3.24 \(\int \frac{1}{(3 x-4 x^2)^{7/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0115283, antiderivative size = 67, normalized size of antiderivative = 1., 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 = {614, 613} \[ -\frac{4096 (3-8 x)}{10935 \sqrt{3 x-4 x^2}}-\frac{128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac{2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \]

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 614

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]

Rule 613

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]

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 = 0.0148466, size = 51, normalized size = 0.76 \[ \frac{2 \left (262144 x^5-491520 x^4+276480 x^3-34560 x^2-3240 x-729\right )}{10935 (3-4 x)^2 x^2 \sqrt{-x (4 x-3)}} \]

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*(3 - 4*x)^2*x^2*Sqrt[-(x*(-3 + 4
*x))])

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Maple [A]  time = 0.048, size = 45, normalized size = 0.7 \begin{align*} -{\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/10935*x*(-3+4*x)*(262144*x^5-491520*x^4+276480*x^3-34560*x^2-3240*x-729)/(-4*x^2+3*x)^(7/2)

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Maxima [A]  time = 1.36279, size = 111, normalized size = 1.66 \begin{align*} \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{align*}

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)

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Fricas [A]  time = 2.24674, size = 180, normalized size = 2.69 \begin{align*} -\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{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-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 = 1.627, size = 66, normalized size = 0.99 \begin{align*} -\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{align*}

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