Optimal. Leaf size=58 \[ \frac {8 (x+1)}{405 \sqrt {x^2+2 x+4}}+\frac {4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac {x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {614, 613} \[ \frac {8 (x+1)}{405 \sqrt {x^2+2 x+4}}+\frac {4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac {x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx &=\frac {1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac {4}{15} \int \frac {1}{\left (4+2 x+x^2\right )^{5/2}} \, dx\\ &=\frac {1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac {4 (1+x)}{135 \left (4+2 x+x^2\right )^{3/2}}+\frac {8}{135} \int \frac {1}{\left (4+2 x+x^2\right )^{3/2}} \, dx\\ &=\frac {1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac {4 (1+x)}{135 \left (4+2 x+x^2\right )^{3/2}}+\frac {8 (1+x)}{405 \sqrt {4+2 x+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.67 \[ \frac {(x+1) \left (8 x^4+32 x^3+108 x^2+152 x+203\right )}{405 \left (x^2+2 x+4\right )^{5/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 39, normalized size = 0.67 \[ \frac {(x+1) \left (8 x^4+32 x^3+108 x^2+152 x+203\right )}{405 \left (x^2+2 x+4\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 98, normalized size = 1.69 \[ \frac {8 \, x^{6} + 48 \, x^{5} + 192 \, x^{4} + 448 \, x^{3} + 768 \, x^{2} + {\left (8 \, x^{5} + 40 \, x^{4} + 140 \, x^{3} + 260 \, x^{2} + 355 \, x + 203\right )} \sqrt {x^{2} + 2 \, x + 4} + 768 \, x + 512}{405 \, {\left (x^{6} + 6 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} + 96 \, x^{2} + 96 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 33, normalized size = 0.57 \[ \frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 5\right )} x + 35\right )} x + 65\right )} x + 355\right )} x + 203}{405 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 38, normalized size = 0.66
method | result | size |
gosper | \(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\) | \(38\) |
trager | \(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\) | \(38\) |
risch | \(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\) | \(38\) |
default | \(\frac {2 x +2}{30 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}+\frac {\frac {4}{135}+\frac {4 x}{135}}{\left (x^{2}+2 x +4\right )^{\frac {3}{2}}}+\frac {\frac {8}{405}+\frac {8 x}{405}}{\sqrt {x^{2}+2 x +4}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 76, normalized size = 1.31 \[ \frac {8 \, x}{405 \, \sqrt {x^{2} + 2 \, x + 4}} + \frac {8}{405 \, \sqrt {x^{2} + 2 \, x + 4}} + \frac {4 \, x}{135 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}}} + \frac {4}{135 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}}} + \frac {x}{15 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} + \frac {1}{15 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 69, normalized size = 1.19 \[ \frac {51\,x+8\,x\,{\left (x^2+2\,x+4\right )}^2+8\,{\left (x^2+2\,x+4\right )}^2+12\,x^2+12\,x\,\left (x^2+2\,x+4\right )+75}{{\left (x^2+2\,x+4\right )}^{3/2}\,\left (405\,x^2+810\,x+1620\right )} \]
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 \[ \int \frac {1}{\left (x^{2} + 2 x + 4\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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