3.1959 \(\int (1-2 x)^{5/2} (3+5 x)^3 \, dx\)

Optimal. Leaf size=53 \[ \frac{125}{104} (1-2 x)^{13/2}-\frac{75}{8} (1-2 x)^{11/2}+\frac{605}{24} (1-2 x)^{9/2}-\frac{1331}{56} (1-2 x)^{7/2} \]

[Out]

(-1331*(1 - 2*x)^(7/2))/56 + (605*(1 - 2*x)^(9/2))/24 - (75*(1 - 2*x)^(11/2))/8 + (125*(1 - 2*x)^(13/2))/104

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Rubi [A]  time = 0.0092651, 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} \[ \frac{125}{104} (1-2 x)^{13/2}-\frac{75}{8} (1-2 x)^{11/2}+\frac{605}{24} (1-2 x)^{9/2}-\frac{1331}{56} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x)^3,x]

[Out]

(-1331*(1 - 2*x)^(7/2))/56 + (605*(1 - 2*x)^(9/2))/24 - (75*(1 - 2*x)^(11/2))/8 + (125*(1 - 2*x)^(13/2))/104

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)^3 \, dx &=\int \left (\frac{1331}{8} (1-2 x)^{5/2}-\frac{1815}{8} (1-2 x)^{7/2}+\frac{825}{8} (1-2 x)^{9/2}-\frac{125}{8} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{1331}{56} (1-2 x)^{7/2}+\frac{605}{24} (1-2 x)^{9/2}-\frac{75}{8} (1-2 x)^{11/2}+\frac{125}{104} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0132644, size = 28, normalized size = 0.53 \[ -\frac{1}{273} (1-2 x)^{7/2} \left (2625 x^3+6300 x^2+5495 x+1838\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(1838 + 5495*x + 6300*x^2 + 2625*x^3))/273

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{2625\,{x}^{3}+6300\,{x}^{2}+5495\,x+1838}{273} \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)^3,x)

[Out]

-1/273*(2625*x^3+6300*x^2+5495*x+1838)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.12376, size = 50, normalized size = 0.94 \begin{align*} \frac{125}{104} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{75}{8} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{605}{24} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{1331}{56} \,{\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)^3,x, algorithm="maxima")

[Out]

125/104*(-2*x + 1)^(13/2) - 75/8*(-2*x + 1)^(11/2) + 605/24*(-2*x + 1)^(9/2) - 1331/56*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.36454, size = 131, normalized size = 2.47 \begin{align*} \frac{1}{273} \,{\left (21000 \, x^{6} + 18900 \, x^{5} - 15890 \, x^{4} - 16061 \, x^{3} + 4614 \, x^{2} + 5533 \, x - 1838\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)^3,x, algorithm="fricas")

[Out]

1/273*(21000*x^6 + 18900*x^5 - 15890*x^4 - 16061*x^3 + 4614*x^2 + 5533*x - 1838)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.99777, size = 100, normalized size = 1.89 \begin{align*} \frac{1000 x^{6} \sqrt{1 - 2 x}}{13} + \frac{900 x^{5} \sqrt{1 - 2 x}}{13} - \frac{2270 x^{4} \sqrt{1 - 2 x}}{39} - \frac{16061 x^{3} \sqrt{1 - 2 x}}{273} + \frac{1538 x^{2} \sqrt{1 - 2 x}}{91} + \frac{5533 x \sqrt{1 - 2 x}}{273} - \frac{1838 \sqrt{1 - 2 x}}{273} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000*x**6*sqrt(1 - 2*x)/13 + 900*x**5*sqrt(1 - 2*x)/13 - 2270*x**4*sqrt(1 - 2*x)/39 - 16061*x**3*sqrt(1 - 2*x)
/273 + 1538*x**2*sqrt(1 - 2*x)/91 + 5533*x*sqrt(1 - 2*x)/273 - 1838*sqrt(1 - 2*x)/273

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Giac [A]  time = 2.23161, size = 88, normalized size = 1.66 \begin{align*} \frac{125}{104} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{75}{8} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{605}{24} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{1331}{56} \,{\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)^3,x, algorithm="giac")

[Out]

125/104*(2*x - 1)^6*sqrt(-2*x + 1) + 75/8*(2*x - 1)^5*sqrt(-2*x + 1) + 605/24*(2*x - 1)^4*sqrt(-2*x + 1) + 133
1/56*(2*x - 1)^3*sqrt(-2*x + 1)