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]

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.58 \[ -\frac {1}{105} (1-2 x)^{5/2} \left (175 x^2+335 x+193\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 29, normalized size = 0.72 \[ -\frac {1}{105} \, {\left (700 \, x^{4} + 640 \, x^{3} - 393 \, x^{2} - 437 \, x + 193\right )} \sqrt {-2 \, x + 1} \]

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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giac [A]  time = 1.07, size = 49, normalized size = 1.22 \[ -\frac {5}{12} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {17}{7} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {77}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \[ -\frac {\left (175 x^{2}+335 x +193\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 28, normalized size = 0.70 \[ -\frac {5}{12} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {17}{7} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {77}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]

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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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \[ -\frac {{\left (1-2\,x\right )}^{5/2}\,\left (2040\,x+175\,{\left (2\,x-1\right )}^2+597\right )}{420} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(2040*x + 175*(2*x - 1)^2 + 597))/420

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sympy [A]  time = 7.33, size = 34, normalized size = 0.85 \[ - \frac {5 \left (1 - 2 x\right )^{\frac {9}{2}}}{12} + \frac {17 \left (1 - 2 x\right )^{\frac {7}{2}}}{7} - \frac {77 \left (1 - 2 x\right )^{\frac {5}{2}}}{20} \]

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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