Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {8 c d \left (c d^2+a e^2\right )}{e^5 \sqrt {d+e x}}+\frac {4 c \left (3 c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}-\frac {8 c^2 d (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \]
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Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {4 c \sqrt {d+e x} \left (a e^2+3 c d^2\right )}{e^5}+\frac {8 c d \left (a e^2+c d^2\right )}{e^5 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 c^2 d (d+e x)^{3/2}}{3 e^5} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{5/2}}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^{3/2}}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 \sqrt {d+e x}}-\frac {4 c^2 d \sqrt {d+e x}}{e^4}+\frac {c^2 (d+e x)^{3/2}}{e^4}\right ) \, dx \\ & = -\frac {2 \left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {8 c d \left (c d^2+a e^2\right )}{e^5 \sqrt {d+e x}}+\frac {4 c \left (3 c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}-\frac {8 c^2 d (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \]
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Time = 1.97 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 c \left (3 c \,x^{2} e^{2}-14 x c d e +30 e^{2} a +73 c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (-12 x c d e +e^{2} a -11 c \,d^{2}\right ) \left (e^{2} a +c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(84\) |
pseudoelliptic | \(\frac {\frac {2 \left (3 e^{4} x^{4}-8 d \,e^{3} x^{3}+48 d^{2} e^{2} x^{2}+192 d^{3} e x +128 d^{4}\right ) c^{2}}{15}+\frac {32 \left (\frac {3}{8} x^{2} e^{2}+\frac {3}{2} d e x +d^{2}\right ) e^{2} a c}{3}-\frac {2 a^{2} e^{4}}{3}}{\left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(91\) |
gosper | \(-\frac {2 \left (-3 c^{2} x^{4} e^{4}+8 x^{3} c^{2} d \,e^{3}-30 x^{2} a c \,e^{4}-48 x^{2} c^{2} d^{2} e^{2}-120 x a c d \,e^{3}-192 x \,c^{2} d^{3} e +5 a^{2} e^{4}-80 a c \,d^{2} e^{2}-128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(106\) |
trager | \(-\frac {2 \left (-3 c^{2} x^{4} e^{4}+8 x^{3} c^{2} d \,e^{3}-30 x^{2} a c \,e^{4}-48 x^{2} c^{2} d^{2} e^{2}-120 x a c d \,e^{3}-192 x \,c^{2} d^{3} e +5 a^{2} e^{4}-80 a c \,d^{2} e^{2}-128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}+\frac {8 d c \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(117\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}+\frac {8 d c \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(117\) |
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Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} e^{4} x^{4} - 8 \, c^{2} d e^{3} x^{3} + 128 \, c^{2} d^{4} + 80 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4} + 6 \, {\left (8 \, c^{2} d^{2} e^{2} + 5 \, a c e^{4}\right )} x^{2} + 24 \, {\left (8 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
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Time = 1.65 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {4 c^{2} d \left (d + e x\right )^{\frac {3}{2}}}{3 e^{4}} + \frac {c^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {4 c d \left (a e^{2} + c d^{2}\right )}{e^{4} \sqrt {d + e x}} + \frac {\sqrt {d + e x} \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{e^{4}} - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{3 e^{4} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d + 30 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {5 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 12 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )}}{15 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (e x + d\right )} c^{2} d^{3} - c^{2} d^{4} + 12 \, {\left (e x + d\right )} a c d e^{2} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{5}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} e^{20} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d e^{20} + 90 \, \sqrt {e x + d} c^{2} d^{2} e^{20} + 30 \, \sqrt {e x + d} a c e^{22}\right )}}{15 \, e^{25}} \]
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Time = 9.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\frac {2\,a^2\,e^4}{3}+\frac {2\,c^2\,d^4}{3}-\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )+\frac {4\,a\,c\,d^2\,e^2}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,\sqrt {d+e\,x}}{e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \]
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