3.6 \(\int (3 x-4 x^2)^{7/2} \, dx\)

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]

(-25515*(3 - 8*x)*Sqrt[3*x - 4*x^2])/4194304 - (945*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/131072 - (21*(3 - 8*x)*(3*x
- 4*x^2)^(5/2))/2048 - ((3 - 8*x)*(3*x - 4*x^2)^(7/2))/64 - (229635*ArcSin[1 - (8*x)/3])/16777216

________________________________________________________________________________________

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.

[In]

Int[(3*x - 4*x^2)^(7/2),x]

[Out]

(-25515*(3 - 8*x)*Sqrt[3*x - 4*x^2])/4194304 - (945*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/131072 - (21*(3 - 8*x)*(3*x
- 4*x^2)^(5/2))/2048 - ((3 - 8*x)*(3*x - 4*x^2)^(7/2))/64 - (229635*ArcSin[1 - (8*x)/3])/16777216

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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.

[In]

Integrate[(3*x - 4*x^2)^(7/2),x]

[Out]

(2*x*(-229635 + 102060*x + 54432*x^2 + 41472*x^3 + 75534336*x^4 - 338558976*x^5 + 581959680*x^6 - 452984832*x^
7 + 134217728*x^8) - 229635*Sqrt[3 - 4*x]*Sqrt[x]*ArcSin[Sqrt[1 - (4*x)/3]])/(8388608*Sqrt[-(x*(-3 + 4*x))])

________________________________________________________________________________________

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.

[In]

int((-4*x^2+3*x)^(7/2),x)

[Out]

-945/131072*(3-8*x)*(-4*x^2+3*x)^(3/2)-21/2048*(3-8*x)*(-4*x^2+3*x)^(5/2)-1/64*(3-8*x)*(-4*x^2+3*x)^(7/2)+2296
35/16777216*arcsin(-1+8/3*x)-25515/4194304*(3-8*x)*(-4*x^2+3*x)^(1/2)

________________________________________________________________________________________

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.

[In]

integrate((-4*x^2+3*x)^(7/2),x, algorithm="maxima")

[Out]

1/8*(-4*x^2 + 3*x)^(7/2)*x - 3/64*(-4*x^2 + 3*x)^(7/2) + 21/256*(-4*x^2 + 3*x)^(5/2)*x - 63/2048*(-4*x^2 + 3*x
)^(5/2) + 945/16384*(-4*x^2 + 3*x)^(3/2)*x - 2835/131072*(-4*x^2 + 3*x)^(3/2) + 25515/524288*sqrt(-4*x^2 + 3*x
)*x - 76545/4194304*sqrt(-4*x^2 + 3*x) - 229635/16777216*arcsin(-8/3*x + 1)

________________________________________________________________________________________

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.

[In]

integrate((-4*x^2+3*x)^(7/2),x, algorithm="fricas")

[Out]

-1/4194304*(33554432*x^7 - 88080384*x^6 + 79429632*x^5 - 25067520*x^4 + 82944*x^3 + 72576*x^2 + 68040*x + 7654
5)*sqrt(-4*x^2 + 3*x) - 229635/8388608*arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

________________________________________________________________________________________

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.

[In]

integrate((-4*x**2+3*x)**(7/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(7/2), x)

________________________________________________________________________________________

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.

[In]

integrate((-4*x^2+3*x)^(7/2),x, algorithm="giac")

[Out]

-1/4194304*(8*(16*(8*(32*(8*(16*(8*x - 21)*x + 303)*x - 765)*x + 81)*x + 567)*x + 8505)*x + 76545)*sqrt(-4*x^2
+ 3*x) + 229635/16777216*arcsin(8/3*x - 1)