Integrand size = 32, antiderivative size = 39 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {656, 623} \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
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Rule 623
Rule 656
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c^2} \\ & = \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {x (d+e x) (2 d+e x)}{2 c \sqrt {c (d+e x)^2}} \]
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Time = 3.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )^{3}}{2 \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(40\) |
default | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )^{3}}{2 \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(40\) |
trager | \(\frac {x \left (e x +2 d \right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{2 c^{2} \left (e x +d \right )}\) | \(43\) |
risch | \(\frac {\left (e x +d \right ) e \,x^{2}}{2 c \sqrt {c \left (e x +d \right )^{2}}}+\frac {\left (e x +d \right ) d x}{c \sqrt {c \left (e x +d \right )^{2}}}\) | \(49\) |
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Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (c^{2} e x + c^{2} d\right )}} \]
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Time = 2.54 (sec) , antiderivative size = 223, normalized size of antiderivative = 5.72 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\begin {cases} \left (\frac {d}{2 c e} + \frac {x}{2 c}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {2 d^{2} \sqrt {c d^{2} + 2 c d e x} + \frac {2 \left (- c d^{2} \sqrt {c d^{2} + 2 c d e x} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3}\right )}{c} + \frac {c^{2} d^{4} \sqrt {c d^{2} + 2 c d e x} - \frac {2 c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5}}{2 c^{2} d^{2}}}{2 c d e} & \text {for}\: c d e \neq 0 \\\frac {d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}}{\sqrt {c d^{2}}} & \text {otherwise} \end {cases}}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (35) = 70\).
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.54 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {e^{2} x^{3}}{2 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} + \frac {3 \, d e x^{2}}{2 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac {d^{3}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {e x^{2} + 2 \, d x}{2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (e x + d\right )} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]
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