3.1874 \(\int (1-2 x)^{3/2} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{36} (1-2 x)^{9/2}+\frac{55}{14} (1-2 x)^{7/2}-\frac{121}{20} (1-2 x)^{5/2} \]

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

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Rubi [A]  time = 0.0073321, 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} \[ -\frac{25}{36} (1-2 x)^{9/2}+\frac{55}{14} (1-2 x)^{7/2}-\frac{121}{20} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

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)^{3/2} (3+5 x)^2 \, dx &=\int \left (\frac{121}{4} (1-2 x)^{3/2}-\frac{55}{2} (1-2 x)^{5/2}+\frac{25}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{121}{20} (1-2 x)^{5/2}+\frac{55}{14} (1-2 x)^{7/2}-\frac{25}{36} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0102137, size = 23, normalized size = 0.57 \[ -\frac{1}{315} (1-2 x)^{5/2} \left (875 x^2+1600 x+887\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(887 + 1600*x + 875*x^2))/315

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{875\,{x}^{2}+1600\,x+887}{315} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(875*x^2+1600*x+887)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.11864, size = 38, normalized size = 0.95 \begin{align*} -\frac{25}{36} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{55}{14} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{121}{20} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(-2*x + 1)^(9/2) + 55/14*(-2*x + 1)^(7/2) - 121/20*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.32179, size = 96, normalized size = 2.4 \begin{align*} -\frac{1}{315} \,{\left (3500 \, x^{4} + 2900 \, x^{3} - 1977 \, x^{2} - 1948 \, x + 887\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(3500*x^4 + 2900*x^3 - 1977*x^2 - 1948*x + 887)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.44187, size = 236, normalized size = 5.9 \begin{align*} \begin{cases} - \frac{20 \sqrt{5} i \left (x + \frac{3}{5}\right )^{4} \sqrt{10 x - 5}}{9} + \frac{220 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{63} - \frac{121 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{525} - \frac{2662 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{7875} - \frac{29282 \sqrt{5} i \sqrt{10 x - 5}}{39375} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{20 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{4}}{9} + \frac{220 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{63} - \frac{121 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{525} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{7875} - \frac{29282 \sqrt{5} \sqrt{5 - 10 x}}{39375} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-20*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/9 + 220*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/63 - 121*sq
rt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/525 - 2662*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/7875 - 29282*sqrt(5)*I*sqrt(
10*x - 5)/39375, 10*Abs(x + 3/5)/11 > 1), (-20*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/9 + 220*sqrt(5)*sqrt(5 - 10
*x)*(x + 3/5)**3/63 - 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/525 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/7875
- 29282*sqrt(5)*sqrt(5 - 10*x)/39375, True))

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Giac [A]  time = 2.41498, size = 66, normalized size = 1.65 \begin{align*} -\frac{25}{36} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{55}{14} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{121}{20} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(2*x - 1)^4*sqrt(-2*x + 1) - 55/14*(2*x - 1)^3*sqrt(-2*x + 1) - 121/20*(2*x - 1)^2*sqrt(-2*x + 1)