3.1.7 \(\int (3 x-4 x^2)^{5/2} \, dx\) [7]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 = {626, 633, 222} \begin {gather*} -\frac {3645 \text {ArcSin}\left (1-\frac {8 x}{3}\right )}{131072}-\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} \end {gather*}

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 222

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]

Rule 626

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 633

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]

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 \text {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*}

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Mathematica [A]
time = 0.08, size = 92, normalized size = 1.16 \begin {gather*} \frac {\sqrt {-x (-3+4 x)} \left (2 \sqrt {x} \sqrt {-3+4 x} \left (-3645-3240 x-3456 x^2+248832 x^3-491520 x^4+262144 x^5\right )+10935 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )\right )}{196608 \sqrt {x} \sqrt {-3+4 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(5/2),x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(2*Sqrt[x]*Sqrt[-3 + 4*x]*(-3645 - 3240*x - 3456*x^2 + 248832*x^3 - 491520*x^4 + 262144
*x^5) + 10935*Log[-2*Sqrt[x] + Sqrt[-3 + 4*x]]))/(196608*Sqrt[x]*Sqrt[-3 + 4*x])

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Maple [A]
time = 0.41, size = 64, normalized size = 0.81

method	result	size
risch	\(-\frac {\left (262144 x^{5}-491520 x^{4}+248832 x^{3}-3456 x^{2}-3240 x -3645\right ) x \left (-3+4 x \right )}{98304 \sqrt {-x \left (-3+4 x \right )}}+\frac {3645 \arcsin \left (-1+\frac {8 x}{3}\right )}{131072}\)	\(53\)
default	\(-\frac {15 \left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{1024}-\frac {\left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}{48}+\frac {3645 \arcsin \left (-1+\frac {8 x}{3}\right )}{131072}-\frac {405 \left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{32768}\)	\(64\)
meijerg	\(-\frac {10935 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (-\frac {1835008}{243} x^{5}+\frac {1146880}{81} x^{4}-7168 x^{3}+\frac {896}{9} x^{2}+\frac {280}{3} x +105\right ) \sqrt {-\frac {4 x}{3}+1}}{30240}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{192}\right )}{1024 \sqrt {\pi }}\)	\(67\)
trager	\(\left (\frac {8}{3} x^{5}-5 x^{4}+\frac {81}{32} x^{3}-\frac {9}{256} x^{2}-\frac {135}{4096} x -\frac {1215}{32768}\right ) \sqrt {-4 x^{2}+3 x}-\frac {3645 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{131072}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2+3*x)^(5/2),x,method=_RETURNVERBOSE)

[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 = 0.48, size = 90, normalized size = 1.14 \begin {gather*} \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 {gather*}

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 = 1.38, size = 58, normalized size = 0.73 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- 4 x^{2} + 3 x\right )^{\frac {5}{2}}\, dx \end {gather*}

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 = 2.01, size = 47, normalized size = 0.59 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.24, size = 63, normalized size = 0.80 \begin {gather*} \frac {3645\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{131072}+\frac {15\,\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{3/2}}{512}+\frac {\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{5/2}}{24}+\frac {405\,\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2}}{2048} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 4*x^2)^(5/2),x)

[Out]

(3645*asin((8*x)/3 - 1))/131072 + (15*(4*x - 3/2)*(3*x - 4*x^2)^(3/2))/512 + ((4*x - 3/2)*(3*x - 4*x^2)^(5/2))
/24 + (405*(x/2 - 3/16)*(3*x - 4*x^2)^(1/2))/2048

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