3.1946 \(\int (1-2 x)^{5/2} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{44} (1-2 x)^{11/2}+\frac{55}{18} (1-2 x)^{9/2}-\frac{121}{28} (1-2 x)^{7/2} \]

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

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Rubi [A]  time = 0.0076592, 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} \[ -\frac{25}{44} (1-2 x)^{11/2}+\frac{55}{18} (1-2 x)^{9/2}-\frac{121}{28} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

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 (1-2 x)^{5/2} (3+5 x)^2 \, dx &=\int \left (\frac{121}{4} (1-2 x)^{5/2}-\frac{55}{2} (1-2 x)^{7/2}+\frac{25}{4} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{121}{28} (1-2 x)^{7/2}+\frac{55}{18} (1-2 x)^{9/2}-\frac{25}{44} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0109682, size = 23, normalized size = 0.57 \[ -\frac{1}{693} (1-2 x)^{7/2} \left (1575 x^2+2660 x+1271\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(1271 + 2660*x + 1575*x^2))/693

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{1575\,{x}^{2}+2660\,x+1271}{693} \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)*(3+5*x)^2,x)

[Out]

-1/693*(1575*x^2+2660*x+1271)*(1-2*x)^(7/2)

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Maxima [A]  time = 2.20678, size = 38, normalized size = 0.95 \begin{align*} -\frac{25}{44} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{55}{18} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{121}{28} \,{\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)*(3+5*x)^2,x, algorithm="maxima")

[Out]

-25/44*(-2*x + 1)^(11/2) + 55/18*(-2*x + 1)^(9/2) - 121/28*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.46786, size = 112, normalized size = 2.8 \begin{align*} \frac{1}{693} \,{\left (12600 \, x^{5} + 2380 \, x^{4} - 12302 \, x^{3} - 867 \, x^{2} + 4966 \, x - 1271\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(12600*x^5 + 2380*x^4 - 12302*x^3 - 867*x^2 + 4966*x - 1271)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.50879, size = 85, normalized size = 2.12 \begin{align*} \frac{200 x^{5} \sqrt{1 - 2 x}}{11} + \frac{340 x^{4} \sqrt{1 - 2 x}}{99} - \frac{12302 x^{3} \sqrt{1 - 2 x}}{693} - \frac{289 x^{2} \sqrt{1 - 2 x}}{231} + \frac{4966 x \sqrt{1 - 2 x}}{693} - \frac{1271 \sqrt{1 - 2 x}}{693} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

200*x**5*sqrt(1 - 2*x)/11 + 340*x**4*sqrt(1 - 2*x)/99 - 12302*x**3*sqrt(1 - 2*x)/693 - 289*x**2*sqrt(1 - 2*x)/
231 + 4966*x*sqrt(1 - 2*x)/693 - 1271*sqrt(1 - 2*x)/693

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Giac [A]  time = 2.72447, size = 66, normalized size = 1.65 \begin{align*} \frac{25}{44} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{55}{18} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{121}{28} \,{\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)*(3+5*x)^2,x, algorithm="giac")

[Out]

25/44*(2*x - 1)^5*sqrt(-2*x + 1) + 55/18*(2*x - 1)^4*sqrt(-2*x + 1) + 121/28*(2*x - 1)^3*sqrt(-2*x + 1)