3.1857 \(\int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx\)

Optimal. Leaf size=105 \[ \frac{3645 (1-2 x)^{19/2}}{2432}-\frac{59049 (1-2 x)^{17/2}}{2176}+\frac{136647}{640} (1-2 x)^{15/2}-\frac{1580985 (1-2 x)^{13/2}}{1664}+\frac{3658095 (1-2 x)^{11/2}}{1408}-\frac{564235}{128} (1-2 x)^{9/2}+\frac{559433}{128} (1-2 x)^{7/2}-\frac{1294139}{640} (1-2 x)^{5/2} \]

[Out]

(-1294139*(1 - 2*x)^(5/2))/640 + (559433*(1 - 2*x)^(7/2))/128 - (564235*(1 - 2*x)^(9/2))/128 + (3658095*(1 - 2
*x)^(11/2))/1408 - (1580985*(1 - 2*x)^(13/2))/1664 + (136647*(1 - 2*x)^(15/2))/640 - (59049*(1 - 2*x)^(17/2))/
2176 + (3645*(1 - 2*x)^(19/2))/2432

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Rubi [A]  time = 0.017916, 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} \[ \frac{3645 (1-2 x)^{19/2}}{2432}-\frac{59049 (1-2 x)^{17/2}}{2176}+\frac{136647}{640} (1-2 x)^{15/2}-\frac{1580985 (1-2 x)^{13/2}}{1664}+\frac{3658095 (1-2 x)^{11/2}}{1408}-\frac{564235}{128} (1-2 x)^{9/2}+\frac{559433}{128} (1-2 x)^{7/2}-\frac{1294139}{640} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1294139*(1 - 2*x)^(5/2))/640 + (559433*(1 - 2*x)^(7/2))/128 - (564235*(1 - 2*x)^(9/2))/128 + (3658095*(1 - 2
*x)^(11/2))/1408 - (1580985*(1 - 2*x)^(13/2))/1664 + (136647*(1 - 2*x)^(15/2))/640 - (59049*(1 - 2*x)^(17/2))/
2176 + (3645*(1 - 2*x)^(19/2))/2432

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)^6 (3+5 x) \, dx &=\int \left (\frac{1294139}{128} (1-2 x)^{3/2}-\frac{3916031}{128} (1-2 x)^{5/2}+\frac{5078115}{128} (1-2 x)^{7/2}-\frac{3658095}{128} (1-2 x)^{9/2}+\frac{1580985}{128} (1-2 x)^{11/2}-\frac{409941}{128} (1-2 x)^{13/2}+\frac{59049}{128} (1-2 x)^{15/2}-\frac{3645}{128} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac{1294139}{640} (1-2 x)^{5/2}+\frac{559433}{128} (1-2 x)^{7/2}-\frac{564235}{128} (1-2 x)^{9/2}+\frac{3658095 (1-2 x)^{11/2}}{1408}-\frac{1580985 (1-2 x)^{13/2}}{1664}+\frac{136647}{640} (1-2 x)^{15/2}-\frac{59049 (1-2 x)^{17/2}}{2176}+\frac{3645 (1-2 x)^{19/2}}{2432}\\ \end{align*}

Mathematica [A]  time = 0.0225294, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{5/2} \left (44304975 x^7+246022920 x^6+607227192 x^5+876286620 x^4+817490880 x^3+512679760 x^2+214047840 x+51677856\right )}{230945} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(51677856 + 214047840*x + 512679760*x^2 + 817490880*x^3 + 876286620*x^4 + 607227192*x^5 + 24
6022920*x^6 + 44304975*x^7))/230945

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{44304975\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+51677856}{230945} \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)^6*(3+5*x),x)

[Out]

-1/230945*(44304975*x^7+246022920*x^6+607227192*x^5+876286620*x^4+817490880*x^3+512679760*x^2+214047840*x+5167
7856)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.0162, size = 99, normalized size = 0.94 \begin{align*} \frac{3645}{2432} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} - \frac{59049}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} + \frac{136647}{640} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{1580985}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{3658095}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{564235}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{559433}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1294139}{640} \,{\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)^6*(3+5*x),x, algorithm="maxima")

[Out]

3645/2432*(-2*x + 1)^(19/2) - 59049/2176*(-2*x + 1)^(17/2) + 136647/640*(-2*x + 1)^(15/2) - 1580985/1664*(-2*x
+ 1)^(13/2) + 3658095/1408*(-2*x + 1)^(11/2) - 564235/128*(-2*x + 1)^(9/2) + 559433/128*(-2*x + 1)^(7/2) - 12
94139/640*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.30517, size = 242, normalized size = 2.3 \begin{align*} -\frac{1}{230945} \,{\left (177219900 \, x^{9} + 806871780 \, x^{8} + 1489122063 \, x^{7} + 1322260632 \, x^{6} + 372044232 \, x^{5} - 342957860 \, x^{4} - 377036800 \, x^{3} - 136800176 \, x^{2} + 7336416 \, x + 51677856\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)^6*(3+5*x),x, algorithm="fricas")

[Out]

-1/230945*(177219900*x^9 + 806871780*x^8 + 1489122063*x^7 + 1322260632*x^6 + 372044232*x^5 - 342957860*x^4 - 3
77036800*x^3 - 136800176*x^2 + 7336416*x + 51677856)*sqrt(-2*x + 1)

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Sympy [A]  time = 19.431, size = 94, normalized size = 0.9 \begin{align*} \frac{3645 \left (1 - 2 x\right )^{\frac{19}{2}}}{2432} - \frac{59049 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} + \frac{136647 \left (1 - 2 x\right )^{\frac{15}{2}}}{640} - \frac{1580985 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} + \frac{3658095 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{564235 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} + \frac{559433 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} - \frac{1294139 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3645*(1 - 2*x)**(19/2)/2432 - 59049*(1 - 2*x)**(17/2)/2176 + 136647*(1 - 2*x)**(15/2)/640 - 1580985*(1 - 2*x)*
*(13/2)/1664 + 3658095*(1 - 2*x)**(11/2)/1408 - 564235*(1 - 2*x)**(9/2)/128 + 559433*(1 - 2*x)**(7/2)/128 - 12
94139*(1 - 2*x)**(5/2)/640

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Giac [A]  time = 1.38646, size = 174, normalized size = 1.66 \begin{align*} -\frac{3645}{2432} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} - \frac{59049}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} - \frac{136647}{640} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{1580985}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{3658095}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{564235}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{559433}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1294139}{640} \,{\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)^6*(3+5*x),x, algorithm="giac")

[Out]

-3645/2432*(2*x - 1)^9*sqrt(-2*x + 1) - 59049/2176*(2*x - 1)^8*sqrt(-2*x + 1) - 136647/640*(2*x - 1)^7*sqrt(-2
*x + 1) - 1580985/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 3658095/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 564235/128*(2*x
- 1)^4*sqrt(-2*x + 1) - 559433/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1294139/640*(2*x - 1)^2*sqrt(-2*x + 1)