3.1 \(\int \frac{(-5+3 x)^2}{(-1+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{9}{4 \sqrt{2 x-1}}+\frac{7}{2 (2 x-1)^{3/2}}-\frac{49}{20 (2 x-1)^{5/2}} \]

[Out]

-49/(20*(-1 + 2*x)^(5/2)) + 7/(2*(-1 + 2*x)^(3/2)) - 9/(4*Sqrt[-1 + 2*x])

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Rubi [A]  time = 0.0074687, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{9}{4 \sqrt{2 x-1}}+\frac{7}{2 (2 x-1)^{3/2}}-\frac{49}{20 (2 x-1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(-5 + 3*x)^2/(-1 + 2*x)^(7/2),x]

[Out]

-49/(20*(-1 + 2*x)^(5/2)) + 7/(2*(-1 + 2*x)^(3/2)) - 9/(4*Sqrt[-1 + 2*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(-5+3 x)^2}{(-1+2 x)^{7/2}} \, dx &=\int \left (\frac{49}{4 (-1+2 x)^{7/2}}-\frac{21}{2 (-1+2 x)^{5/2}}+\frac{9}{4 (-1+2 x)^{3/2}}\right ) \, dx\\ &=-\frac{49}{20 (-1+2 x)^{5/2}}+\frac{7}{2 (-1+2 x)^{3/2}}-\frac{9}{4 \sqrt{-1+2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0117336, size = 23, normalized size = 0.57 \[ \frac{-45 x^2+80 x-41}{5 (2 x-1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 3*x)^2/(-1 + 2*x)^(7/2),x]

[Out]

(-41 + 80*x - 45*x^2)/(5*(-1 + 2*x)^(5/2))

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{45\,{x}^{2}-80\,x+41}{5} \left ( 2\,x-1 \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5+3*x)^2/(2*x-1)^(7/2),x)

[Out]

-1/5*(45*x^2-80*x+41)/(2*x-1)^(5/2)

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Maxima [A]  time = 0.921544, size = 31, normalized size = 0.78 \begin{align*} -\frac{45 \,{\left (2 \, x - 1\right )}^{2} - 140 \, x + 119}{20 \,{\left (2 \, x - 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+3*x)^2/(-1+2*x)^(7/2),x, algorithm="maxima")

[Out]

-1/20*(45*(2*x - 1)^2 - 140*x + 119)/(2*x - 1)^(5/2)

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Fricas [A]  time = 1.83398, size = 92, normalized size = 2.3 \begin{align*} -\frac{{\left (45 \, x^{2} - 80 \, x + 41\right )} \sqrt{2 \, x - 1}}{5 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+3*x)^2/(-1+2*x)^(7/2),x, algorithm="fricas")

[Out]

-1/5*(45*x^2 - 80*x + 41)*sqrt(2*x - 1)/(8*x^3 - 12*x^2 + 6*x - 1)

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Sympy [B]  time = 1.35946, size = 116, normalized size = 2.9 \begin{align*} - \frac{45 x^{2}}{20 x^{2} \sqrt{2 x - 1} - 20 x \sqrt{2 x - 1} + 5 \sqrt{2 x - 1}} + \frac{80 x}{20 x^{2} \sqrt{2 x - 1} - 20 x \sqrt{2 x - 1} + 5 \sqrt{2 x - 1}} - \frac{41}{20 x^{2} \sqrt{2 x - 1} - 20 x \sqrt{2 x - 1} + 5 \sqrt{2 x - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+3*x)**2/(-1+2*x)**(7/2),x)

[Out]

-45*x**2/(20*x**2*sqrt(2*x - 1) - 20*x*sqrt(2*x - 1) + 5*sqrt(2*x - 1)) + 80*x/(20*x**2*sqrt(2*x - 1) - 20*x*s
qrt(2*x - 1) + 5*sqrt(2*x - 1)) - 41/(20*x**2*sqrt(2*x - 1) - 20*x*sqrt(2*x - 1) + 5*sqrt(2*x - 1))

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Giac [A]  time = 1.12353, size = 31, normalized size = 0.78 \begin{align*} -\frac{45 \,{\left (2 \, x - 1\right )}^{2} - 140 \, x + 119}{20 \,{\left (2 \, x - 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+3*x)^2/(-1+2*x)^(7/2),x, algorithm="giac")

[Out]

-1/20*(45*(2*x - 1)^2 - 140*x + 119)/(2*x - 1)^(5/2)