Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}-\frac {(4 b) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a} \\ & = -\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^2} \\ & = -\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (8 b^{2} x^{2}-4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(35\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (8 b^{2} x^{2}-4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(35\) |
default | \(-\frac {2 \sqrt {b x +a}}{5 a \,x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\) | \(55\) |
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Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )} \sqrt {b x + a}}{15 \, a^{3} x^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (63) = 126\).
Time = 4.69 (sec) , antiderivative size = 287, normalized size of antiderivative = 4.22 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=- \frac {6 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {4 a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {24 a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (\frac {15 \, \sqrt {b x + a} b^{2}}{\sqrt {x}} - \frac {10 \, {\left (b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{15 \, a^{3}} \]
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Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (\frac {15 \, b^{5}}{a} + 4 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{5}}{a^{3}} - \frac {5 \, b^{5}}{a^{2}}\right )} {\left (b x + a\right )}\right )} \sqrt {b x + a} b}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
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Time = 0.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2}{5\,a}+\frac {16\,b^2\,x^2}{15\,a^3}-\frac {8\,b\,x}{15\,a^2}\right )}{x^{5/2}} \]
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