Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {8 c d \left (c d^2+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {4 c \left (3 c d^2+a e^2\right )}{e^5 \sqrt {d+e x}}-\frac {8 c^2 d \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \]
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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} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt {d+e x}}+\frac {8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac {8 c^2 d \sqrt {d+e x}}{e^5} \]
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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)^{7/2}}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^{5/2}}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^{3/2}}-\frac {4 c^2 d}{e^4 \sqrt {d+e x}}+\frac {c^2 \sqrt {d+e x}}{e^4}\right ) \, dx \\ & = -\frac {2 \left (c d^2+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {8 c d \left (c d^2+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {4 c \left (3 c d^2+a e^2\right )}{e^5 \sqrt {d+e x}}-\frac {8 c^2 d \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (3 a^2 e^4+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]
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Time = 2.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (5 e^{4} x^{4}-40 d \,e^{3} x^{3}-240 d^{2} e^{2} x^{2}-320 d^{3} e x -128 d^{4}\right ) c^{2}}{15}-\frac {32 \left (\frac {15}{8} x^{2} e^{2}+\frac {5}{2} d e x +d^{2}\right ) e^{2} a c}{15}-\frac {2 a^{2} e^{4}}{5}}{\left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(91\) |
gosper | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}+40 x^{3} c^{2} d \,e^{3}+30 x^{2} a c \,e^{4}+240 x^{2} c^{2} d^{2} e^{2}+40 x a c d \,e^{3}+320 x \,c^{2} d^{3} e +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(106\) |
trager | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}+40 x^{3} c^{2} d \,e^{3}+30 x^{2} a c \,e^{4}+240 x^{2} c^{2} d^{2} e^{2}+40 x a c d \,e^{3}+320 x \,c^{2} d^{3} e +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c \left (e^{2} a +3 c \,d^{2}\right )}{\sqrt {e x +d}}+\frac {8 d c \left (e^{2} a +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(110\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c \left (e^{2} a +3 c \,d^{2}\right )}{\sqrt {e x +d}}+\frac {8 d c \left (e^{2} a +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(110\) |
risch | \(-\frac {2 c^{2} \left (-e x +11 d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (30 x^{2} a c \,e^{4}+90 x^{2} c^{2} d^{2} e^{2}+40 x a c d \,e^{3}+160 x \,c^{2} d^{3} e +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+73 c^{2} d^{4}\right )}{15 e^{5} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(125\) |
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Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} - 128 \, c^{2} d^{4} - 16 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 30 \, {\left (8 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} - 40 \, {\left (8 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (119) = 238\).
Time = 0.53 (sec) , antiderivative size = 592, normalized size of antiderivative = 4.81 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 12 \, \sqrt {e x + d} c^{2} d\right )}}{e^{4}} - \frac {3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 30 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{2} - 20 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \]
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Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (90 \, {\left (e x + d\right )}^{2} c^{2} d^{2} - 20 \, {\left (e x + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 30 \, {\left (e x + d\right )}^{2} a c e^{2} - 20 \, {\left (e x + d\right )} a c d e^{2} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{5}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e^{10} - 12 \, \sqrt {e x + d} c^{2} d e^{10}\right )}}{3 \, e^{15}} \]
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Time = 9.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2\,\left (3\,a^2\,e^4+16\,a\,c\,d^2\,e^2+40\,a\,c\,d\,e^3\,x+30\,a\,c\,e^4\,x^2+128\,c^2\,d^4+320\,c^2\,d^3\,e\,x+240\,c^2\,d^2\,e^2\,x^2+40\,c^2\,d\,e^3\,x^3-5\,c^2\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \]
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