Optimal. Leaf size=85 \[ \frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {40, 39}
\begin {gather*} \frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {2}{15} \int \frac {1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx\\ &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {2}{135} \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\\ &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.09, size = 149, normalized size = 1.75 \begin {gather*} \frac {x \left (15-80 x^2+128 x^4\right ) \left (3363+32 x^5-2378 \sqrt {2+4 x}+x \left (13930-7472 \sqrt {2+4 x}\right )-80 x^4 \left (-19+2 \sqrt {2+4 x}\right )-80 x^3 \left (-121+28 \sqrt {2+4 x}\right )-8 x^2 \left (-2375+894 \sqrt {2+4 x}\right )\right )}{810 \sqrt {3-6 x} (1-2 x)^2 \left (-4+3 \sqrt {2+4 x}+2 x \left (-4+\sqrt {2+4 x}\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 98, normalized size = 1.15
method | result | size |
gosper | \(-\frac {\left (2 x -1\right ) \left (1+2 x \right ) x \left (128 x^{4}-80 x^{2}+15\right )}{15 \left (3-6 x \right )^{\frac {7}{2}} \left (2+4 x \right )^{\frac {7}{2}}}\) | \(40\) |
default | \(\frac {1}{60 \left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {5}{2}}}+\frac {1}{108 \left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {5}{2}}}+\frac {1}{81 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {5}{2}}}-\frac {\sqrt {3-6 x}}{405 \left (2+4 x \right )^{\frac {5}{2}}}-\frac {\sqrt {3-6 x}}{1215 \left (2+4 x \right )^{\frac {3}{2}}}-\frac {\sqrt {3-6 x}}{2430 \sqrt {2+4 x}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 37, normalized size = 0.44 \begin {gather*} \frac {x}{405 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{135 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} + \frac {x}{30 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.24, size = 49, normalized size = 0.58 \begin {gather*} -\frac {{\left (128 \, x^{5} - 80 \, x^{3} + 15 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{19440 \, {\left (64 \, x^{6} - 48 \, x^{4} + 12 \, x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs.
\(2 (61) = 122\).
time = 1.76, size = 205, normalized size = 2.41 \begin {gather*} \frac {1}{39813120} \, \sqrt {6} {\left (\frac {3 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{5}}{{\left (2 \, x + 1\right )}^{\frac {5}{2}}} + \frac {85 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{3}}{{\left (2 \, x + 1\right )}^{\frac {3}{2}}} + \frac {2130 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}}{\sqrt {2 \, x + 1}}\right )} - \frac {{\left ({\left (64 \, \sqrt {6} {\left (2 \, x + 1\right )} - 275 \, \sqrt {6}\right )} {\left (2 \, x + 1\right )} + 300 \, \sqrt {6}\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1}}{622080 \, {\left (2 \, x - 1\right )}^{3}} - \frac {\sqrt {6} {\left (\frac {2130 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{4}}{{\left (2 \, x + 1\right )}^{2}} + \frac {85 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{2}}{2 \, x + 1} + 3\right )} {\left (2 \, x + 1\right )}^{\frac {5}{2}}}{39813120 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.45, size = 66, normalized size = 0.78 \begin {gather*} -\frac {15\,x\,\sqrt {3-6\,x}-80\,x^3\,\sqrt {3-6\,x}+128\,x^5\,\sqrt {3-6\,x}}{\left (\left (6\,x-3\right )\,\left (240\,x+360\right )+1440\right )\,\sqrt {4\,x+2}\,{\left (6\,x-3\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________