Optimal. Leaf size=56 \[ \frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {654, 622, 31}
\begin {gather*} \frac {(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}}+\frac {1}{4} e \sqrt {4 x^2+12 x+9} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 622
Rule 654
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\sqrt {9+12 x+4 x^2}} \, dx\\ &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {((2 d-3 e) (6+4 x)) \int \frac {1}{6+4 x} \, dx}{2 \sqrt {9+12 x+4 x^2}}\\ &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 42, normalized size = 0.75 \begin {gather*} \frac {(3+2 x) (e (3+2 x)+(2 d-3 e) \log (3+2 x))}{4 \sqrt {(3+2 x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.56, size = 40, normalized size = 0.71
method | result | size |
meijerg | \(\frac {3 e \left (\frac {2 x}{3}-\ln \left (1+\frac {2 x}{3}\right )\right )}{4}+\frac {d \ln \left (1+\frac {2 x}{3}\right )}{2}\) | \(26\) |
default | \(\frac {\left (2 x +3\right ) \left (2 \ln \left (2 x +3\right ) d -3 e \ln \left (2 x +3\right )+2 e x \right )}{4 \sqrt {\left (2 x +3\right )^{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (2 x +3\right )^{2}}\, e x}{4 x +6}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {3 e}{2}+d \right ) \ln \left (2 x +3\right )}{4 x +6}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 32, normalized size = 0.57 \begin {gather*} \frac {1}{2} \, d \log \left (x + \frac {3}{2}\right ) - \frac {3}{4} \, e \log \left (x + \frac {3}{2}\right ) + \frac {1}{4} \, \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 = 3.24, size = 22, normalized size = 0.39 \begin {gather*} \frac {1}{2} \, x e + \frac {1}{4} \, {\left (2 \, d - 3 \, e\right )} \log \left (2 \, x + 3\right ) \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 \frac {d + e x}{\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 = 0.62, size = 39, normalized size = 0.70 \begin {gather*} \frac {x e}{2 \, \mathrm {sgn}\left (2 \, x + 3\right )} + \frac {{\left (2 \, d - 3 \, e\right )} \log \left ({\left | 2 \, x + 3 \right |}\right )}{4 \, \mathrm {sgn}\left (2 \, x + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.50, size = 46, normalized size = 0.82 \begin {gather*} \frac {e\,\sqrt {4\,x^2+12\,x+9}}{4}-\frac {3\,e\,\ln \left (x+\frac {\left |2\,x+3\right |}{2}+\frac {3}{2}\right )}{4}+\frac {d\,\ln \left (4\,x+6\right )\,\mathrm {sign}\left (8\,x+12\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________