3.1945 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^2 \, dx\)

Optimal. Leaf size=53 \[ \frac{75}{104} (1-2 x)^{13/2}-\frac{505}{88} (1-2 x)^{11/2}+\frac{1133}{72} (1-2 x)^{9/2}-\frac{121}{8} (1-2 x)^{7/2} \]

[Out]

(-121*(1 - 2*x)^(7/2))/8 + (1133*(1 - 2*x)^(9/2))/72 - (505*(1 - 2*x)^(11/2))/88 + (75*(1 - 2*x)^(13/2))/104

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Rubi [A]  time = 0.0106357, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ \frac{75}{104} (1-2 x)^{13/2}-\frac{505}{88} (1-2 x)^{11/2}+\frac{1133}{72} (1-2 x)^{9/2}-\frac{121}{8} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*(1 - 2*x)^(7/2))/8 + (1133*(1 - 2*x)^(9/2))/72 - (505*(1 - 2*x)^(11/2))/88 + (75*(1 - 2*x)^(13/2))/104

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^2 \, dx &=\int \left (\frac{847}{8} (1-2 x)^{5/2}-\frac{1133}{8} (1-2 x)^{7/2}+\frac{505}{8} (1-2 x)^{9/2}-\frac{75}{8} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{121}{8} (1-2 x)^{7/2}+\frac{1133}{72} (1-2 x)^{9/2}-\frac{505}{88} (1-2 x)^{11/2}+\frac{75}{104} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0135188, size = 28, normalized size = 0.53 \[ -\frac{(1-2 x)^{7/2} \left (7425 x^3+18405 x^2+16531 x+5671\right )}{1287} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(5671 + 16531*x + 18405*x^2 + 7425*x^3))/1287

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Maple [A]  time = 0.002, size = 25, normalized size = 0.5 \begin{align*} -{\frac{7425\,{x}^{3}+18405\,{x}^{2}+16531\,x+5671}{1287} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1287*(7425*x^3+18405*x^2+16531*x+5671)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.36408, size = 50, normalized size = 0.94 \begin{align*} \frac{75}{104} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{505}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1133}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{121}{8} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/104*(-2*x + 1)^(13/2) - 505/88*(-2*x + 1)^(11/2) + 1133/72*(-2*x + 1)^(9/2) - 121/8*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.48552, size = 135, normalized size = 2.55 \begin{align*} \frac{1}{1287} \,{\left (59400 \, x^{6} + 58140 \, x^{5} - 44062 \, x^{4} - 49999 \, x^{3} + 12729 \, x^{2} + 17495 \, x - 5671\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1287*(59400*x^6 + 58140*x^5 - 44062*x^4 - 49999*x^3 + 12729*x^2 + 17495*x - 5671)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.93246, size = 100, normalized size = 1.89 \begin{align*} \frac{600 x^{6} \sqrt{1 - 2 x}}{13} + \frac{6460 x^{5} \sqrt{1 - 2 x}}{143} - \frac{44062 x^{4} \sqrt{1 - 2 x}}{1287} - \frac{49999 x^{3} \sqrt{1 - 2 x}}{1287} + \frac{4243 x^{2} \sqrt{1 - 2 x}}{429} + \frac{17495 x \sqrt{1 - 2 x}}{1287} - \frac{5671 \sqrt{1 - 2 x}}{1287} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

600*x**6*sqrt(1 - 2*x)/13 + 6460*x**5*sqrt(1 - 2*x)/143 - 44062*x**4*sqrt(1 - 2*x)/1287 - 49999*x**3*sqrt(1 -
2*x)/1287 + 4243*x**2*sqrt(1 - 2*x)/429 + 17495*x*sqrt(1 - 2*x)/1287 - 5671*sqrt(1 - 2*x)/1287

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Giac [A]  time = 2.0376, size = 88, normalized size = 1.66 \begin{align*} \frac{75}{104} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{505}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1133}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{121}{8} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/104*(2*x - 1)^6*sqrt(-2*x + 1) + 505/88*(2*x - 1)^5*sqrt(-2*x + 1) + 1133/72*(2*x - 1)^4*sqrt(-2*x + 1) + 1
21/8*(2*x - 1)^3*sqrt(-2*x + 1)