Integrand size = 15, antiderivative size = 87 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=\frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {32 b^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {32 b^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {2}{a x^{5/2} \sqrt {a+b x}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {2}{a x^{5/2} \sqrt {a+b x}}+\frac {6 \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{a} \\ & = \frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}-\frac {(24 b) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a^2} \\ & = \frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}+\frac {\left (16 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{5 a^3} \\ & = \frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {32 b^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \left (a^3-2 a^2 b x+8 a b^2 x^2+16 b^3 x^3\right )}{5 a^4 x^{5/2} \sqrt {a+b x}} \]
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Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \left (16 b^{3} x^{3}+8 a \,b^{2} x^{2}-2 a^{2} b x +a^{3}\right )}{5 x^{\frac {5}{2}} \sqrt {b x +a}\, a^{4}}\) | \(44\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (11 b^{2} x^{2}-3 a b x +a^{2}\right )}{5 a^{4} x^{\frac {5}{2}}}-\frac {2 b^{3} \sqrt {x}}{a^{4} \sqrt {b x +a}}\) | \(52\) |
default | \(-\frac {2}{5 a \,x^{\frac {5}{2}} \sqrt {b x +a}}-\frac {6 b \left (-\frac {2}{3 a \,x^{\frac {3}{2}} \sqrt {b x +a}}-\frac {4 b \left (-\frac {2}{a \sqrt {x}\, \sqrt {b x +a}}-\frac {4 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\right )}{3 a}\right )}{5 a}\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (16 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {b x + a} \sqrt {x}}{5 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (82) = 164\).
Time = 6.50 (sec) , antiderivative size = 348, normalized size of antiderivative = 4.00 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=- \frac {2 a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {10 a^{3} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {60 a^{2} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {80 a b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {32 b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, b^{3} \sqrt {x}}{\sqrt {b x + a} a^{4}} - \frac {2 \, {\left (\frac {15 \, \sqrt {b x + a} b^{2}}{\sqrt {x}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{5 \, a^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {4 \, b^{\frac {9}{2}}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{3} {\left | b \right |}} - \frac {2 \, {\left (\frac {15 \, b^{6}}{a^{2} {\left | b \right |}} + {\left (b x + a\right )} {\left (\frac {11 \, {\left (b x + a\right )} b^{6}}{a^{4} {\left | b \right |}} - \frac {25 \, b^{6}}{a^{3} {\left | b \right |}}\right )}\right )} \sqrt {b x + a}}{5 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}}} \]
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Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2}{5\,a\,b}-\frac {4\,x}{5\,a^2}+\frac {16\,b\,x^2}{5\,a^3}+\frac {32\,b^2\,x^3}{5\,a^4}\right )}{x^{7/2}+\frac {a\,x^{5/2}}{b}} \]
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