3.1863 \(\int (1-2 x)^{3/2} (3+5 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{14} (1-2 x)^{7/2}-\frac{11}{10} (1-2 x)^{5/2} \]

[Out]

(-11*(1 - 2*x)^(5/2))/10 + (5*(1 - 2*x)^(7/2))/14

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Rubi [A]  time = 0.0051879, 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} \[ \frac{5}{14} (1-2 x)^{7/2}-\frac{11}{10} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x),x]

[Out]

(-11*(1 - 2*x)^(5/2))/10 + (5*(1 - 2*x)^(7/2))/14

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) \, dx &=\int \left (\frac{11}{2} (1-2 x)^{3/2}-\frac{5}{2} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{11}{10} (1-2 x)^{5/2}+\frac{5}{14} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0084097, size = 18, normalized size = 0.67 \[ -\frac{1}{35} (1-2 x)^{5/2} (25 x+26) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(26 + 25*x))/35

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*} -{\frac{25\,x+26}{35} \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),x)

[Out]

-1/35*(25*x+26)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.2933, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{14} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{11}{10} \,{\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),x, algorithm="maxima")

[Out]

5/14*(-2*x + 1)^(7/2) - 11/10*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.40494, size = 70, normalized size = 2.59 \begin{align*} -\frac{1}{35} \,{\left (100 \, x^{3} + 4 \, x^{2} - 79 \, x + 26\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),x, algorithm="fricas")

[Out]

-1/35*(100*x^3 + 4*x^2 - 79*x + 26)*sqrt(-2*x + 1)

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Sympy [B]  time = 0.357208, size = 54, normalized size = 2. \begin{align*} - \frac{20 x^{3} \sqrt{1 - 2 x}}{7} - \frac{4 x^{2} \sqrt{1 - 2 x}}{35} + \frac{79 x \sqrt{1 - 2 x}}{35} - \frac{26 \sqrt{1 - 2 x}}{35} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*x**3*sqrt(1 - 2*x)/7 - 4*x**2*sqrt(1 - 2*x)/35 + 79*x*sqrt(1 - 2*x)/35 - 26*sqrt(1 - 2*x)/35

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Giac [A]  time = 2.02132, size = 45, normalized size = 1.67 \begin{align*} -\frac{5}{14} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{11}{10} \,{\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),x, algorithm="giac")

[Out]

-5/14*(2*x - 1)^3*sqrt(-2*x + 1) - 11/10*(2*x - 1)^2*sqrt(-2*x + 1)