3.1934 \(\int (1-2 x)^{5/2} (3+5 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{18} (1-2 x)^{9/2}-\frac{11}{14} (1-2 x)^{7/2} \]

[Out]

(-11*(1 - 2*x)^(7/2))/14 + (5*(1 - 2*x)^(9/2))/18

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Rubi [A]  time = 0.0048761, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{5}{18} (1-2 x)^{9/2}-\frac{11}{14} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x),x]

[Out]

(-11*(1 - 2*x)^(7/2))/14 + (5*(1 - 2*x)^(9/2))/18

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) \, dx &=\int \left (\frac{11}{2} (1-2 x)^{5/2}-\frac{5}{2} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{11}{14} (1-2 x)^{7/2}+\frac{5}{18} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0086448, size = 18, normalized size = 0.67 \[ -\frac{1}{63} (1-2 x)^{7/2} (35 x+32) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(32 + 35*x))/63

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Maple [A]  time = 0., size = 15, normalized size = 0.6 \begin{align*} -{\frac{35\,x+32}{63} \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),x)

[Out]

-1/63*(35*x+32)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.17762, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{18} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{11}{14} \,{\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),x, algorithm="maxima")

[Out]

5/18*(-2*x + 1)^(9/2) - 11/14*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.52107, size = 86, normalized size = 3.19 \begin{align*} \frac{1}{63} \,{\left (280 \, x^{4} - 164 \, x^{3} - 174 \, x^{2} + 157 \, x - 32\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),x, algorithm="fricas")

[Out]

1/63*(280*x^4 - 164*x^3 - 174*x^2 + 157*x - 32)*sqrt(-2*x + 1)

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Sympy [B]  time = 1.16103, size = 70, normalized size = 2.59 \begin{align*} \frac{40 x^{4} \sqrt{1 - 2 x}}{9} - \frac{164 x^{3} \sqrt{1 - 2 x}}{63} - \frac{58 x^{2} \sqrt{1 - 2 x}}{21} + \frac{157 x \sqrt{1 - 2 x}}{63} - \frac{32 \sqrt{1 - 2 x}}{63} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40*x**4*sqrt(1 - 2*x)/9 - 164*x**3*sqrt(1 - 2*x)/63 - 58*x**2*sqrt(1 - 2*x)/21 + 157*x*sqrt(1 - 2*x)/63 - 32*s
qrt(1 - 2*x)/63

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Giac [A]  time = 2.10712, size = 45, normalized size = 1.67 \begin{align*} \frac{5}{18} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{11}{14} \,{\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),x, algorithm="giac")

[Out]

5/18*(2*x - 1)^4*sqrt(-2*x + 1) + 11/14*(2*x - 1)^3*sqrt(-2*x + 1)