Optimal. Leaf size=50 \[ \frac {1}{24} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac {1}{28} e \left (9+12 x+4 x^2\right )^{7/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {654, 623}
\begin {gather*} \frac {1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac {1}{28} e \left (4 x^2+12 x+9\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \left (9+12 x+4 x^2\right )^{5/2} \, dx &=\frac {1}{28} e \left (9+12 x+4 x^2\right )^{7/2}+\frac {1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^{5/2} \, dx\\ &=\frac {1}{24} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac {1}{28} e \left (9+12 x+4 x^2\right )^{7/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 81, normalized size = 1.62 \begin {gather*} \frac {x \sqrt {(3+2 x)^2} \left (14 d \left (729+1215 x+1080 x^2+540 x^3+144 x^4+16 x^5\right )+3 e x \left (1701+3780 x+3780 x^2+2016 x^3+560 x^4+64 x^5\right )\right )}{42 (3+2 x)} \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. \(85\) vs.
\(2(42)=84\).
time = 0.56, size = 86, normalized size = 1.72
method | result | size |
gosper | \(\frac {x \left (192 x^{6} e +224 x^{5} d +1680 x^{5} e +2016 d \,x^{4}+6048 x^{4} e +7560 d \,x^{3}+11340 e \,x^{3}+15120 d \,x^{2}+11340 e \,x^{2}+17010 d x +5103 e x +10206 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{42 \left (2 x +3\right )^{5}}\) | \(86\) |
default | \(\frac {x \left (192 x^{6} e +224 x^{5} d +1680 x^{5} e +2016 d \,x^{4}+6048 x^{4} e +7560 d \,x^{3}+11340 e \,x^{3}+15120 d \,x^{2}+11340 e \,x^{2}+17010 d x +5103 e x +10206 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{42 \left (2 x +3\right )^{5}}\) | \(86\) |
risch | \(\frac {32 \sqrt {\left (2 x +3\right )^{2}}\, e \,x^{7}}{7 \left (2 x +3\right )}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (32 d +240 e \right ) x^{6}}{12 x +18}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (240 d +720 e \right ) x^{5}}{10 x +15}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (720 d +1080 e \right ) x^{4}}{8 x +12}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (1080 d +810 e \right ) x^{3}}{6 x +9}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (810 d +243 e \right ) x^{2}}{4 x +6}+\frac {243 \sqrt {\left (2 x +3\right )^{2}}\, d x}{2 x +3}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 81, normalized size = 1.62 \begin {gather*} \frac {1}{6} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} d x + \frac {1}{28} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {7}{2}} e - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} x e + \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} d - \frac {3}{8} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.18, size = 70, normalized size = 1.40 \begin {gather*} \frac {16}{3} \, d x^{6} + 48 \, d x^{5} + 180 \, d x^{4} + 360 \, d x^{3} + 405 \, d x^{2} + 243 \, d x + \frac {1}{14} \, {\left (64 \, x^{7} + 560 \, x^{6} + 2016 \, x^{5} + 3780 \, x^{4} + 3780 \, x^{3} + 1701 \, x^{2}\right )} e \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 \left (d + e x\right ) \left (\left (2 x + 3\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs.
\(2 (44) = 88\).
time = 0.94, size = 165, normalized size = 3.30 \begin {gather*} \frac {32}{7} \, x^{7} e \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {16}{3} \, d x^{6} \mathrm {sgn}\left (2 \, x + 3\right ) + 40 \, x^{6} e \mathrm {sgn}\left (2 \, x + 3\right ) + 48 \, d x^{5} \mathrm {sgn}\left (2 \, x + 3\right ) + 144 \, x^{5} e \mathrm {sgn}\left (2 \, x + 3\right ) + 180 \, d x^{4} \mathrm {sgn}\left (2 \, x + 3\right ) + 270 \, x^{4} e \mathrm {sgn}\left (2 \, x + 3\right ) + 360 \, d x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + 270 \, x^{3} e \mathrm {sgn}\left (2 \, x + 3\right ) + 405 \, d x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {243}{2} \, x^{2} e \mathrm {sgn}\left (2 \, x + 3\right ) + 243 \, d x \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {243}{56} \, {\left (14 \, d - 3 \, e\right )} \mathrm {sgn}\left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \left (d+e\,x\right )\,{\left (4\,x^2+12\,x+9\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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