Integrand size = 18, antiderivative size = 149 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 a^4 (A b-a B) \sqrt {a+b x}}{b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{5/2}}{5 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 (A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6} \]
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Time = 0.04 (sec) , antiderivative size = 149, 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} \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 a^4 \sqrt {a+b x} (A b-a B)}{b^6}-\frac {2 a^3 (a+b x)^{3/2} (4 A b-5 a B)}{3 b^6}+\frac {4 a^2 (a+b x)^{5/2} (3 A b-5 a B)}{5 b^6}+\frac {2 (a+b x)^{9/2} (A b-5 a B)}{9 b^6}-\frac {4 a (a+b x)^{7/2} (2 A b-5 a B)}{7 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^4 (-A b+a B)}{b^5 \sqrt {a+b x}}+\frac {a^3 (-4 A b+5 a B) \sqrt {a+b x}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{3/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{5/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{7/2}}{b^5}+\frac {B (a+b x)^{9/2}}{b^5}\right ) \, dx \\ & = \frac {2 a^4 (A b-a B) \sqrt {a+b x}}{b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{5/2}}{5 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 (A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.71 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (-1280 a^5 B+128 a^4 b (11 A+5 B x)+35 b^5 x^4 (11 A+9 B x)-32 a^3 b^2 x (22 A+15 B x)+16 a^2 b^3 x^2 (33 A+25 B x)-10 a b^4 x^3 (44 A+35 B x)\right )}{3465 b^6} \]
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Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(\frac {256 \left (\frac {35 x^{4} \left (\frac {9 B x}{11}+A \right ) b^{5}}{128}-\frac {5 x^{3} a \left (\frac {35 B x}{44}+A \right ) b^{4}}{16}+\frac {3 x^{2} \left (\frac {25 B x}{33}+A \right ) a^{2} b^{3}}{8}-\frac {x \left (\frac {15 B x}{22}+A \right ) a^{3} b^{2}}{2}+a^{4} \left (\frac {5 B x}{11}+A \right ) b -\frac {10 a^{5} B}{11}\right ) \sqrt {b x +a}}{315 b^{6}}\) | \(92\) |
gosper | \(\frac {2 \sqrt {b x +a}\, \left (315 b^{5} B \,x^{5}+385 A \,b^{5} x^{4}-350 B a \,b^{4} x^{4}-440 A a \,b^{4} x^{3}+400 B \,a^{2} b^{3} x^{3}+528 A \,a^{2} b^{3} x^{2}-480 B \,a^{3} b^{2} x^{2}-704 a^{3} b^{2} A x +640 a^{4} b B x +1408 a^{4} b A -1280 a^{5} B \right )}{3465 b^{6}}\) | \(119\) |
trager | \(\frac {2 \sqrt {b x +a}\, \left (315 b^{5} B \,x^{5}+385 A \,b^{5} x^{4}-350 B a \,b^{4} x^{4}-440 A a \,b^{4} x^{3}+400 B \,a^{2} b^{3} x^{3}+528 A \,a^{2} b^{3} x^{2}-480 B \,a^{3} b^{2} x^{2}-704 a^{3} b^{2} A x +640 a^{4} b B x +1408 a^{4} b A -1280 a^{5} B \right )}{3465 b^{6}}\) | \(119\) |
risch | \(\frac {2 \sqrt {b x +a}\, \left (315 b^{5} B \,x^{5}+385 A \,b^{5} x^{4}-350 B a \,b^{4} x^{4}-440 A a \,b^{4} x^{3}+400 B \,a^{2} b^{3} x^{3}+528 A \,a^{2} b^{3} x^{2}-480 B \,a^{3} b^{2} x^{2}-704 a^{3} b^{2} A x +640 a^{4} b B x +1408 a^{4} b A -1280 a^{5} B \right )}{3465 b^{6}}\) | \(119\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{4} \left (A b -B a \right ) \sqrt {b x +a}}{b^{6}}\) | \(137\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{4} \left (A b -B a \right ) \sqrt {b x +a}}{b^{6}}\) | \(137\) |
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Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, B b^{5} x^{5} - 1280 \, B a^{5} + 1408 \, A a^{4} b - 35 \, {\left (10 \, B a b^{4} - 11 \, A b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{3465 \, b^{6}} \]
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Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {11}{2}}}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (A b - 5 B a\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{3 b} + \frac {\sqrt {a + b x} \left (A a^{4} b - B a^{5}\right )}{b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, {\left (b x + a\right )}^{\frac {11}{2}} B - 385 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 3465 \, {\left (B a^{5} - A a^{4} b\right )} \sqrt {b x + a}\right )}}{3465 \, b^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A}{b^{4}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B}{b^{5}}\right )}}{3465 \, b} \]
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Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,\sqrt {a+b\,x}}{b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6} \]
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