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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} -\frac {9}{4 \sqrt {2 x-1}}+\frac {7}{2 (2 x-1)^{3/2}}-\frac {49}{20 (2 x-1)^{5/2}} \end {gather*}

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 45

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 = 0.02, size = 23, normalized size = 0.58 \begin {gather*} \frac {-41+80 x-45 x^2}{5 (-1+2 x)^{5/2}} \end {gather*}

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))

________________________________________________________________________________________

Mathics [A]
time = 2.91, size = 31, normalized size = 0.78 \begin {gather*} \frac {-41+80 x-45 x^2}{5 \sqrt {-1+2 x} \left (1-4 x+4 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41 + 80 x - 45 x ^ 2) / (5 Sqrt[-1 + 2 x] (1 - 4 x + 4 x ^ 2))

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 29, normalized size = 0.72

method	result	size
gosper	\(-\frac {45 x^{2}-80 x +41}{5 \left (2 x -1\right )^{\frac {5}{2}}}\)	\(20\)
trager	\(-\frac {45 x^{2}-80 x +41}{5 \left (2 x -1\right )^{\frac {5}{2}}}\)	\(20\)
risch	\(-\frac {45 x^{2}-80 x +41}{5 \left (2 x -1\right )^{\frac {5}{2}}}\)	\(20\)
derivativedivides	\(-\frac {49}{20 \left (2 x -1\right )^{\frac {5}{2}}}+\frac {7}{2 \left (2 x -1\right )^{\frac {3}{2}}}-\frac {9}{4 \sqrt {2 x -1}}\)	\(29\)
default	\(-\frac {49}{20 \left (2 x -1\right )^{\frac {5}{2}}}+\frac {7}{2 \left (2 x -1\right )^{\frac {3}{2}}}-\frac {9}{4 \sqrt {2 x -1}}\)	\(29\)
meijerg	\(-\frac {20 \left (-\mathrm {signum}\left (x -\frac {1}{2}\right )\right )^{\frac {7}{2}} \left (\frac {3 \sqrt {\pi }}{4}-\frac {3 \sqrt {\pi }}{4 \left (1-2 x \right )^{\frac {5}{2}}}\right )}{3 \sqrt {\pi }\, \mathrm {signum}\left (x -\frac {1}{2}\right )^{\frac {7}{2}}}-\frac {3 \left (-\mathrm {signum}\left (x -\frac {1}{2}\right )\right )^{\frac {7}{2}} \left (2 \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (120 x^{2}-80 x +16\right )}{8 \left (1-2 x \right )^{\frac {5}{2}}}\right )}{5 \sqrt {\pi }\, \mathrm {signum}\left (x -\frac {1}{2}\right )^{\frac {7}{2}}}-\frac {4 \left (-\mathrm {signum}\left (x -\frac {1}{2}\right )\right )^{\frac {7}{2}} \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-40 x +8\right )}{16 \left (1-2 x \right )^{\frac {5}{2}}}\right )}{\sqrt {\pi }\, \mathrm {signum}\left (x -\frac {1}{2}\right )^{\frac {7}{2}}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x-5)^2/(2*x-1)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.24, size = 23, normalized size = 0.58 \begin {gather*} -\frac {45 \, {\left (2 \, x - 1\right )}^{2} - 140 \, x + 119}{20 \, {\left (2 \, x - 1\right )}^{\frac {5}{2}}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.31, size = 36, normalized size = 0.90 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (34) = 68\)
time = 0.34, size = 116, normalized size = 2.90 \begin {gather*} - \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 {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} \frac {-45 \left (2 x-1\right )^{2}+70 \left (2 x-1\right )-49}{20 \sqrt {2 x-1} \left (2 x-1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.35, size = 23, normalized size = 0.58 \begin {gather*} -\frac {\frac {9\,{\left (2\,x-1\right )}^2}{4}-7\,x+\frac {119}{20}}{{\left (2\,x-1\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 5)^2/(2*x - 1)^(7/2),x)

[Out]

-((9*(2*x - 1)^2)/4 - 7*x + 119/20)/(2*x - 1)^(5/2)

________________________________________________________________________________________