Integrand size = 18, antiderivative size = 147 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=-\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}-\frac {2 a^3 (4 A b-5 a B) \sqrt {a+b x}}{b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{3/2}}{3 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {2 (A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6} \]
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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} \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=-\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}-\frac {2 a^3 \sqrt {a+b x} (4 A b-5 a B)}{b^6}+\frac {4 a^2 (a+b x)^{3/2} (3 A b-5 a B)}{3 b^6}+\frac {2 (a+b x)^{7/2} (A b-5 a B)}{7 b^6}-\frac {4 a (a+b x)^{5/2} (2 A b-5 a B)}{5 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6} \]
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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)^{3/2}}+\frac {a^3 (-4 A b+5 a B)}{b^5 \sqrt {a+b x}}-\frac {2 a^2 (-3 A b+5 a B) \sqrt {a+b x}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{3/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{5/2}}{b^5}+\frac {B (a+b x)^{7/2}}{b^5}\right ) \, dx \\ & = -\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}-\frac {2 a^3 (4 A b-5 a B) \sqrt {a+b x}}{b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{3/2}}{3 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {2 (A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2560 a^5 B-256 a^4 b (9 A-5 B x)+32 a^2 b^3 x^2 (9 A+5 B x)-64 a^3 b^2 x (18 A+5 B x)+10 b^5 x^4 (9 A+7 B x)-4 a b^4 x^3 (36 A+25 B x)}{315 b^6 \sqrt {a+b x}} \]
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Time = 0.53 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\left (70 B \,x^{5}+90 A \,x^{4}\right ) b^{5}-144 x^{3} \left (\frac {25 B x}{36}+A \right ) a \,b^{4}+288 x^{2} \left (\frac {5 B x}{9}+A \right ) a^{2} b^{3}-1152 x \left (\frac {5 B x}{18}+A \right ) a^{3} b^{2}-2304 a^{4} \left (-\frac {5 B x}{9}+A \right ) b +2560 a^{5} B}{315 \sqrt {b x +a}\, b^{6}}\) | \(96\) |
gosper | \(-\frac {2 \left (-35 b^{5} B \,x^{5}-45 A \,b^{5} x^{4}+50 B a \,b^{4} x^{4}+72 A a \,b^{4} x^{3}-80 B \,a^{2} b^{3} x^{3}-144 A \,a^{2} b^{3} x^{2}+160 B \,a^{3} b^{2} x^{2}+576 a^{3} b^{2} A x -640 a^{4} b B x +1152 a^{4} b A -1280 a^{5} B \right )}{315 \sqrt {b x +a}\, b^{6}}\) | \(119\) |
trager | \(-\frac {2 \left (-35 b^{5} B \,x^{5}-45 A \,b^{5} x^{4}+50 B a \,b^{4} x^{4}+72 A a \,b^{4} x^{3}-80 B \,a^{2} b^{3} x^{3}-144 A \,a^{2} b^{3} x^{2}+160 B \,a^{3} b^{2} x^{2}+576 a^{3} b^{2} A x -640 a^{4} b B x +1152 a^{4} b A -1280 a^{5} B \right )}{315 \sqrt {b x +a}\, b^{6}}\) | \(119\) |
risch | \(-\frac {2 \left (-35 B \,x^{4} b^{4}-45 A \,x^{3} b^{4}+85 B \,x^{3} a \,b^{3}+117 A \,x^{2} a \,b^{3}-165 B \,x^{2} a^{2} b^{2}-261 A x \,a^{2} b^{2}+325 B x \,a^{3} b +837 A \,a^{3} b -965 B \,a^{4}\right ) \sqrt {b x +a}}{315 b^{6}}-\frac {2 a^{4} \left (A b -B a \right )}{b^{6} \sqrt {b x +a}}\) | \(119\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 A b \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {10 B a \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {8 A a b \left (b x +a \right )^{\frac {5}{2}}}{5}+4 B \,a^{2} \left (b x +a \right )^{\frac {5}{2}}+4 A \,a^{2} b \left (b x +a \right )^{\frac {3}{2}}-\frac {20 B \,a^{3} \left (b x +a \right )^{\frac {3}{2}}}{3}-8 A \,a^{3} b \sqrt {b x +a}+10 a^{4} B \sqrt {b x +a}-\frac {2 a^{4} \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{6}}\) | \(138\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 A b \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {10 B a \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {8 A a b \left (b x +a \right )^{\frac {5}{2}}}{5}+4 B \,a^{2} \left (b x +a \right )^{\frac {5}{2}}+4 A \,a^{2} b \left (b x +a \right )^{\frac {3}{2}}-\frac {20 B \,a^{3} \left (b x +a \right )^{\frac {3}{2}}}{3}-8 A \,a^{3} b \sqrt {b x +a}+10 a^{4} B \sqrt {b x +a}-\frac {2 a^{4} \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{6}}\) | \(138\) |
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Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, B b^{5} x^{5} + 1280 \, B a^{5} - 1152 \, A a^{4} b - 5 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} x^{4} + 8 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} - 16 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{315 \, {\left (b^{7} x + a b^{6}\right )}} \]
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Time = 1.69 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {9}{2}}}{9 b} + \frac {a^{4} \left (- A b + B a\right )}{b \sqrt {a + b x}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (A b - 5 B a\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{3 b} + \frac {\sqrt {a + b x} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} B - 45 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 126 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 210 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \sqrt {b x + a}}{b} + \frac {315 \, {\left (B a^{5} - A a^{4} b\right )}}{\sqrt {b x + a} b}\right )}}{315 \, b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.13 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (B a^{5} - A a^{4} b\right )}}{\sqrt {b x + a} b^{6}} + \frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} B b^{48} - 225 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{48} + 630 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{48} - 1050 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{48} + 1575 \, \sqrt {b x + a} B a^{4} b^{48} + 45 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{49} - 252 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{49} + 630 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{49} - 1260 \, \sqrt {b x + a} A a^{3} b^{49}\right )}}{315 \, b^{54}} \]
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Time = 0.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {2\,B\,a^5-2\,A\,a^4\,b}{b^6\,\sqrt {a+b\,x}}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,\sqrt {a+b\,x}}{b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6} \]
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