Optimal. Leaf size=211 \[ -\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {793, 659, 192, 191} \[ \frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rule 793
Rubi steps
\begin {align*} \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {32 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {224 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d^2 e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^3 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^5 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^7 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 137, normalized size = 0.65 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^9-20 d^8 e x+3200 d^7 e^2 x^2+4320 d^6 e^3 x^3-1280 d^5 e^4 x^4-6208 d^4 e^5 x^5-3072 d^3 e^6 x^6+1792 d^2 e^7 x^7+2048 d e^8 x^8+512 e^9 x^9\right )}{6435 d^9 e^2 (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.34, size = 316, normalized size = 1.50 \[ -\frac {5 \, e^{10} x^{10} + 20 \, d e^{9} x^{9} + 15 \, d^{2} e^{8} x^{8} - 40 \, d^{3} e^{7} x^{7} - 70 \, d^{4} e^{6} x^{6} + 70 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 15 \, d^{8} e^{2} x^{2} - 20 \, d^{9} e x - 5 \, d^{10} + {\left (512 \, e^{9} x^{9} + 2048 \, d e^{8} x^{8} + 1792 \, d^{2} e^{7} x^{7} - 3072 \, d^{3} e^{6} x^{6} - 6208 \, d^{4} e^{5} x^{5} - 1280 \, d^{5} e^{4} x^{4} + 4320 \, d^{6} e^{3} x^{3} + 3200 \, d^{7} e^{2} x^{2} - 20 \, d^{8} e x - 5 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{9} e^{12} x^{10} + 4 \, d^{10} e^{11} x^{9} + 3 \, d^{11} e^{10} x^{8} - 8 \, d^{12} e^{9} x^{7} - 14 \, d^{13} e^{8} x^{6} + 14 \, d^{15} e^{6} x^{4} + 8 \, d^{16} e^{5} x^{3} - 3 \, d^{17} e^{4} x^{2} - 4 \, d^{18} e^{3} x - d^{19} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 132, normalized size = 0.63 \[ -\frac {\left (-e x +d \right ) \left (-512 e^{9} x^{9}-2048 e^{8} x^{8} d -1792 e^{7} x^{7} d^{2}+3072 e^{6} x^{6} d^{3}+6208 e^{5} x^{5} d^{4}+1280 x^{4} d^{5} e^{4}-4320 x^{3} d^{6} e^{3}-3200 x^{2} d^{7} e^{2}+20 d^{8} x e +5 d^{9}\right )}{6435 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{9} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 405, normalized size = 1.92 \[ \frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {4}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} + \frac {64 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e} + \frac {256 \, x}{6435 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} e} + \frac {512 \, x}{6435 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.19, size = 252, normalized size = 1.19 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {41}{41184\,d^6\,e^2}+\frac {256\,x}{6435\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {47}{1716\,d^4\,e^2}-\frac {1369\,x}{34320\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^3\,e^2\,{\left (d+e\,x\right )}^7}+\frac {25\,\sqrt {d^2-e^2\,x^2}}{2288\,d^4\,e^2\,{\left (d+e\,x\right )}^6}+\frac {125\,\sqrt {d^2-e^2\,x^2}}{20592\,d^5\,e^2\,{\left (d+e\,x\right )}^5}-\frac {41\,\sqrt {d^2-e^2\,x^2}}{41184\,d^6\,e^2\,{\left (d+e\,x\right )}^4}+\frac {512\,x\,\sqrt {d^2-e^2\,x^2}}{6435\,d^9\,e\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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