Optimal. Leaf size=47 \[ \frac {1-2 x}{6 \left (2 x^2-8 x+1\right )^{3/2}}-\frac {2 (2-x)}{21 \sqrt {2 x^2-8 x+1}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 47, 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 = {638, 613} \[ \frac {1-2 x}{6 \left (2 x^2-8 x+1\right )^{3/2}}-\frac {2 (2-x)}{21 \sqrt {2 x^2-8 x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 613
Rule 638
Rubi steps
\begin {align*} \int \frac {1+3 x}{\left (1-8 x+2 x^2\right )^{5/2}} \, dx &=\frac {1-2 x}{6 \left (1-8 x+2 x^2\right )^{3/2}}-\frac {2}{3} \int \frac {1}{\left (1-8 x+2 x^2\right )^{3/2}} \, dx\\ &=\frac {1-2 x}{6 \left (1-8 x+2 x^2\right )^{3/2}}-\frac {2 (2-x)}{21 \sqrt {1-8 x+2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 33, normalized size = 0.70 \[ \frac {8 x^3-48 x^2+54 x-1}{42 \left (2 x^2-8 x+1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.37, size = 33, normalized size = 0.70 \[ \frac {8 x^3-48 x^2+54 x-1}{42 \left (2 x^2-8 x+1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.78, size = 73, normalized size = 1.55 \[ -\frac {4 \, x^{4} - 32 \, x^{3} + 68 \, x^{2} - {\left (8 \, x^{3} - 48 \, x^{2} + 54 \, x - 1\right )} \sqrt {2 \, x^{2} - 8 \, x + 1} - 16 \, x + 1}{42 \, {\left (4 \, x^{4} - 32 \, x^{3} + 68 \, x^{2} - 16 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.70, size = 27, normalized size = 0.57 \[ \frac {2 \, {\left (4 \, {\left (x - 6\right )} x + 27\right )} x - 1}{42 \, {\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.45, size = 30, normalized size = 0.64
method | result | size |
gosper | \(\frac {8 x^{3}-48 x^{2}+54 x -1}{42 \left (2 x^{2}-8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(\frac {8 x^{3}-48 x^{2}+54 x -1}{42 \left (2 x^{2}-8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
risch | \(\frac {8 x^{3}-48 x^{2}+54 x -1}{42 \left (2 x^{2}-8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {4 x -8}{12 \left (2 x^{2}-8 x +1\right )^{\frac {3}{2}}}+\frac {4 x -8}{42 \sqrt {2 x^{2}-8 x +1}}-\frac {1}{2 \left (2 x^{2}-8 x +1\right )^{\frac {3}{2}}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 59, normalized size = 1.26 \[ \frac {2 \, x}{21 \, \sqrt {2 \, x^{2} - 8 \, x + 1}} - \frac {4}{21 \, \sqrt {2 \, x^{2} - 8 \, x + 1}} - \frac {x}{3 \, {\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac {3}{2}}} + \frac {1}{6 \, {\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.36, size = 29, normalized size = 0.62 \[ \frac {8\,x^3-48\,x^2+54\,x-1}{42\,{\left (2\,x^2-8\,x+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 x + 1}{\left (2 x^{2} - 8 x + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________