3.1790 \(\int \sqrt{1-2 x} (2+3 x)^6 (3+5 x) \, dx\)

Optimal. Leaf size=105 \[ \frac{3645 (1-2 x)^{17/2}}{2176}-\frac{19683}{640} (1-2 x)^{15/2}+\frac{409941 (1-2 x)^{13/2}}{1664}-\frac{1580985 (1-2 x)^{11/2}}{1408}+\frac{406455}{128} (1-2 x)^{9/2}-\frac{725445}{128} (1-2 x)^{7/2}+\frac{3916031}{640} (1-2 x)^{5/2}-\frac{1294139}{384} (1-2 x)^{3/2} \]

[Out]

(-1294139*(1 - 2*x)^(3/2))/384 + (3916031*(1 - 2*x)^(5/2))/640 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2
*x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/6
40 + (3645*(1 - 2*x)^(17/2))/2176

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Rubi [A]  time = 0.019877, antiderivative size = 105, normalized size of antiderivative = 1., 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} \[ \frac{3645 (1-2 x)^{17/2}}{2176}-\frac{19683}{640} (1-2 x)^{15/2}+\frac{409941 (1-2 x)^{13/2}}{1664}-\frac{1580985 (1-2 x)^{11/2}}{1408}+\frac{406455}{128} (1-2 x)^{9/2}-\frac{725445}{128} (1-2 x)^{7/2}+\frac{3916031}{640} (1-2 x)^{5/2}-\frac{1294139}{384} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

(-1294139*(1 - 2*x)^(3/2))/384 + (3916031*(1 - 2*x)^(5/2))/640 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2
*x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/6
40 + (3645*(1 - 2*x)^(17/2))/2176

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 \sqrt{1-2 x} (2+3 x)^6 (3+5 x) \, dx &=\int \left (\frac{1294139}{128} \sqrt{1-2 x}-\frac{3916031}{128} (1-2 x)^{3/2}+\frac{5078115}{128} (1-2 x)^{5/2}-\frac{3658095}{128} (1-2 x)^{7/2}+\frac{1580985}{128} (1-2 x)^{9/2}-\frac{409941}{128} (1-2 x)^{11/2}+\frac{59049}{128} (1-2 x)^{13/2}-\frac{3645}{128} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{1294139}{384} (1-2 x)^{3/2}+\frac{3916031}{640} (1-2 x)^{5/2}-\frac{725445}{128} (1-2 x)^{7/2}+\frac{406455}{128} (1-2 x)^{9/2}-\frac{1580985 (1-2 x)^{11/2}}{1408}+\frac{409941 (1-2 x)^{13/2}}{1664}-\frac{19683}{640} (1-2 x)^{15/2}+\frac{3645 (1-2 x)^{17/2}}{2176}\\ \end{align*}

Mathematica [A]  time = 0.0217708, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{3/2} \left (7818525 x^7+44409222 x^6+113196204 x^5+171389520 x^4+172440720 x^3+122662080 x^2+64000896 x+23667392\right )}{36465} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

-((1 - 2*x)^(3/2)*(23667392 + 64000896*x + 122662080*x^2 + 172440720*x^3 + 171389520*x^4 + 113196204*x^5 + 444
09222*x^6 + 7818525*x^7))/36465

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{7818525\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+23667392}{36465} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x)

[Out]

-1/36465*(7818525*x^7+44409222*x^6+113196204*x^5+171389520*x^4+172440720*x^3+122662080*x^2+64000896*x+23667392
)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.17905, size = 99, normalized size = 0.94 \begin{align*} \frac{3645}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{19683}{640} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{409941}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{1580985}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{406455}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{725445}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{3916031}{640} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1294139}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

3645/2176*(-2*x + 1)^(17/2) - 19683/640*(-2*x + 1)^(15/2) + 409941/1664*(-2*x + 1)^(13/2) - 1580985/1408*(-2*x
+ 1)^(11/2) + 406455/128*(-2*x + 1)^(9/2) - 725445/128*(-2*x + 1)^(7/2) + 3916031/640*(-2*x + 1)^(5/2) - 1294
139/384*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.33512, size = 209, normalized size = 1.99 \begin{align*} \frac{1}{36465} \,{\left (15637050 \, x^{8} + 80999919 \, x^{7} + 181983186 \, x^{6} + 229582836 \, x^{5} + 173491920 \, x^{4} + 72883440 \, x^{3} + 5339712 \, x^{2} - 16666112 \, x - 23667392\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/36465*(15637050*x^8 + 80999919*x^7 + 181983186*x^6 + 229582836*x^5 + 173491920*x^4 + 72883440*x^3 + 5339712*
x^2 - 16666112*x - 23667392)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.57671, size = 94, normalized size = 0.9 \begin{align*} \frac{3645 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} - \frac{19683 \left (1 - 2 x\right )^{\frac{15}{2}}}{640} + \frac{409941 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{1580985 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{406455 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} - \frac{725445 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} + \frac{3916031 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} - \frac{1294139 \left (1 - 2 x\right )^{\frac{3}{2}}}{384} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

3645*(1 - 2*x)**(17/2)/2176 - 19683*(1 - 2*x)**(15/2)/640 + 409941*(1 - 2*x)**(13/2)/1664 - 1580985*(1 - 2*x)*
*(11/2)/1408 + 406455*(1 - 2*x)**(9/2)/128 - 725445*(1 - 2*x)**(7/2)/128 + 3916031*(1 - 2*x)**(5/2)/640 - 1294
139*(1 - 2*x)**(3/2)/384

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Giac [A]  time = 2.41913, size = 165, normalized size = 1.57 \begin{align*} \frac{3645}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{19683}{640} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{409941}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{1580985}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{406455}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{725445}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{3916031}{640} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1294139}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

3645/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 19683/640*(2*x - 1)^7*sqrt(-2*x + 1) + 409941/1664*(2*x - 1)^6*sqrt(-2*
x + 1) + 1580985/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 406455/128*(2*x - 1)^4*sqrt(-2*x + 1) + 725445/128*(2*x - 1
)^3*sqrt(-2*x + 1) + 3916031/640*(2*x - 1)^2*sqrt(-2*x + 1) - 1294139/384*(-2*x + 1)^(3/2)