3.1.24 \(\int \frac {1}{(3 x-4 x^2)^{7/2}} \, dx\) [24]

Optimal. Leaf size=67 \[ -\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}} \]

[Out]

-2/45*(3-8*x)/(-4*x^2+3*x)^(5/2)-128/1215*(3-8*x)/(-4*x^2+3*x)^(3/2)-4096/10935*(3-8*x)/(-4*x^2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {628, 627} \begin {gather*} -\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*x - 4*x^2)^(-7/2),x]

[Out]

(-2*(3 - 8*x))/(45*(3*x - 4*x^2)^(5/2)) - (128*(3 - 8*x))/(1215*(3*x - 4*x^2)^(3/2)) - (4096*(3 - 8*x))/(10935
*Sqrt[3*x - 4*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rubi steps

\begin {align*} \int \frac {1}{\left (3 x-4 x^2\right )^{7/2}} \, dx &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}+\frac {64}{45} \int \frac {1}{\left (3 x-4 x^2\right )^{5/2}} \, dx\\ &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}+\frac {2048 \int \frac {1}{\left (3 x-4 x^2\right )^{3/2}} \, dx}{1215}\\ &=-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}-\frac {128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac {4096 (3-8 x)}{10935 \sqrt {3 x-4 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 41, normalized size = 0.61 \begin {gather*} \frac {2 \left (-729-3240 x-34560 x^2+276480 x^3-491520 x^4+262144 x^5\right )}{10935 (-x (-3+4 x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(-7/2),x]

[Out]

(2*(-729 - 3240*x - 34560*x^2 + 276480*x^3 - 491520*x^4 + 262144*x^5))/(10935*(-(x*(-3 + 4*x)))^(5/2))

________________________________________________________________________________________

Maple [A]
time = 0.38, size = 56, normalized size = 0.84

method	result	size
meijerg	\(-\frac {2 \sqrt {3}\, \left (-\frac {262144}{243} x^{5}+\frac {163840}{81} x^{4}-\frac {10240}{9} x^{3}+\frac {1280}{9} x^{2}+\frac {40}{3} x +3\right )}{1215 x^{\frac {5}{2}} \left (-\frac {4 x}{3}+1\right )^{\frac {5}{2}}}\)	\(41\)
gosper	\(-\frac {2 x \left (-3+4 x \right ) \left (262144 x^{5}-491520 x^{4}+276480 x^{3}-34560 x^{2}-3240 x -729\right )}{10935 \left (-4 x^{2}+3 x \right )^{\frac {7}{2}}}\)	\(45\)
trager	\(-\frac {2 \left (262144 x^{5}-491520 x^{4}+276480 x^{3}-34560 x^{2}-3240 x -729\right ) \sqrt {-4 x^{2}+3 x}}{10935 \left (-3+4 x \right )^{3} x^{3}}\)	\(49\)
default	\(-\frac {2 \left (3-8 x \right )}{45 \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}-\frac {128 \left (3-8 x \right )}{1215 \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}-\frac {4096 \left (3-8 x \right )}{10935 \sqrt {-4 x^{2}+3 x}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+3*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/45*(3-8*x)/(-4*x^2+3*x)^(5/2)-128/1215*(3-8*x)/(-4*x^2+3*x)^(3/2)-4096/10935*(3-8*x)/(-4*x^2+3*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 82, normalized size = 1.22 \begin {gather*} \frac {32768 \, x}{10935 \, \sqrt {-4 \, x^{2} + 3 \, x}} - \frac {4096}{3645 \, \sqrt {-4 \, x^{2} + 3 \, x}} + \frac {1024 \, x}{1215 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} - \frac {128}{405 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} + \frac {16 \, x}{45 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} - \frac {2}{15 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(7/2),x, algorithm="maxima")

[Out]

32768/10935*x/sqrt(-4*x^2 + 3*x) - 4096/3645/sqrt(-4*x^2 + 3*x) + 1024/1215*x/(-4*x^2 + 3*x)^(3/2) - 128/405/(
-4*x^2 + 3*x)^(3/2) + 16/45*x/(-4*x^2 + 3*x)^(5/2) - 2/15/(-4*x^2 + 3*x)^(5/2)

________________________________________________________________________________________

Fricas [A]
time = 1.76, size = 61, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (262144 \, x^{5} - 491520 \, x^{4} + 276480 \, x^{3} - 34560 \, x^{2} - 3240 \, x - 729\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{10935 \, {\left (64 \, x^{6} - 144 \, x^{5} + 108 \, x^{4} - 27 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(7/2),x, algorithm="fricas")

[Out]

-2/10935*(262144*x^5 - 491520*x^4 + 276480*x^3 - 34560*x^2 - 3240*x - 729)*sqrt(-4*x^2 + 3*x)/(64*x^6 - 144*x^
5 + 108*x^4 - 27*x^3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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(1/(-4*x**2+3*x)**(7/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(-7/2), x)

________________________________________________________________________________________

Giac [A]
time = 1.82, size = 49, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x - 15\right )} x + 135\right )} x - 135\right )} x - 405\right )} x - 729\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{10935 \, {\left (4 \, x^{2} - 3 \, x\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(7/2),x, algorithm="giac")

[Out]

-2/10935*(8*(32*(8*(16*(8*x - 15)*x + 135)*x - 135)*x - 405)*x - 729)*sqrt(-4*x^2 + 3*x)/(4*x^2 - 3*x)^3

________________________________________________________________________________________

Mupad [B]
time = 0.20, size = 73, normalized size = 1.09 \begin {gather*} -\frac {6480\,x-9216\,x\,\left (3\,x-4\,x^2\right )-32768\,x\,{\left (3\,x-4\,x^2\right )}^2+12288\,{\left (3\,x-4\,x^2\right )}^2-13824\,x^2+1458}{{\left (3\,x-4\,x^2\right )}^{3/2}\,\left (32805\,x-43740\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x - 4*x^2)^(7/2),x)

[Out]

-(6480*x - 9216*x*(3*x - 4*x^2) - 32768*x*(3*x - 4*x^2)^2 + 12288*(3*x - 4*x^2)^2 - 13824*x^2 + 1458)/((3*x -
4*x^2)^(3/2)*(32805*x - 43740*x^2))

________________________________________________________________________________________