Integrand size = 18, antiderivative size = 120 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=-\frac {2 a^3 (A b-a B) \sqrt {a+b x}}{b^5}+\frac {2 a^2 (3 A b-4 a B) (a+b x)^{3/2}}{3 b^5}-\frac {6 a (A b-2 a B) (a+b x)^{5/2}}{5 b^5}+\frac {2 (A b-4 a B) (a+b x)^{7/2}}{7 b^5}+\frac {2 B (a+b x)^{9/2}}{9 b^5} \]
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Time = 0.03 (sec) , antiderivative size = 120, 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} \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=-\frac {2 a^3 \sqrt {a+b x} (A b-a B)}{b^5}+\frac {2 a^2 (a+b x)^{3/2} (3 A b-4 a B)}{3 b^5}+\frac {2 (a+b x)^{7/2} (A b-4 a B)}{7 b^5}-\frac {6 a (a+b x)^{5/2} (A b-2 a B)}{5 b^5}+\frac {2 B (a+b x)^{9/2}}{9 b^5} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 (-A b+a B)}{b^4 \sqrt {a+b x}}-\frac {a^2 (-3 A b+4 a B) \sqrt {a+b x}}{b^4}+\frac {3 a (-A b+2 a B) (a+b x)^{3/2}}{b^4}+\frac {(A b-4 a B) (a+b x)^{5/2}}{b^4}+\frac {B (a+b x)^{7/2}}{b^4}\right ) \, dx \\ & = -\frac {2 a^3 (A b-a B) \sqrt {a+b x}}{b^5}+\frac {2 a^2 (3 A b-4 a B) (a+b x)^{3/2}}{3 b^5}-\frac {6 a (A b-2 a B) (a+b x)^{5/2}}{5 b^5}+\frac {2 (A b-4 a B) (a+b x)^{7/2}}{7 b^5}+\frac {2 B (a+b x)^{9/2}}{9 b^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (128 a^4 B+24 a^2 b^2 x (3 A+2 B x)-16 a^3 b (9 A+4 B x)+5 b^4 x^3 (9 A+7 B x)-2 a b^3 x^2 (27 A+20 B x)\right )}{315 b^5} \]
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Time = 0.55 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(-\frac {32 \sqrt {b x +a}\, \left (-\frac {5 x^{3} \left (\frac {7 B x}{9}+A \right ) b^{4}}{16}+\frac {3 x^{2} \left (\frac {20 B x}{27}+A \right ) a \,b^{3}}{8}-\frac {x \left (\frac {2 B x}{3}+A \right ) a^{2} b^{2}}{2}+a^{3} \left (\frac {4 B x}{9}+A \right ) b -\frac {8 B \,a^{4}}{9}\right )}{35 b^{5}}\) | \(75\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-35 B \,x^{4} b^{4}-45 A \,x^{3} b^{4}+40 B \,x^{3} a \,b^{3}+54 A \,x^{2} a \,b^{3}-48 B \,x^{2} a^{2} b^{2}-72 A x \,a^{2} b^{2}+64 B x \,a^{3} b +144 A \,a^{3} b -128 B \,a^{4}\right )}{315 b^{5}}\) | \(95\) |
trager | \(-\frac {2 \sqrt {b x +a}\, \left (-35 B \,x^{4} b^{4}-45 A \,x^{3} b^{4}+40 B \,x^{3} a \,b^{3}+54 A \,x^{2} a \,b^{3}-48 B \,x^{2} a^{2} b^{2}-72 A x \,a^{2} b^{2}+64 B x \,a^{3} b +144 A \,a^{3} b -128 B \,a^{4}\right )}{315 b^{5}}\) | \(95\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-35 B \,x^{4} b^{4}-45 A \,x^{3} b^{4}+40 B \,x^{3} a \,b^{3}+54 A \,x^{2} a \,b^{3}-48 B \,x^{2} a^{2} b^{2}-72 A x \,a^{2} b^{2}+64 B x \,a^{3} b +144 A \,a^{3} b -128 B \,a^{4}\right )}{315 b^{5}}\) | \(95\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A b -4 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 a^{2} B -3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-a^{3} B +3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{3} \left (A b -B a \right ) \sqrt {b x +a}}{b^{5}}\) | \(110\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-A b +4 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 \left (-3 a^{2} B +3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 \left (a^{3} B -3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{3} \left (A b -B a \right ) \sqrt {b x +a}}{b^{5}}\) | \(110\) |
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Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, B b^{4} x^{4} + 128 \, B a^{4} - 144 \, A a^{3} b - 5 \, {\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x + a}}{315 \, b^{5}} \]
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Time = 0.59 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {9}{2}}}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (A b - 4 B a\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \cdot \left (3 A a^{2} b - 4 B a^{3}\right )}{3 b} + \frac {\sqrt {a + b x} \left (- A a^{3} b + B a^{4}\right )}{b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{4}}{4} + \frac {B x^{5}}{5}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} B - 45 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 189 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {b x + a}\right )}}{315 \, b^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B}{b^{4}}\right )}}{315 \, b} \]
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Time = 0.39 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (A+B x)}{\sqrt {a+b x}} \, dx=\frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,\sqrt {a+b\,x}}{b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^5} \]
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