Optimal. Leaf size=181 \[ \frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \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)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{11 d}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^2}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^3}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^4}\\ &=\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \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}{99 d^6}\\ &=\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 115, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-28 d^7-13 d^6 e x+72 d^5 e^2 x^2+122 d^4 e^3 x^3+32 d^3 e^4 x^4-72 d^2 e^5 x^5-64 d e^6 x^6-16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \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. \(319\) vs.
\(2(157)=314\).
time = 0.48, size = 320, normalized size = 1.77
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (16 e^{7} x^{7}+64 e^{6} x^{6} d +72 e^{5} x^{5} d^{2}-32 d^{3} e^{4} x^{4}-122 d^{4} e^{3} x^{3}-72 d^{5} e^{2} x^{2}+13 x \,d^{6} e +28 d^{7}\right )}{99 \left (e x +d \right )^{3} d^{8} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(110\) |
trager | \(-\frac {\left (16 e^{7} x^{7}+64 e^{6} x^{6} d +72 e^{5} x^{5} d^{2}-32 d^{3} e^{4} x^{4}-122 d^{4} e^{3} x^{3}-72 d^{5} e^{2} x^{2}+13 x \,d^{6} e +28 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{99 d^{8} \left (e x +d \right )^{6} \left (-e x +d \right )^{2} e}\) | \(112\) |
default | \(\frac {-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{4} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {7 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {2 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}\right )}{3 d}\right )}{11 d}}{e^{4}}\) | \(320\) |
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. 351 vs.
\(2 (151) = 302\).
time = 0.30, size = 351, normalized size = 1.94 \begin {gather*} -\frac {1}{11 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x^{4} e^{5} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x^{3} e^{4} + 6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x^{2} e^{3} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {7}{99 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x^{3} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x^{2} e^{3} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} + \frac {8 \, x}{99 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{99 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.16, size = 250, normalized size = 1.38 \begin {gather*} -\frac {28 \, x^{8} e^{8} + 112 \, d x^{7} e^{7} + 112 \, d^{2} x^{6} e^{6} - 112 \, d^{3} x^{5} e^{5} - 280 \, d^{4} x^{4} e^{4} - 112 \, d^{5} x^{3} e^{3} + 112 \, d^{6} x^{2} e^{2} + 112 \, d^{7} x e + 28 \, d^{8} + {\left (16 \, x^{7} e^{7} + 64 \, d x^{6} e^{6} + 72 \, d^{2} x^{5} e^{5} - 32 \, d^{3} x^{4} e^{4} - 122 \, d^{4} x^{3} e^{3} - 72 \, d^{5} x^{2} e^{2} + 13 \, d^{6} x e + 28 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{99 \, {\left (d^{8} x^{8} e^{9} + 4 \, d^{9} x^{7} e^{8} + 4 \, d^{10} x^{6} e^{7} - 4 \, d^{11} x^{5} e^{6} - 10 \, d^{12} x^{4} e^{5} - 4 \, d^{13} x^{3} e^{4} + 4 \, d^{14} x^{2} e^{3} + 4 \, d^{15} x e^{2} + d^{16} 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 {5}{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 = 0.81, size = 197, normalized size = 1.09 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {215\,x}{1584\,d^6}-\frac {91}{792\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{44\,d^3\,e\,{\left (d+e\,x\right )}^6}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{99\,d^4\,e\,{\left (d+e\,x\right )}^5}-\frac {79\,\sqrt {d^2-e^2\,x^2}}{1584\,d^5\,e\,{\left (d+e\,x\right )}^4}-\frac {29\,\sqrt {d^2-e^2\,x^2}}{528\,d^6\,e\,{\left (d+e\,x\right )}^3}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{99\,d^8\,\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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