Optimal. Leaf size=75 \[ \frac {2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} \frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac {2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{3/2}}{e^2}+\frac {c (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.73 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (7 e (-2 b d+5 a e+3 b e x)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 59, normalized size = 0.79
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 x^{2} c \,e^{2}+21 b \,e^{2} x -12 c d e x +35 e^{2} a -14 b d e +8 c \,d^{2}\right )}{105 e^{3}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\) | \(59\) |
default | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\) | \(59\) |
trager | \(\frac {2 \left (15 e^{3} c \,x^{3}+21 b \,e^{3} x^{2}+3 c d \,e^{2} x^{2}+35 a \,e^{3} x +7 b d \,e^{2} x -4 c \,d^{2} e x +35 a d \,e^{2}-14 b \,d^{2} e +8 c \,d^{3}\right ) \sqrt {e x +d}}{105 e^{3}}\) | \(85\) |
risch | \(\frac {2 \left (15 e^{3} c \,x^{3}+21 b \,e^{3} x^{2}+3 c d \,e^{2} x^{2}+35 a \,e^{3} x +7 b d \,e^{2} x -4 c \,d^{2} e x +35 a d \,e^{2}-14 b \,d^{2} e +8 c \,d^{3}\right ) \sqrt {e x +d}}{105 e^{3}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 62, normalized size = 0.83 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c - 21 \, {\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.79, size = 77, normalized size = 1.03 \begin {gather*} \frac {2}{105} \, {\left (8 \, c d^{3} + {\left (15 \, c x^{3} + 21 \, b x^{2} + 35 \, a x\right )} e^{3} + {\left (3 \, c d x^{2} + 7 \, b d x + 35 \, a d\right )} e^{2} - 2 \, {\left (2 \, c d^{2} x + 7 \, b d^{2}\right )} e\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.35, size = 71, normalized size = 0.95 \begin {gather*} \frac {2 \left (\frac {c \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{2}}\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. 200 vs.
\(2 (64) = 128\).
time = 0.94, size = 200, normalized size = 2.67 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b 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 )} c d e^{\left (-2\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 )} b e^{\left (-1\right )} + 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 e^{\left (-2\right )} + 105 \, \sqrt {x e + d} a d + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 58, normalized size = 0.77 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,c\,{\left (d+e\,x\right )}^2+35\,a\,e^2+35\,c\,d^2+21\,b\,e\,\left (d+e\,x\right )-42\,c\,d\,\left (d+e\,x\right )-35\,b\,d\,e\right )}{105\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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