3.1862 \(\int (1-2 x)^{3/2} (2+3 x) (3+5 x) \, dx\)

Optimal. Leaf size=40 \[ -\frac{5}{12} (1-2 x)^{9/2}+\frac{17}{7} (1-2 x)^{7/2}-\frac{77}{20} (1-2 x)^{5/2} \]

[Out]

(-77*(1 - 2*x)^(5/2))/20 + (17*(1 - 2*x)^(7/2))/7 - (5*(1 - 2*x)^(9/2))/12

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Rubi [A]  time = 0.0081724, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{5}{12} (1-2 x)^{9/2}+\frac{17}{7} (1-2 x)^{7/2}-\frac{77}{20} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77*(1 - 2*x)^(5/2))/20 + (17*(1 - 2*x)^(7/2))/7 - (5*(1 - 2*x)^(9/2))/12

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x) \, dx &=\int \left (\frac{77}{4} (1-2 x)^{3/2}-17 (1-2 x)^{5/2}+\frac{15}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{77}{20} (1-2 x)^{5/2}+\frac{17}{7} (1-2 x)^{7/2}-\frac{5}{12} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0104367, size = 23, normalized size = 0.57 \[ -\frac{1}{105} (1-2 x)^{5/2} \left (175 x^2+335 x+193\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(193 + 335*x + 175*x^2))/105

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{175\,{x}^{2}+335\,x+193}{105} \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)*(2+3*x)*(3+5*x),x)

[Out]

-1/105*(175*x^2+335*x+193)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.04005, size = 38, normalized size = 0.95 \begin{align*} -\frac{5}{12} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{17}{7} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{77}{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)*(2+3*x)*(3+5*x),x, algorithm="maxima")

[Out]

-5/12*(-2*x + 1)^(9/2) + 17/7*(-2*x + 1)^(7/2) - 77/20*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.34833, size = 90, normalized size = 2.25 \begin{align*} -\frac{1}{105} \,{\left (700 \, x^{4} + 640 \, x^{3} - 393 \, x^{2} - 437 \, x + 193\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(700*x^4 + 640*x^3 - 393*x^2 - 437*x + 193)*sqrt(-2*x + 1)

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Sympy [A]  time = 5.73067, size = 34, normalized size = 0.85 \begin{align*} - \frac{5 \left (1 - 2 x\right )^{\frac{9}{2}}}{12} + \frac{17 \left (1 - 2 x\right )^{\frac{7}{2}}}{7} - \frac{77 \left (1 - 2 x\right )^{\frac{5}{2}}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*(1 - 2*x)**(9/2)/12 + 17*(1 - 2*x)**(7/2)/7 - 77*(1 - 2*x)**(5/2)/20

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Giac [A]  time = 2.72287, size = 66, normalized size = 1.65 \begin{align*} -\frac{5}{12} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{17}{7} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{77}{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)*(2+3*x)*(3+5*x),x, algorithm="giac")

[Out]

-5/12*(2*x - 1)^4*sqrt(-2*x + 1) - 17/7*(2*x - 1)^3*sqrt(-2*x + 1) - 77/20*(2*x - 1)^2*sqrt(-2*x + 1)