3.1.6 \(\int (3 x-4 x^2)^{7/2} \, dx\) [6]

Optimal. Leaf size=101 \[ -\frac {25515 (3-8 x) \sqrt {3 x-4 x^2}}{4194304}-\frac {945 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{131072}-\frac {21 (3-8 x) \left (3 x-4 x^2\right )^{5/2}}{2048}-\frac {1}{64} (3-8 x) \left (3 x-4 x^2\right )^{7/2}-\frac {229635 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{16777216} \]

[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)

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Rubi [A]
time = 0.02, 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 = {626, 633, 222} \begin {gather*} -\frac {229635 \text {ArcSin}\left (1-\frac {8 x}{3}\right )}{16777216}-\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} \end {gather*}

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 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 )^{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 \text {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*}

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Mathematica [A]
time = 0.10, size = 102, normalized size = 1.01 \begin {gather*} \frac {\sqrt {-x (-3+4 x)} \left (-2 \sqrt {x} \sqrt {-3+4 x} \left (76545+68040 x+72576 x^2+82944 x^3-25067520 x^4+79429632 x^5-88080384 x^6+33554432 x^7\right )+229635 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )\right )}{8388608 \sqrt {x} \sqrt {-3+4 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(7/2),x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(-2*Sqrt[x]*Sqrt[-3 + 4*x]*(76545 + 68040*x + 72576*x^2 + 82944*x^3 - 25067520*x^4 + 79
429632*x^5 - 88080384*x^6 + 33554432*x^7) + 229635*Log[-2*Sqrt[x] + Sqrt[-3 + 4*x]]))/(8388608*Sqrt[x]*Sqrt[-3
+ 4*x])

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Maple [A]
time = 0.42, size = 82, normalized size = 0.81

method	result	size
risch	\(\frac {\left (33554432 x^{7}-88080384 x^{6}[phone] x^{5}-25067520 x^{4}+82944 x^{3}+72576 x^{2}+68040 x +76545\right ) x \left (-3+4 x \right )}{4194304 \sqrt {-x \left (-3+4 x \right )}}+\frac {229635 \arcsin \left (-1+\frac {8 x}{3}\right )}{16777216}\)	\(63\)
meijerg	\(-\frac {688905 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (\frac {33554432}{243} x^{7}-\frac {29360128}{81} x^{6}+\frac {26476544}{81} x^{5}-\frac {2785280}{27} x^{4}+\frac {1024}{3} x^{3}+\frac {896}{3} x^{2}+280 x +315\right ) \sqrt {-\frac {4 x}{3}+1}}{1451520}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{3072}\right )}{8192 \sqrt {\pi }}\)	\(77\)
default	\(-\frac {945 \left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{131072}-\frac {21 \left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}{2048}-\frac {\left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {7}{2}}}{64}+\frac {229635 \arcsin \left (-1+\frac {8 x}{3}\right )}{16777216}-\frac {25515 \left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{4194304}\)	\(82\)
trager	\(\left (-8 x^{7}+21 x^{6}-\frac {303}{16} x^{5}+\frac {765}{128} x^{4}-\frac {81}{4096} x^{3}-\frac {567}{32768} x^{2}-\frac {8505}{524288} x -\frac {76545}{4194304}\right ) \sqrt {-4 x^{2}+3 x}+\frac {229635 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{16777216}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]
time = 0.48, size = 117, normalized size = 1.16 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 1.51, size = 68, normalized size = 0.67 \begin {gather*} -\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 {gather*}

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)

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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 {7}{2}}\, dx \end {gather*}

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)

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Giac [A]
time = 1.38, size = 57, normalized size = 0.56 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.17, size = 81, normalized size = 0.80 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 4*x^2)^(7/2),x)

[Out]

(229635*asin((8*x)/3 - 1))/16777216 + (945*(4*x - 3/2)*(3*x - 4*x^2)^(3/2))/65536 + (21*(4*x - 3/2)*(3*x - 4*x
^2)^(5/2))/1024 + ((4*x - 3/2)*(3*x - 4*x^2)^(7/2))/32 + (25515*(x/2 - 3/16)*(3*x - 4*x^2)^(1/2))/262144

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