Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0141922, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {640, 607} \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 640
Rule 607
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx &=-\frac{e}{20 \left (9+12 x+4 x^2\right )^{5/2}}+\frac{1}{2} (2 d-3 e) \int \frac{1}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx\\ &=-\frac{e}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac{2 d-3 e}{24 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0152839, size = 34, normalized size = 0.65 \[ \frac{-10 d-3 (4 e x+e)}{120 (2 x+3)^5 \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.08, size = 28, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 3+2\,x \right ) \left ( 12\,ex+10\,d+3\,e \right ) }{120} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.64083, size = 49, normalized size = 0.94 \begin{align*} -\frac{e}{20 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}}} - \frac{d}{12 \,{\left (2 \, x + 3\right )}^{6}} + \frac{e}{8 \,{\left (2 \, x + 3\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48923, size = 131, normalized size = 2.52 \begin{align*} -\frac{12 \, e x + 10 \, d + 3 \, e}{120 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]