Optimal. Leaf size=101 \[ -\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {229635 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{16777216} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {229635 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{16777216} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 612
Rule 619
Rubi steps
\begin {align*} \int \left (3 x-4 x^2\right )^{7/2} \, dx &=-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}+\frac {63}{128} \int \left (3 x-4 x^2\right )^{5/2} \, dx\\ &=-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}+\frac {945 \int \left (3 x-4 x^2\right )^{3/2} \, dx}{4096}\\ &=-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}+\frac {25515 \int \sqrt {3 x-4 x^2} \, dx}{262144}\\ &=-\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}+\frac {229635 \int \frac {1}{\sqrt {3 x-4 x^2}} \, dx}{8388608}\\ &=-\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac {76545 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,3-8 x\right )}{16777216}\\ &=-\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac {229635 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{16777216}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 88, normalized size = 0.87 \[ \frac {2 x \left (134217728 x^8-452984832 x^7+581959680 x^6-338558976 x^5+75534336 x^4+41472 x^3+54432 x^2+102060 x-229635\right )-229635 \sqrt {3-4 x} \sqrt {x} \sin ^{-1}\left (\sqrt {1-\frac {4 x}{3}}\right )}{8388608 \sqrt {-x (4 x-3)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 68, normalized size = 0.67 \[ -\frac {1}{4194304} \, {\left (33554432 \, x^{7} - 88080384 \, x^{6} + 79429632 \, x^{5} - 25067520 \, x^{4} + 82944 \, x^{3} + 72576 \, x^{2} + 68040 \, x + 76545\right )} \sqrt {-4 \, x^{2} + 3 \, x} - \frac {229635}{8388608} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.47, size = 57, normalized size = 0.56 \[ -\frac {1}{4194304} \, {\left (8 \, {\left (16 \, {\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x - 21\right )} x + 303\right )} x - 765\right )} x + 81\right )} x + 567\right )} x + 8505\right )} x + 76545\right )} \sqrt {-4 \, x^{2} + 3 \, x} + \frac {229635}{16777216} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 82, normalized size = 0.81 \[ \frac {229635 \arcsin \left (\frac {8 x}{3}-1\right )}{16777216}-\frac {945 \left (-8 x +3\right ) \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{131072}-\frac {21 \left (-8 x +3\right ) \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}{2048}-\frac {\left (-8 x +3\right ) \left (-4 x^{2}+3 x \right )^{\frac {7}{2}}}{64}-\frac {25515 \left (-8 x +3\right ) \sqrt {-4 x^{2}+3 x}}{4194304} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.06, size = 117, normalized size = 1.16 \[ \frac {1}{8} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {7}{2}} x - \frac {3}{64} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {7}{2}} + \frac {21}{256} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}} x - \frac {63}{2048} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}} + \frac {945}{16384} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} x - \frac {2835}{131072} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} + \frac {25515}{524288} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {76545}{4194304} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {229635}{16777216} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 81, normalized size = 0.80 \[ \frac {229635\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{16777216}+\frac {945\,\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{3/2}}{65536}+\frac {21\,\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{5/2}}{1024}+\frac {\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{7/2}}{32}+\frac {25515\,\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2}}{262144} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- 4 x^{2} + 3 x\right )^{\frac {7}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________