Optimal. Leaf size=106 \[ \frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197}
\begin {gather*} -\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {6 x}{35 d^3 \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
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d^3}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{35 d^5}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 104, normalized size = 0.98 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^6+30 d^5 e x+30 d^4 e^2 x^2-40 d^3 e^3 x^3-40 d^2 e^4 x^4+16 d e^5 x^5+16 e^6 x^6\right )}{35 d^7 e (d-e x)^3 (d+e x)^4} \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. \(222\) vs.
\(2(90)=180\).
time = 0.48, size = 223, normalized size = 2.10
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-16 e^{6} x^{6}-16 d \,e^{5} x^{5}+40 e^{4} x^{4} d^{2}+40 d^{3} e^{3} x^{3}-30 d^{4} e^{2} x^{2}-30 d^{5} e x +5 d^{6}\right )}{35 d^{7} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(92\) |
trager | \(-\frac {\left (-16 e^{6} x^{6}-16 d \,e^{5} x^{5}+40 e^{4} x^{4} d^{2}+40 d^{3} e^{3} x^{3}-30 d^{4} e^{2} x^{2}-30 d^{5} e x +5 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{7} \left (e x +d \right )^{4} \left (-e x +d \right )^{3} e}\) | \(101\) |
default | \(\frac {-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}}{e}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 100, normalized size = 0.94 \begin {gather*} -\frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e\right )}} + \frac {6 \, x}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, x}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {16 \, x}{35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs.
\(2 (86) = 172\).
time = 2.37, size = 220, normalized size = 2.08 \begin {gather*} -\frac {5 \, x^{7} e^{7} + 5 \, d x^{6} e^{6} - 15 \, d^{2} x^{5} e^{5} - 15 \, d^{3} x^{4} e^{4} + 15 \, d^{4} x^{3} e^{3} + 15 \, d^{5} x^{2} e^{2} - 5 \, d^{6} x e - 5 \, d^{7} + {\left (16 \, x^{6} e^{6} + 16 \, d x^{5} e^{5} - 40 \, d^{2} x^{4} e^{4} - 40 \, d^{3} x^{3} e^{3} + 30 \, d^{4} x^{2} e^{2} + 30 \, d^{5} x e - 5 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{35 \, {\left (d^{7} x^{7} e^{8} + d^{8} x^{6} e^{7} - 3 \, d^{9} x^{5} e^{6} - 3 \, d^{10} x^{4} e^{5} + 3 \, d^{11} x^{3} e^{4} + 3 \, d^{12} x^{2} e^{3} - d^{13} x e^{2} - d^{14} e\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 {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, 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 = 0.63, size = 155, normalized size = 1.46 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {17\,x}{70\,d^3}-\frac {1}{7\,d^2\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{35\,d^5}+\frac {1}{56\,d^4\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^4\,e\,{\left (d+e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{35\,d^7\,\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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