Integrand size = 15, antiderivative size = 93 \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=-\frac {\sqrt {(a+b x)^5}}{a x}+\frac {3 b \left (2 \sqrt {a+b x} \left (a+\frac {1}{3} (a+b x)\right )+a^{3/2} \log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )\right )}{2 a} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1973, 43, 52, 65, 214} \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=-\frac {3 b \sqrt {(a+b x)^3} \text {arctanh}\left (\sqrt {\frac {b x}{a}+1}\right )}{a \left (\frac {b x}{a}+1\right )^{3/2}}+\frac {3 b \sqrt {(a+b x)^3}}{a+b x}-\frac {\sqrt {(a+b x)^3}}{x} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {(a+b x)^3} \int \frac {\left (1+\frac {b x}{a}\right )^{3/2}}{x^2} \, dx}{\left (1+\frac {b x}{a}\right )^{3/2}} \\ & = -\frac {\sqrt {(a+b x)^3}}{x}+\frac {\left (3 b \sqrt {(a+b x)^3}\right ) \int \frac {\sqrt {1+\frac {b x}{a}}}{x} \, dx}{2 a \left (1+\frac {b x}{a}\right )^{3/2}} \\ & = -\frac {\sqrt {(a+b x)^3}}{x}+\frac {3 b \sqrt {(a+b x)^3}}{a+b x}+\frac {\left (3 b \sqrt {(a+b x)^3}\right ) \int \frac {1}{x \sqrt {1+\frac {b x}{a}}} \, dx}{2 a \left (1+\frac {b x}{a}\right )^{3/2}} \\ & = -\frac {\sqrt {(a+b x)^3}}{x}+\frac {3 b \sqrt {(a+b x)^3}}{a+b x}+\frac {\left (3 \sqrt {(a+b x)^3}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {a x^2}{b}} \, dx,x,\sqrt {1+\frac {b x}{a}}\right )}{\left (1+\frac {b x}{a}\right )^{3/2}} \\ & = -\frac {\sqrt {(a+b x)^3}}{x}+\frac {3 b \sqrt {(a+b x)^3}}{a+b x}-\frac {3 b \sqrt {(a+b x)^3} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=-\frac {\sqrt {(a+b x)^3} \left ((a-2 b x) \sqrt {a+b x}+3 \sqrt {a} b x \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{x (a+b x)^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {\sqrt {\left (b x +a \right )^{3}}\, \left (-2 b x \sqrt {b x +a}\, \sqrt {a}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a b x +\sqrt {b x +a}\, a^{\frac {3}{2}}\right )}{\left (b x +a \right )^{\frac {3}{2}} x \sqrt {a}}\) | \(68\) |
risch | \(-\frac {a \sqrt {\left (b x +a \right )^{3}}}{\left (b x +a \right ) x}+\frac {b \left (4 \sqrt {b x +a}-6 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right ) \sqrt {\left (b x +a \right )^{3}}}{2 \left (b x +a \right )^{\frac {3}{2}}}\) | \(70\) |
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Time = 0.25 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} + a b x\right )} \sqrt {a} \log \left (\frac {b^{2} x^{2} + 3 \, a b x + 2 \, a^{2} - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {a}}{b x^{2} + a x}\right ) + 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (2 \, b x - a\right )}}{2 \, {\left (b x^{2} + a x\right )}}, \frac {3 \, {\left (b^{2} x^{2} + a b x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {-a}}{a b x + a^{2}}\right ) + \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (2 \, b x - a\right )}}{b x^{2} + a x}\right ] \]
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\[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{3}}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=\int { \frac {\sqrt {{\left (b x + a\right )}^{3}}}{x^{2}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=\frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b x + a} b^{2} - \frac {\sqrt {b x + a} a b}{x}}{b} \]
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Timed out. \[ \int \frac {\sqrt {(a+b x)^3}}{x^2} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^3}}{x^2} \,d x \]
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