Integrand size = 15, antiderivative size = 32 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 32, 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 = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^2}{x^7} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=\frac {-10 a^2-24 a b \sqrt {x}-15 b^2 x}{30 x^3} \]
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Time = 5.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a^{2}}{3 x^{3}}-\frac {4 a b}{5 x^{\frac {5}{2}}}-\frac {b^{2}}{2 x^{2}}\) | \(25\) |
default | \(-\frac {a^{2}}{3 x^{3}}-\frac {4 a b}{5 x^{\frac {5}{2}}}-\frac {b^{2}}{2 x^{2}}\) | \(25\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a^{2} x^{2}+3 b^{2} x^{2}+2 a^{2} x +3 b^{2} x +2 a^{2}\right )}{6 x^{3}}-\frac {4 a b}{5 x^{\frac {5}{2}}}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=- \frac {a^{2}}{3 x^{3}} - \frac {4 a b}{5 x^{\frac {5}{2}}} - \frac {b^{2}}{2 x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx=-\frac {15\,b^2\,x+10\,a^2+24\,a\,b\,\sqrt {x}}{30\,x^3} \]
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