Optimal. Leaf size=211 \[ \frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.07, 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 = {807, 673, 198,
197} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 673
Rule 807
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 = 0.58, size = 137, normalized size = 0.65 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(707\) vs.
\(2(183)=366\).
time = 0.07, size = 708, normalized size = 3.36
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-512 e^{9} x^{9}-2048 d \,e^{8} x^{8}-1792 e^{7} x^{7} d^{2}+3072 e^{6} x^{6} d^{3}+6208 e^{5} x^{5} d^{4}+1280 x^{4} d^{5} e^{4}-4320 d^{6} e^{3} x^{3}-3200 x^{2} d^{7} e^{2}+20 d^{8} x e +5 d^{9}\right )}{6435 \left (e x +d \right )^{3} d^{9} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(132\) |
trager | \(-\frac {\left (-512 e^{9} x^{9}-2048 d \,e^{8} x^{8}-1792 e^{7} x^{7} d^{2}+3072 e^{6} x^{6} d^{3}+6208 e^{5} x^{5} d^{4}+1280 x^{4} d^{5} e^{4}-4320 d^{6} e^{3} x^{3}-3200 x^{2} d^{7} e^{2}+20 d^{8} x e +5 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6435 d^{9} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e^{2}}\) | \(134\) |
default | \(\frac {-\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 e^{2} \left (x +\frac {d}{e}\right )+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 e^{2} \left (x +\frac {d}{e}\right )+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 e^{2} \left (x +\frac {d}{e}\right )+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}}{e^{4}}-\frac {d \left (-\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 e^{2} \left (x +\frac {d}{e}\right )+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 e^{2} \left (x +\frac {d}{e}\right )+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 e^{2} \left (x +\frac {d}{e}\right )+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}\right )}{e^{5}}\) | \(708\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 371 vs.
\(2 (173) = 346\).
time = 0.31, size = 371, normalized size = 1.76 \begin {gather*} \frac {1}{13 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4} e^{6} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{5} + 6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{4} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{3} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {4}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{5} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{3} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{4} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{3} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{3} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} + \frac {64 \, x e^{\left (-1\right )}}{2145 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}} + \frac {256 \, x e^{\left (-1\right )}}{6435 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7}} + \frac {512 \, x e^{\left (-1\right )}}{6435 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.86, size = 291, normalized size = 1.38 \begin {gather*} -\frac {5 \, x^{10} e^{10} + 20 \, d x^{9} e^{9} + 15 \, d^{2} x^{8} e^{8} - 40 \, d^{3} x^{7} e^{7} - 70 \, d^{4} x^{6} e^{6} + 70 \, d^{6} x^{4} e^{4} + 40 \, d^{7} x^{3} e^{3} - 15 \, d^{8} x^{2} e^{2} - 20 \, d^{9} x e - 5 \, d^{10} + {\left (512 \, x^{9} e^{9} + 2048 \, d x^{8} e^{8} + 1792 \, d^{2} x^{7} e^{7} - 3072 \, d^{3} x^{6} e^{6} - 6208 \, d^{4} x^{5} e^{5} - 1280 \, d^{5} x^{4} e^{4} + 4320 \, d^{6} x^{3} e^{3} + 3200 \, d^{7} x^{2} e^{2} - 20 \, d^{8} x e - 5 \, d^{9}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{9} x^{10} e^{12} + 4 \, d^{10} x^{9} e^{11} + 3 \, d^{11} x^{8} e^{10} - 8 \, d^{12} x^{7} e^{9} - 14 \, d^{13} x^{6} e^{8} + 14 \, d^{15} x^{4} e^{6} + 8 \, d^{16} x^{3} e^{5} - 3 \, d^{17} x^{2} e^{4} - 4 \, d^{18} x e^{3} - d^{19} e^{2}\right )}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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