Integrand size = 39, antiderivative size = 46 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {662} \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rule 662
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{c d \sqrt {(a e+c d x) (d+e x)}} \]
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Time = 0.53 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{\sqrt {e x +d}\, \left (c d x +a e \right ) c d}\) | \(42\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{c d \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(50\) |
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Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x} \]
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\[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.39 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2}{\sqrt {c d x + a e} c d} \]
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Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{2}}{\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d {\left | e \right |}} + \frac {2 \, e^{2}}{\sqrt {-c d^{2} e + a e^{3}} c d {\left | e \right |}} \]
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Time = 12.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.78 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2\,\sqrt {d+e\,x}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{c^2\,d^2\,e\,\left (\frac {a}{c}+x^2+\frac {x\,\left (c^2\,d^3+a\,c\,d\,e^2\right )}{c^2\,d^2\,e}\right )} \]
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