Integrand size = 15, antiderivative size = 86 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )+5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}} \]
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+\frac {1}{3} (5 b) \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx \\ & = -\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx \\ & = 5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+\frac {1}{2} \left (5 a b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = 5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = 5 b^2 \sqrt {x} \sqrt {a+b x}-\frac {10 b (a+b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a+b x)^{5/2}}{3 x^{3/2}}+5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-2 a^2-14 a b x+3 b^2 x^2\right )}{3 x^{3/2}}+10 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-3 b^{2} x^{2}+14 a b x +2 a^{2}\right )}{3 x^{\frac {3}{2}}}+\frac {5 a \,b^{\frac {3}{2}} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{2 \sqrt {x}\, \sqrt {b x +a}}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=\left [\frac {15 \, a b^{\frac {3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{6 \, x^{2}}, -\frac {15 \, a \sqrt {-b} b x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, x^{2}}\right ] \]
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Time = 3.89 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )}}{2} + 5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )} + b^{\frac {5}{2}} x \sqrt {\frac {a}{b x} + 1} \]
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Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=-\frac {5}{2} \, a b^{\frac {3}{2}} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {4 \, \sqrt {b x + a} a b}{\sqrt {x}} - \frac {\sqrt {b x + a} a b^{2}}{{\left (b - \frac {b x + a}{x}\right )} \sqrt {x}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{3 \, x^{\frac {3}{2}}} \]
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Time = 77.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=-\frac {{\left (15 \, a b^{\frac {3}{2}} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) - \frac {{\left (15 \, a^{2} b^{3} + {\left (3 \, {\left (b x + a\right )} b^{3} - 20 \, a b^{3}\right )} {\left (b x + a\right )}\right )} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}}\right )} b}{3 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^{5/2}} \,d x \]
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