Optimal. Leaf size=43 \[ \frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} \frac {2}{5} (d+e x)^{5/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{3/2}}{e}+\frac {c d (d+e x)^{5/2}}{e}\right ) \, dx\\ &=\frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (7 a e^2+c d (-2 d+5 e x)\right )}{35 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 39, normalized size = 0.91
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 c d e x +7 e^{2} a -2 c \,d^{2}\right )}{35 e^{2}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\) | \(39\) |
default | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\) | \(39\) |
trager | \(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\) | \(75\) |
risch | \(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 38, normalized size = 0.88 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c d - 7 \, {\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 70, normalized size = 1.63 \begin {gather*} \frac {2}{35} \, {\left (c d^{3} x e - 2 \, c d^{4} + 7 \, a x^{2} e^{4} + {\left (5 \, c d x^{3} + 14 \, a d x\right )} e^{3} + {\left (8 \, c d^{2} x^{2} + 7 \, a d^{2}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.30, size = 41, normalized size = 0.95 \begin {gather*} \frac {2 \left (\frac {c d \left (d + e x\right )^{\frac {7}{2}}}{7 e} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} \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. 212 vs.
\(2 (36) = 72\).
time = 1.81, size = 212, normalized size = 4.93 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{3} e^{\left (-1\right )} + 14 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{2} e^{\left (-1\right )} + 105 \, \sqrt {x e + d} a d^{2} e + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d e + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a e\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (7\,a\,e^2-7\,c\,d^2+5\,c\,d\,\left (d+e\,x\right )\right )}{35\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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