Integrand size = 18, antiderivative size = 147 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {2 a^4 (A b-a B)}{3 b^6 (a+b x)^{3/2}}+\frac {2 a^3 (4 A b-5 a B)}{b^6 \sqrt {a+b x}}+\frac {4 a^2 (3 A b-5 a B) \sqrt {a+b x}}{b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac {2 (A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {2 B (a+b x)^{7/2}}{7 b^6} \]
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Time = 0.04 (sec) , antiderivative size = 147, 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} \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {2 a^4 (A b-a B)}{3 b^6 (a+b x)^{3/2}}+\frac {2 a^3 (4 A b-5 a B)}{b^6 \sqrt {a+b x}}+\frac {4 a^2 \sqrt {a+b x} (3 A b-5 a B)}{b^6}-\frac {4 a (a+b x)^{3/2} (2 A b-5 a B)}{3 b^6}+\frac {2 (a+b x)^{5/2} (A b-5 a B)}{5 b^6}+\frac {2 B (a+b x)^{7/2}}{7 b^6} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^4 (-A b+a B)}{b^5 (a+b x)^{5/2}}+\frac {a^3 (-4 A b+5 a B)}{b^5 (a+b x)^{3/2}}-\frac {2 a^2 (-3 A b+5 a B)}{b^5 \sqrt {a+b x}}+\frac {2 a (-2 A b+5 a B) \sqrt {a+b x}}{b^5}+\frac {(A b-5 a B) (a+b x)^{3/2}}{b^5}+\frac {B (a+b x)^{5/2}}{b^5}\right ) \, dx \\ & = -\frac {2 a^4 (A b-a B)}{3 b^6 (a+b x)^{3/2}}+\frac {2 a^3 (4 A b-5 a B)}{b^6 \sqrt {a+b x}}+\frac {4 a^2 (3 A b-5 a B) \sqrt {a+b x}}{b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac {2 (A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {2 B (a+b x)^{7/2}}{7 b^6} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {-2560 a^5 B+256 a^4 b (7 A-15 B x)+192 a^3 b^2 x (14 A-5 B x)+6 b^5 x^4 (7 A+5 B x)+32 a^2 b^3 x^2 (21 A+5 B x)-4 a b^4 x^3 (28 A+15 B x)}{105 b^6 (a+b x)^{3/2}} \]
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Time = 0.54 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\left (30 B \,x^{5}+42 A \,x^{4}\right ) b^{5}-112 x^{3} \left (\frac {15 B x}{28}+A \right ) a \,b^{4}+672 x^{2} \left (\frac {5 B x}{21}+A \right ) a^{2} b^{3}+2688 x \,a^{3} \left (-\frac {5 B x}{14}+A \right ) b^{2}+1792 a^{4} \left (-\frac {15 B x}{7}+A \right ) b -2560 a^{5} B}{105 \left (b x +a \right )^{\frac {3}{2}} b^{6}}\) | \(96\) |
risch | \(\frac {2 \left (15 b^{3} B \,x^{3}+21 A \,b^{3} x^{2}-60 B a \,b^{2} x^{2}-98 a \,b^{2} A x +185 a^{2} b B x +511 a^{2} b A -790 a^{3} B \right ) \sqrt {b x +a}}{105 b^{6}}+\frac {2 a^{3} \left (12 A \,b^{2} x -15 B a b x +11 a b A -14 a^{2} B \right )}{3 b^{6} \left (b x +a \right )^{\frac {3}{2}}}\) | \(112\) |
gosper | \(\frac {\frac {2}{7} b^{5} B \,x^{5}+\frac {2}{5} A \,b^{5} x^{4}-\frac {4}{7} B a \,b^{4} x^{4}-\frac {16}{15} A a \,b^{4} x^{3}+\frac {32}{21} B \,a^{2} b^{3} x^{3}+\frac {32}{5} A \,a^{2} b^{3} x^{2}-\frac {64}{7} B \,a^{3} b^{2} x^{2}+\frac {128}{5} a^{3} b^{2} A x -\frac {256}{7} a^{4} b B x +\frac {256}{15} a^{4} b A -\frac {512}{21} a^{5} B}{\left (b x +a \right )^{\frac {3}{2}} b^{6}}\) | \(119\) |
trager | \(\frac {\frac {2}{7} b^{5} B \,x^{5}+\frac {2}{5} A \,b^{5} x^{4}-\frac {4}{7} B a \,b^{4} x^{4}-\frac {16}{15} A a \,b^{4} x^{3}+\frac {32}{21} B \,a^{2} b^{3} x^{3}+\frac {32}{5} A \,a^{2} b^{3} x^{2}-\frac {64}{7} B \,a^{3} b^{2} x^{2}+\frac {128}{5} a^{3} b^{2} A x -\frac {256}{7} a^{4} b B x +\frac {256}{15} a^{4} b A -\frac {512}{21} a^{5} B}{\left (b x +a \right )^{\frac {3}{2}} b^{6}}\) | \(119\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}-2 B a \left (b x +a \right )^{\frac {5}{2}}-\frac {8 A b a \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {20 B \,a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}+12 A \,a^{2} b \sqrt {b x +a}-20 B \,a^{3} \sqrt {b x +a}+\frac {2 a^{3} \left (4 A b -5 B a \right )}{\sqrt {b x +a}}-\frac {2 a^{4} \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{6}}\) | \(131\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}-2 B a \left (b x +a \right )^{\frac {5}{2}}-\frac {8 A b a \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {20 B \,a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}+12 A \,a^{2} b \sqrt {b x +a}-20 B \,a^{3} \sqrt {b x +a}+\frac {2 a^{3} \left (4 A b -5 B a \right )}{\sqrt {b x +a}}-\frac {2 a^{4} \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{6}}\) | \(131\) |
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Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.96 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (15 \, B b^{5} x^{5} - 1280 \, B a^{5} + 896 \, A a^{4} b - 3 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{4} + 8 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 192 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{105 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
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Time = 1.74 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {7}{2}}}{7 b} + \frac {a^{4} \left (- A b + B a\right )}{3 b \left (a + b x\right )^{\frac {3}{2}}} - \frac {a^{3} \left (- 4 A b + 5 B a\right )}{b \sqrt {a + b x}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b - 5 B a\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{3 b} + \frac {\sqrt {a + b x} \left (6 A a^{2} b - 10 B a^{3}\right )}{b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (b x + a\right )}^{\frac {7}{2}} B - 21 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 70 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 210 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {b x + a}}{b} + \frac {35 \, {\left (B a^{5} - A a^{4} b - 3 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} b}\right )}}{105 \, b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.07 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (b x + a\right )} B a^{4} - B a^{5} - 12 \, {\left (b x + a\right )} A a^{3} b + A a^{4} b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{6}} + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} B b^{36} - 105 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{36} + 350 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{36} - 1050 \, \sqrt {b x + a} B a^{3} b^{36} + 21 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{37} - 140 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{37} + 630 \, \sqrt {b x + a} A a^{2} b^{37}\right )}}{105 \, b^{42}} \]
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Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}-\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,\left (a+b\,x\right )-\frac {2\,B\,a^5}{3}+\frac {2\,A\,a^4\,b}{3}}{b^6\,{\left (a+b\,x\right )}^{3/2}}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,\sqrt {a+b\,x}}{b^6} \]
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