Optimal. Leaf size=67 \[ -\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {614, 613} \[ -\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (3 x-4 x^2\right )^{7/2}} \, dx &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}+\frac {64}{45} \int \frac {1}{\left (3 x-4 x^2\right )^{5/2}} \, dx\\ &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}+\frac {2048 \int \frac {1}{\left (3 x-4 x^2\right )^{3/2}} \, dx}{1215}\\ &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 51, normalized size = 0.76 \[ \frac {2 \left (262144 x^5-491520 x^4+276480 x^3-34560 x^2-3240 x-729\right )}{10935 (3-4 x)^2 x^2 \sqrt {-x (4 x-3)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 61, normalized size = 0.91 \[ -\frac {2 \, {\left (262144 \, x^{5} - 491520 \, x^{4} + 276480 \, x^{3} - 34560 \, x^{2} - 3240 \, x - 729\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{10935 \, {\left (64 \, x^{6} - 144 \, x^{5} + 108 \, x^{4} - 27 \, x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.50, size = 49, normalized size = 0.73 \[ -\frac {2 \, {\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x - 15\right )} x + 135\right )} x - 135\right )} x - 405\right )} x - 729\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{10935 \, {\left (4 \, x^{2} - 3 \, x\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 45, normalized size = 0.67 \[ -\frac {2 \left (4 x -3\right ) \left (262144 x^{5}-491520 x^{4}+276480 x^{3}-34560 x^{2}-3240 x -729\right ) x}{10935 \left (-4 x^{2}+3 x \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 82, normalized size = 1.22 \[ \frac {32768 \, x}{10935 \, \sqrt {-4 \, x^{2} + 3 \, x}} - \frac {4096}{3645 \, \sqrt {-4 \, x^{2} + 3 \, x}} + \frac {1024 \, x}{1215 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} - \frac {128}{405 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} + \frac {16 \, x}{45 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} - \frac {2}{15 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 73, normalized size = 1.09 \[ -\frac {6480\,x-9216\,x\,\left (3\,x-4\,x^2\right )-32768\,x\,{\left (3\,x-4\,x^2\right )}^2+12288\,{\left (3\,x-4\,x^2\right )}^2-13824\,x^2+1458}{{\left (3\,x-4\,x^2\right )}^{3/2}\,\left (32805\,x-43740\,x^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- 4 x^{2} + 3 x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________