3.1807 \(\int \sqrt{1-2 x} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{28} (1-2 x)^{7/2}+\frac{11}{2} (1-2 x)^{5/2}-\frac{121}{12} (1-2 x)^{3/2} \]

[Out]

(-121*(1 - 2*x)^(3/2))/12 + (11*(1 - 2*x)^(5/2))/2 - (25*(1 - 2*x)^(7/2))/28

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Rubi [A]  time = 0.0072807, 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} \[ -\frac{25}{28} (1-2 x)^{7/2}+\frac{11}{2} (1-2 x)^{5/2}-\frac{121}{12} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(3/2))/12 + (11*(1 - 2*x)^(5/2))/2 - (25*(1 - 2*x)^(7/2))/28

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 \sqrt{1-2 x} (3+5 x)^2 \, dx &=\int \left (\frac{121}{4} \sqrt{1-2 x}-\frac{55}{2} (1-2 x)^{3/2}+\frac{25}{4} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{121}{12} (1-2 x)^{3/2}+\frac{11}{2} (1-2 x)^{5/2}-\frac{25}{28} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0093514, size = 23, normalized size = 0.57 \[ -\frac{1}{21} (1-2 x)^{3/2} \left (75 x^2+156 x+115\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(115 + 156*x + 75*x^2))/21

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{75\,{x}^{2}+156\,x+115}{21} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(75*x^2+156*x+115)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.15597, size = 38, normalized size = 0.95 \begin{align*} -\frac{25}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{11}{2} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{121}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/28*(-2*x + 1)^(7/2) + 11/2*(-2*x + 1)^(5/2) - 121/12*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.52365, size = 73, normalized size = 1.82 \begin{align*} \frac{1}{21} \,{\left (150 \, x^{3} + 237 \, x^{2} + 74 \, x - 115\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(150*x^3 + 237*x^2 + 74*x - 115)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.24947, size = 187, normalized size = 4.68 \begin{align*} \begin{cases} \frac{10 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{7} - \frac{11 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{35} - \frac{242 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{525} - \frac{2662 \sqrt{5} i \sqrt{10 x - 5}}{2625} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{10 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{7} - \frac{11 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{35} - \frac{242 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{525} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x}}{2625} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((10*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/7 - 11*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/35 - 242*sqrt
(5)*I*(x + 3/5)*sqrt(10*x - 5)/525 - 2662*sqrt(5)*I*sqrt(10*x - 5)/2625, 10*Abs(x + 3/5)/11 > 1), (10*sqrt(5)*
sqrt(5 - 10*x)*(x + 3/5)**3/7 - 11*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/35 - 242*sqrt(5)*sqrt(5 - 10*x)*(x + 3/
5)/525 - 2662*sqrt(5)*sqrt(5 - 10*x)/2625, True))

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Giac [A]  time = 2.0016, size = 57, normalized size = 1.42 \begin{align*} \frac{25}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{11}{2} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{121}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/28*(2*x - 1)^3*sqrt(-2*x + 1) + 11/2*(2*x - 1)^2*sqrt(-2*x + 1) - 121/12*(-2*x + 1)^(3/2)