Integrand size = 24, antiderivative size = 205 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
[In]
[Out]
Rule 197
Rule 198
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {9 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d} \\ & = -\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {72 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2} \\ & = -\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3} \\ & = -\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4} \\ & = \frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {192 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6} \\ & = \frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8} \\ & = \frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-128 e^{9} x^{9}-512 d \,e^{8} x^{8}-448 d^{2} e^{7} x^{7}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 d^{6} e^{3} x^{3}-800 x^{2} d^{7} e^{2}+5 x \,d^{8} e +180 d^{9}\right )}{715 \left (e x +d \right )^{3} d^{10} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(132\) |
trager | \(-\frac {\left (-128 e^{9} x^{9}-512 d \,e^{8} x^{8}-448 d^{2} e^{7} x^{7}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 d^{6} e^{3} x^{3}-800 x^{2} d^{7} e^{2}+5 x \,d^{8} e +180 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{715 d^{10} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e}\) | \(134\) |
default | \(\frac {-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}}{e^{4}}\) | \(379\) |
[In]
[Out]
none
Time = 0.73 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} + {\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{715 \, {\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (177) = 354\).
Time = 0.20 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {9}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} + \frac {48 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {64 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8}} + \frac {128 \, x}{715 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10}} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4}} \,d x } \]
[In]
[Out]
Time = 11.90 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{715\,d^8}+\frac {189}{4576\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1139\,x}{5720\,d^6}-\frac {427}{2288\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^4\,e\,{\left (d+e\,x\right )}^7}-\frac {51\,\sqrt {d^2-e^2\,x^2}}{2288\,d^5\,e\,{\left (d+e\,x\right )}^6}-\frac {19\,\sqrt {d^2-e^2\,x^2}}{572\,d^6\,e\,{\left (d+e\,x\right )}^5}-\frac {189\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{715\,d^{10}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
[In]
[Out]