A \,b^{5} x^{4}-3150 B a \,b^{4} x^{4}-3640 A a \,b^{4} x^{3}+2800 B \,a^{2} b^{3} x^{3}+3120 A \,a^{2} b^{3} x^{2}-2400 B \,a^{3} b^{2} x^{2}-2496 a^{3} b^{2} A x +1920 a^{4} b B x +1664 a^{4} b A -1280 a^{5} B \right )}{45045 b^{6}}\) | \(119\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{4} B -4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{6}}\) | \(138\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{4} B -4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{6}}\) | \(138\) |
trager | \(\frac {2 \left (3465 B \,b^{6} x^{6}+4095 A \,b^{6} x^{5}+315 B a \,b^{5} x^{5}+455 A a \,b^{5} x^{4}-350 B \,a^{2} b^{4} x^{4}-520 A \,a^{2} b^{4} x^{3}+400 B \,a^{3} b^{3} x^{3}+624 A \,a^{3} b^{3} x^{2}-480 B \,a^{4} b^{2} x^{2}-832 A \,a^{4} b^{2} x +640 B \,a^{5} b x +1664 A \,a^{5} b -1280 B \,a^{6}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(143\) |
risch | \(\frac {2 \left (3465 B \,b^{6} x^{6}+4095 A \,b^{6} x^{5}+315 B a \,b^{5} x^{5}+455 A a \,b^{5} x^{4}-350 B \,a^{2} b^{4} x^{4}-520 A \,a^{2} b^{4} x^{3}+400 B \,a^{3} b^{3} x^{3}+624 A \,a^{3} b^{3} x^{2}-480 B \,a^{4} b^{2} x^{2}-832 A \,a^{4} b^{2} x +640 B \,a^{5} b x +1664 A \,a^{5} b -1280 B \,a^{6}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 4095 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 15015 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{45045 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 143, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (3465 \, B b^{6} x^{6} - 1280 \, B a^{6} + 1664 \, A a^{5} b + 315 \, {\left (B a b^{5} + 13 \, A b^{6}\right )} x^{5} - 35 \, {\left (10 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.65, size = 150, normalized size = 0.99 \begin {gather*} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {13}{2}}}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (A b - 5 B a\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A a^{4} b - B a^{5}\right )}{3 b}\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. 304 vs.
\(2 (128) = 256\).
time = 1.46, size = 304, normalized size = 2.01 \begin {gather*} \frac {2 \, {\left (\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 )} A 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 )} B a}{b^{5}} + \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^{4}} + \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^{5}}\right )}}{45045 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 137, normalized size = 0.91 \begin {gather*} \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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