Optimal. Leaf size=50 \[ \frac {1}{8} (2 d-3 e) (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2} \]
[Out]
________________________________________________________________________________________
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}{8} (2 x+3) \sqrt {4 x^2+12 x+9} (2 d-3 e)+\frac {1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 623
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx &=\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{2} (2 d-3 e) \int \sqrt {9+12 x+4 x^2} \, dx\\ &=\frac {1}{8} (2 d-3 e) (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 38, normalized size = 0.76 \begin {gather*} \frac {x \sqrt {(3+2 x)^2} (6 d (3+x)+e x (9+4 x))}{6 (3+2 x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.50, size = 27, normalized size = 0.54
method | result | size |
default | \(\frac {\mathrm {csgn}\left (2 x +3\right ) \left (2 x +3\right )^{2} \left (4 e x +6 d -3 e \right )}{24}\) | \(27\) |
gosper | \(\frac {x \left (4 e \,x^{2}+6 d x +9 e x +18 d \right ) \sqrt {\left (2 x +3\right )^{2}}}{12 x +18}\) | \(38\) |
risch | \(\frac {2 \sqrt {\left (2 x +3\right )^{2}}\, e \,x^{3}}{3 \left (2 x +3\right )}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (2 d +3 e \right ) x^{2}}{4 x +6}+\frac {3 \sqrt {\left (2 x +3\right )^{2}}\, d x}{2 x +3}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 81, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d x + \frac {1}{12} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} e - \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} x e + \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d - \frac {9}{8} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.95, size = 25, normalized size = 0.50 \begin {gather*} d x^{2} + 3 \, d x + \frac {1}{6} \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right ) \sqrt {\left (2 x + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.77, size = 64, normalized size = 1.28 \begin {gather*} \frac {2}{3} \, x^{3} e \mathrm {sgn}\left (2 \, x + 3\right ) + d x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {3}{2} \, x^{2} e \mathrm {sgn}\left (2 \, x + 3\right ) + 3 \, d x \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {9}{8} \, {\left (2 \, d - e\right )} \mathrm {sgn}\left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.61, size = 30, normalized size = 0.60 \begin {gather*} \frac {\left (2\,x+3\right )\,\left (6\,d-3\,e+4\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________