Integrand size = 20, antiderivative size = 84 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{5 a x^{5/2}}+\frac {2 (4 A b-5 a B) \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {4 b (4 A b-5 a B) \sqrt {a+b x}}{15 a^3 \sqrt {x}} \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {4 b \sqrt {a+b x} (4 A b-5 a B)}{15 a^3 \sqrt {x}}+\frac {2 \sqrt {a+b x} (4 A b-5 a B)}{15 a^2 x^{3/2}}-\frac {2 A \sqrt {a+b x}}{5 a x^{5/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b x}}{5 a x^{5/2}}+\frac {\left (2 \left (-2 A b+\frac {5 a B}{2}\right )\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a} \\ & = -\frac {2 A \sqrt {a+b x}}{5 a x^{5/2}}+\frac {2 (4 A b-5 a B) \sqrt {a+b x}}{15 a^2 x^{3/2}}+\frac {(2 b (4 A b-5 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^2} \\ & = -\frac {2 A \sqrt {a+b x}}{5 a x^{5/2}}+\frac {2 (4 A b-5 a B) \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {4 b (4 A b-5 a B) \sqrt {a+b x}}{15 a^3 \sqrt {x}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (3 a^2 A-4 a A b x+5 a^2 B x+8 A b^2 x^2-10 a b B x^2\right )}{15 a^3 x^{5/2}} \]
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Time = 0.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} x^{2}-10 B a b \,x^{2}-4 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(53\) |
default | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} x^{2}-10 B a b \,x^{2}-4 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(53\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} x^{2}-10 B a b \,x^{2}-4 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(53\) |
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Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (3 \, A a^{2} - 2 \, {\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt {b x + a}}{15 \, a^{3} x^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (82) = 164\).
Time = 4.51 (sec) , antiderivative size = 342, normalized size of antiderivative = 4.07 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=- \frac {6 A a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {4 A a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 A a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {24 A a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 A b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {2 B \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 a x} + \frac {4 B b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=\frac {4 \, \sqrt {b x^{2} + a x} B b}{3 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{2}}{15 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B}{3 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b}{15 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{5 \, a x^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=\frac {2 \, \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} {\left (b x + a\right )}}{a^{3}} - \frac {5 \, {\left (5 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )}}{a^{3}}\right )} + \frac {15 \, {\left (B a^{3} b^{4} - A a^{2} b^{5}\right )}}{a^{3}}\right )} b}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
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Time = 0.82 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5\,a}+\frac {x^2\,\left (16\,A\,b^2-20\,B\,a\,b\right )}{15\,a^3}+\frac {x\,\left (10\,B\,a^2-8\,A\,a\,b\right )}{15\,a^3}\right )}{x^{5/2}} \]
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