3.1886 \(\int (1-2 x)^{3/2} (3+5 x)^3 \, dx\)

Optimal. Leaf size=53 \[ \frac{125}{88} (1-2 x)^{11/2}-\frac{275}{24} (1-2 x)^{9/2}+\frac{1815}{56} (1-2 x)^{7/2}-\frac{1331}{40} (1-2 x)^{5/2} \]

[Out]

(-1331*(1 - 2*x)^(5/2))/40 + (1815*(1 - 2*x)^(7/2))/56 - (275*(1 - 2*x)^(9/2))/24 + (125*(1 - 2*x)^(11/2))/88

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Rubi [A]  time = 0.0088765, antiderivative size = 53, 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} \[ \frac{125}{88} (1-2 x)^{11/2}-\frac{275}{24} (1-2 x)^{9/2}+\frac{1815}{56} (1-2 x)^{7/2}-\frac{1331}{40} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x)^3,x]

[Out]

(-1331*(1 - 2*x)^(5/2))/40 + (1815*(1 - 2*x)^(7/2))/56 - (275*(1 - 2*x)^(9/2))/24 + (125*(1 - 2*x)^(11/2))/88

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)^3 \, dx &=\int \left (\frac{1331}{8} (1-2 x)^{3/2}-\frac{1815}{8} (1-2 x)^{5/2}+\frac{825}{8} (1-2 x)^{7/2}-\frac{125}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{1331}{40} (1-2 x)^{5/2}+\frac{1815}{56} (1-2 x)^{7/2}-\frac{275}{24} (1-2 x)^{9/2}+\frac{125}{88} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0123255, size = 28, normalized size = 0.53 \[ -\frac{(1-2 x)^{5/2} \left (13125 x^3+33250 x^2+31775 x+12592\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(12592 + 31775*x + 33250*x^2 + 13125*x^3))/1155

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Maple [A]  time = 0.002, size = 25, normalized size = 0.5 \begin{align*} -{\frac{13125\,{x}^{3}+33250\,{x}^{2}+31775\,x+12592}{1155} \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)^3,x)

[Out]

-1/1155*(13125*x^3+33250*x^2+31775*x+12592)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.1271, size = 50, normalized size = 0.94 \begin{align*} \frac{125}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{275}{24} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1815}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1331}{40} \,{\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)^3,x, algorithm="maxima")

[Out]

125/88*(-2*x + 1)^(11/2) - 275/24*(-2*x + 1)^(9/2) + 1815/56*(-2*x + 1)^(7/2) - 1331/40*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.35214, size = 120, normalized size = 2.26 \begin{align*} -\frac{1}{1155} \,{\left (52500 \, x^{5} + 80500 \, x^{4} + 7225 \, x^{3} - 43482 \, x^{2} - 18593 \, x + 12592\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)^3,x, algorithm="fricas")

[Out]

-1/1155*(52500*x^5 + 80500*x^4 + 7225*x^3 - 43482*x^2 - 18593*x + 12592)*sqrt(-2*x + 1)

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Sympy [B]  time = 2.00834, size = 286, normalized size = 5.4 \begin{align*} \begin{cases} - \frac{100 \sqrt{5} i \left (x + \frac{3}{5}\right )^{5} \sqrt{10 x - 5}}{11} + \frac{40 \sqrt{5} i \left (x + \frac{3}{5}\right )^{4} \sqrt{10 x - 5}}{3} - \frac{11 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{21} - \frac{121 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{175} - \frac{2662 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{2625} - \frac{29282 \sqrt{5} i \sqrt{10 x - 5}}{13125} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{100 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{5}}{11} + \frac{40 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{4}}{3} - \frac{11 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{21} - \frac{121 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{175} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{2625} - \frac{29282 \sqrt{5} \sqrt{5 - 10 x}}{13125} & \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)**3,x)

[Out]

Piecewise((-100*sqrt(5)*I*(x + 3/5)**5*sqrt(10*x - 5)/11 + 40*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/3 - 11*sqr
t(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/21 - 121*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/175 - 2662*sqrt(5)*I*(x + 3/
5)*sqrt(10*x - 5)/2625 - 29282*sqrt(5)*I*sqrt(10*x - 5)/13125, 10*Abs(x + 3/5)/11 > 1), (-100*sqrt(5)*sqrt(5 -
10*x)*(x + 3/5)**5/11 + 40*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/3 - 11*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**3/21
- 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/175 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/2625 - 29282*sqrt(5)*sqr
t(5 - 10*x)/13125, True))

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Giac [A]  time = 2.144, size = 88, normalized size = 1.66 \begin{align*} -\frac{125}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{275}{24} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1815}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1331}{40} \,{\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)^3,x, algorithm="giac")

[Out]

-125/88*(2*x - 1)^5*sqrt(-2*x + 1) - 275/24*(2*x - 1)^4*sqrt(-2*x + 1) - 1815/56*(2*x - 1)^3*sqrt(-2*x + 1) -
1331/40*(2*x - 1)^2*sqrt(-2*x + 1)