Integrand size = 18, antiderivative size = 151 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 a^4 (A b-a B) (a+b x)^{5/2}}{5 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{9/2}}{9 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 (A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \]
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Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 a^4 (a+b x)^{5/2} (A b-a B)}{5 b^6}-\frac {2 a^3 (a+b x)^{7/2} (4 A b-5 a B)}{7 b^6}+\frac {4 a^2 (a+b x)^{9/2} (3 A b-5 a B)}{9 b^6}+\frac {2 (a+b x)^{13/2} (A b-5 a B)}{13 b^6}-\frac {4 a (a+b x)^{11/2} (2 A b-5 a B)}{11 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^4 (-A b+a B) (a+b x)^{3/2}}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^{5/2}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{7/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{9/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{11/2}}{b^5}+\frac {B (a+b x)^{13/2}}{b^5}\right ) \, dx \\ & = \frac {2 a^4 (A b-a B) (a+b x)^{5/2}}{5 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{9/2}}{9 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 (A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 (a+b x)^{5/2} \left (-256 a^5 B+1680 a^2 b^3 x^2 (A+B x)+128 a^4 b (3 A+5 B x)-160 a^3 b^2 x (6 A+7 B x)-210 a b^4 x^3 (12 A+11 B x)+231 b^5 x^4 (15 A+13 B x)\right )}{45045 b^6} \]
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Time = 0.54 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(\frac {256 \left (b x +a \right )^{\frac {5}{2}} \left (\frac {1155 x^{4} \left (\frac {13 B x}{15}+A \right ) b^{5}}{128}-\frac {105 x^{3} \left (\frac {11 B x}{12}+A \right ) a \,b^{4}}{16}+\frac {35 a^{2} x^{2} \left (B x +A \right ) b^{3}}{8}-\frac {5 x \left (\frac {7 B x}{6}+A \right ) a^{3} b^{2}}{2}+a^{4} \left (\frac {5 B x}{3}+A \right ) b -\frac {2 a^{5} B}{3}\right )}{15015 b^{6}}\) | \(91\) |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (3003 b^{5} B \,x^{5}+3465 A \,b^{5} x^{4}-2310 B a \,b^{4} x^{4}-2520 A a \,b^{4} x^{3}+1680 B \,a^{2} b^{3} x^{3}+1680 A \,a^{2} b^{3} x^{2}-1120 B \,a^{3} b^{2} x^{2}-960 a^{3} b^{2} A x +640 a^{4} b B x +384 a^{4} b A -256 a^{5} B \right )}{45045 b^{6}}\) | \(119\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{6}}\) | \(138\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{6}}\) | \(138\) |
trager | \(\frac {2 \left (3003 B \,b^{7} x^{7}+3465 A \,b^{7} x^{6}+3696 B a \,b^{6} x^{6}+4410 A a \,b^{6} x^{5}+63 B \,a^{2} b^{5} x^{5}+105 A \,a^{2} b^{5} x^{4}-70 B \,a^{3} b^{4} x^{4}-120 A \,a^{3} b^{4} x^{3}+80 B \,a^{4} b^{3} x^{3}+144 A \,a^{4} b^{3} x^{2}-96 B \,a^{5} b^{2} x^{2}-192 A \,a^{5} b^{2} x +128 B \,a^{6} b x +384 A \,a^{6} b -256 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(167\) |
risch | \(\frac {2 \left (3003 B \,b^{7} x^{7}+3465 A \,b^{7} x^{6}+3696 B a \,b^{6} x^{6}+4410 A a \,b^{6} x^{5}+63 B \,a^{2} b^{5} x^{5}+105 A \,a^{2} b^{5} x^{4}-70 B \,a^{3} b^{4} x^{4}-120 A \,a^{3} b^{4} x^{3}+80 B \,a^{4} b^{3} x^{3}+144 A \,a^{4} b^{3} x^{2}-96 B \,a^{5} b^{2} x^{2}-192 A \,a^{5} b^{2} x +128 B \,a^{6} b x +384 A \,a^{6} b -256 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(167\) |
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Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, B b^{7} x^{7} - 256 \, B a^{7} + 384 \, A a^{6} b + 231 \, {\left (16 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \, {\left (B a^{2} b^{5} + 70 \, A a b^{6}\right )} x^{5} - 35 \, {\left (2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 40 \, {\left (2 \, B a^{4} b^{3} - 3 \, A a^{3} b^{4}\right )} x^{3} - 48 \, {\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )} x^{2} + 64 \, {\left (2 \, B a^{6} b - 3 \, A a^{5} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{6}} \]
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Time = 0.79 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {15}{2}}}{15 b} + \frac {\left (a + b x\right )^{\frac {13}{2}} \left (A b - 5 B a\right )}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A a^{4} b - B a^{5}\right )}{5 b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{5}}{5} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, {\left (b x + a\right )}^{\frac {15}{2}} B - 3465 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} + 8190 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 10010 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 9009 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{45045 \, b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (128) = 256\).
Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 3.27 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\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^{2}}{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^{2}}{b^{5}} + \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 )} A a}{b^{4}} + \frac {30 \, {\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 a}{b^{5}} + \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 )} A}{b^{4}} + \frac {7 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} B}{b^{5}}\right )}}{45045 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{15/2}}{15\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6} \]
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