3.7 \(\int (3 x-4 x^2)^{5/2} \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{3645 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{131072} \]

[Out]

(-405*(3 - 8*x)*Sqrt[3*x - 4*x^2])/32768 - (15*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/1024 - ((3 - 8*x)*(3*x - 4*x^2)^
(5/2))/48 - (3645*ArcSin[1 - (8*x)/3])/131072

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Rubi [A]  time = 0.0192173, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{3645 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{131072} \]

Antiderivative was successfully verified.

[In]

Int[(3*x - 4*x^2)^(5/2),x]

[Out]

(-405*(3 - 8*x)*Sqrt[3*x - 4*x^2])/32768 - (15*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/1024 - ((3 - 8*x)*(3*x - 4*x^2)^
(5/2))/48 - (3645*ArcSin[1 - (8*x)/3])/131072

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 )^{5/2} \, dx &=-\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac{15}{32} \int \left (3 x-4 x^2\right )^{3/2} \, dx\\ &=-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac{405 \int \sqrt{3 x-4 x^2} \, dx}{2048}\\ &=-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac{3645 \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx}{65536}\\ &=-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{1215 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )}{131072}\\ &=-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{3645 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{131072}\\ \end{align*}

Mathematica [A]  time = 0.0526547, size = 78, normalized size = 0.99 \[ \frac{2 x \left (-1048576 x^6+2752512 x^5-2469888 x^4+760320 x^3+2592 x^2+4860 x-10935\right )-10935 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{196608 \sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(5/2),x]

[Out]

(2*x*(-10935 + 4860*x + 2592*x^2 + 760320*x^3 - 2469888*x^4 + 2752512*x^5 - 1048576*x^6) - 10935*Sqrt[3 - 4*x]
*Sqrt[x]*ArcSin[Sqrt[1 - (4*x)/3]])/(196608*Sqrt[-(x*(-3 + 4*x))])

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Maple [A]  time = 0.044, size = 64, normalized size = 0.8 \begin{align*} -{\frac{45-120\,x}{1024} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3-8\,x}{48} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{5}{2}}}}+{\frac{3645}{131072}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{1215-3240\,x}{32768}\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)^(5/2),x)

[Out]

-15/1024*(3-8*x)*(-4*x^2+3*x)^(3/2)-1/48*(3-8*x)*(-4*x^2+3*x)^(5/2)+3645/131072*arcsin(-1+8/3*x)-405/32768*(3-
8*x)*(-4*x^2+3*x)^(1/2)

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Maxima [A]  time = 1.71395, size = 122, normalized size = 1.54 \begin{align*} \frac{1}{6} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}} x - \frac{1}{16} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}} + \frac{15}{128} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} x - \frac{45}{1024} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{405}{4096} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{1215}{32768} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{3645}{131072} \, \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)^(5/2),x, algorithm="maxima")

[Out]

1/6*(-4*x^2 + 3*x)^(5/2)*x - 1/16*(-4*x^2 + 3*x)^(5/2) + 15/128*(-4*x^2 + 3*x)^(3/2)*x - 45/1024*(-4*x^2 + 3*x
)^(3/2) + 405/4096*sqrt(-4*x^2 + 3*x)*x - 1215/32768*sqrt(-4*x^2 + 3*x) - 3645/131072*arcsin(-8/3*x + 1)

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Fricas [A]  time = 2.07362, size = 189, normalized size = 2.39 \begin{align*} \frac{1}{98304} \,{\left (262144 \, x^{5} - 491520 \, x^{4} + 248832 \, x^{3} - 3456 \, x^{2} - 3240 \, x - 3645\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{3645}{65536} \, \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)^(5/2),x, algorithm="fricas")

[Out]

1/98304*(262144*x^5 - 491520*x^4 + 248832*x^3 - 3456*x^2 - 3240*x - 3645)*sqrt(-4*x^2 + 3*x) - 3645/65536*arct
an(1/2*sqrt(-4*x^2 + 3*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- 4 x^{2} + 3 x\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2+3*x)**(5/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(5/2), x)

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Giac [A]  time = 1.29414, size = 63, normalized size = 0.8 \begin{align*} \frac{1}{98304} \,{\left (8 \,{\left (16 \,{\left (8 \,{\left (32 \,{\left (8 \, x - 15\right )} x + 243\right )} x - 27\right )} x - 405\right )} x - 3645\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{3645}{131072} \, \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)^(5/2),x, algorithm="giac")

[Out]

1/98304*(8*(16*(8*(32*(8*x - 15)*x + 243)*x - 27)*x - 405)*x - 3645)*sqrt(-4*x^2 + 3*x) + 3645/131072*arcsin(8
/3*x - 1)