Optimal. Leaf size=101 \[ -\frac{1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac{21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac{945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac{25515 (3-8 x) \sqrt{3 x-4 x^2}}{4194304}-\frac{229635 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{16777216} \]
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Rubi [A] time = 0.0275681, antiderivative size = 101, normalized size of antiderivative = 1., 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.
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Rule 612
Rule 619
Rule 216
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 = 0.0722242, 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.
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Maple [A] time = 0.045, size = 82, normalized size = 0.8 \begin{align*} -{\frac{2835-7560\,x}{131072} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}-{\frac{63-168\,x}{2048} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{5}{2}}}}-{\frac{3-8\,x}{64} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{7}{2}}}}+{\frac{229635}{16777216}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{76545-204120\,x}{4194304}\sqrt{-4\,{x}^{2}+3\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72491, size = 158, normalized size = 1.56 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23987, size = 247, normalized size = 2.45 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- 4 x^{2} + 3 x\right )^{\frac{7}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30064, size = 77, normalized size = 0.76 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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