3.1860 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x) \, dx\)

Optimal. Leaf size=66 \[ -\frac {135}{208} (1-2 x)^{13/2}+\frac {621}{88} (1-2 x)^{11/2}-\frac {119}{4} (1-2 x)^{9/2}+\frac {469}{8} (1-2 x)^{7/2}-\frac {3773}{80} (1-2 x)^{5/2} \]

[Out]

-3773/80*(1-2*x)^(5/2)+469/8*(1-2*x)^(7/2)-119/4*(1-2*x)^(9/2)+621/88*(1-2*x)^(11/2)-135/208*(1-2*x)^(13/2)

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {135}{208} (1-2 x)^{13/2}+\frac {621}{88} (1-2 x)^{11/2}-\frac {119}{4} (1-2 x)^{9/2}+\frac {469}{8} (1-2 x)^{7/2}-\frac {3773}{80} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3773*(1 - 2*x)^(5/2))/80 + (469*(1 - 2*x)^(7/2))/8 - (119*(1 - 2*x)^(9/2))/4 + (621*(1 - 2*x)^(11/2))/88 - (
135*(1 - 2*x)^(13/2))/208

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 (3+5 x) \, dx &=\int \left (\frac {3773}{16} (1-2 x)^{3/2}-\frac {3283}{8} (1-2 x)^{5/2}+\frac {1071}{4} (1-2 x)^{7/2}-\frac {621}{8} (1-2 x)^{9/2}+\frac {135}{16} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {3773}{80} (1-2 x)^{5/2}+\frac {469}{8} (1-2 x)^{7/2}-\frac {119}{4} (1-2 x)^{9/2}+\frac {621}{88} (1-2 x)^{11/2}-\frac {135}{208} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \[ -\frac {1}{715} (1-2 x)^{5/2} \left (7425 x^4+25515 x^3+35675 x^2+25310 x+8494\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/715*((1 - 2*x)^(5/2)*(8494 + 25310*x + 35675*x^2 + 25515*x^3 + 7425*x^4))

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fricas [A]  time = 0.98, size = 39, normalized size = 0.59 \[ -\frac {1}{715} \, {\left (29700 \, x^{6} + 72360 \, x^{5} + 48065 \, x^{4} - 15945 \, x^{3} - 31589 \, x^{2} - 8666 \, x + 8494\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/715*(29700*x^6 + 72360*x^5 + 48065*x^4 - 15945*x^3 - 31589*x^2 - 8666*x + 8494)*sqrt(-2*x + 1)

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giac [A]  time = 1.60, size = 81, normalized size = 1.23 \[ -\frac {135}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {621}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {119}{4} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {469}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3773}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135/208*(2*x - 1)^6*sqrt(-2*x + 1) - 621/88*(2*x - 1)^5*sqrt(-2*x + 1) - 119/4*(2*x - 1)^4*sqrt(-2*x + 1) - 4
69/8*(2*x - 1)^3*sqrt(-2*x + 1) - 3773/80*(2*x - 1)^2*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \[ -\frac {\left (7425 x^{4}+25515 x^{3}+35675 x^{2}+25310 x +8494\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{715} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/715*(7425*x^4+25515*x^3+35675*x^2+25310*x+8494)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.46, size = 46, normalized size = 0.70 \[ -\frac {135}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {621}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {119}{4} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {469}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3773}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135/208*(-2*x + 1)^(13/2) + 621/88*(-2*x + 1)^(11/2) - 119/4*(-2*x + 1)^(9/2) + 469/8*(-2*x + 1)^(7/2) - 3773
/80*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \[ \frac {469\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {3773\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {119\,{\left (1-2\,x\right )}^{9/2}}{4}+\frac {621\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {135\,{\left (1-2\,x\right )}^{13/2}}{208} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(469*(1 - 2*x)^(7/2))/8 - (3773*(1 - 2*x)^(5/2))/80 - (119*(1 - 2*x)^(9/2))/4 + (621*(1 - 2*x)^(11/2))/88 - (1
35*(1 - 2*x)^(13/2))/208

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sympy [A]  time = 13.53, size = 58, normalized size = 0.88 \[ - \frac {135 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} + \frac {621 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {119 \left (1 - 2 x\right )^{\frac {9}{2}}}{4} + \frac {469 \left (1 - 2 x\right )^{\frac {7}{2}}}{8} - \frac {3773 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135*(1 - 2*x)**(13/2)/208 + 621*(1 - 2*x)**(11/2)/88 - 119*(1 - 2*x)**(9/2)/4 + 469*(1 - 2*x)**(7/2)/8 - 3773
*(1 - 2*x)**(5/2)/80

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