Integrand size = 15, antiderivative size = 89 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=\frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {15}{4} a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 89, 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} \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=\frac {15}{4} a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}+\frac {15}{4} a b \sqrt {x} \sqrt {a+b x} \]
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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}}{\sqrt {x}}+(5 b) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx \\ & = \frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} (15 a b) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx \\ & = \frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{8} \left (15 a^2 b\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = \frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = \frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )}{4 \sqrt {x}}+\frac {15}{2} a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-2 b^{2} x^{2}-9 a b x +8 a^{2}\right )}{4 \sqrt {x}}+\frac {15 a^{2} \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{8 \sqrt {x}\, \sqrt {b x +a}}\) | \(84\) |
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Time = 0.24 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=\left [\frac {15 \, a^{2} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{8 \, x}, -\frac {15 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{4 \, x}\right ] \]
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Time = 5.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=- \frac {2 a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + \frac {a^{\frac {3}{2}} b \sqrt {x}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {11 \sqrt {a} b^{2} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4} + \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
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Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=-\frac {15}{8} \, a^{2} \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a} a^{2}}{\sqrt {x}} - \frac {\frac {7 \, \sqrt {b x + a} a^{2} b^{2}}{\sqrt {x}} - \frac {9 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x + a\right )} b}{x} + \frac {{\left (b x + a\right )}^{2}}{x^{2}}\right )}} \]
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Time = 75.94 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=-\frac {{\left (\frac {15 \, a^{2} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} - \frac {{\left ({\left (2 \, b x + 7 \, a\right )} {\left (b x + a\right )} - 15 \, a^{2}\right )} \sqrt {b x + a}}{\sqrt {{\left (b x + a\right )} b - a b}}\right )} b^{2}}{4 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^{3/2}} \,d x \]
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