3.1928 \(\int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx\)

Optimal. Leaf size=105 \[ \frac{1215}{896} (1-2 x)^{21/2}-\frac{59049 (1-2 x)^{19/2}}{2432}+\frac{409941 (1-2 x)^{17/2}}{2176}-\frac{105399}{128} (1-2 x)^{15/2}+\frac{3658095 (1-2 x)^{13/2}}{1664}-\frac{5078115 (1-2 x)^{11/2}}{1408}+\frac{3916031 (1-2 x)^{9/2}}{1152}-\frac{184877}{128} (1-2 x)^{7/2} \]

[Out]

(-184877*(1 - 2*x)^(7/2))/128 + (3916031*(1 - 2*x)^(9/2))/1152 - (5078115*(1 - 2*x)^(11/2))/1408 + (3658095*(1
- 2*x)^(13/2))/1664 - (105399*(1 - 2*x)^(15/2))/128 + (409941*(1 - 2*x)^(17/2))/2176 - (59049*(1 - 2*x)^(19/2
))/2432 + (1215*(1 - 2*x)^(21/2))/896

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Rubi [A]  time = 0.018027, 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} \[ \frac{1215}{896} (1-2 x)^{21/2}-\frac{59049 (1-2 x)^{19/2}}{2432}+\frac{409941 (1-2 x)^{17/2}}{2176}-\frac{105399}{128} (1-2 x)^{15/2}+\frac{3658095 (1-2 x)^{13/2}}{1664}-\frac{5078115 (1-2 x)^{11/2}}{1408}+\frac{3916031 (1-2 x)^{9/2}}{1152}-\frac{184877}{128} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-184877*(1 - 2*x)^(7/2))/128 + (3916031*(1 - 2*x)^(9/2))/1152 - (5078115*(1 - 2*x)^(11/2))/1408 + (3658095*(1
- 2*x)^(13/2))/1664 - (105399*(1 - 2*x)^(15/2))/128 + (409941*(1 - 2*x)^(17/2))/2176 - (59049*(1 - 2*x)^(19/2
))/2432 + (1215*(1 - 2*x)^(21/2))/896

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)^{5/2} (2+3 x)^6 (3+5 x) \, dx &=\int \left (\frac{1294139}{128} (1-2 x)^{5/2}-\frac{3916031}{128} (1-2 x)^{7/2}+\frac{5078115}{128} (1-2 x)^{9/2}-\frac{3658095}{128} (1-2 x)^{11/2}+\frac{1580985}{128} (1-2 x)^{13/2}-\frac{409941}{128} (1-2 x)^{15/2}+\frac{59049}{128} (1-2 x)^{17/2}-\frac{3645}{128} (1-2 x)^{19/2}\right ) \, dx\\ &=-\frac{184877}{128} (1-2 x)^{7/2}+\frac{3916031 (1-2 x)^{9/2}}{1152}-\frac{5078115 (1-2 x)^{11/2}}{1408}+\frac{3658095 (1-2 x)^{13/2}}{1664}-\frac{105399}{128} (1-2 x)^{15/2}+\frac{409941 (1-2 x)^{17/2}}{2176}-\frac{59049 (1-2 x)^{19/2}}{2432}+\frac{1215}{896} (1-2 x)^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0241881, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{7/2} \left (505076715 x^7+2753997246 x^6+6628858236 x^5+9228315096 x^4+8157896208 x^3+4700947104 x^2+1706820416 x+323646080\right )}{2909907} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(323646080 + 1706820416*x + 4700947104*x^2 + 8157896208*x^3 + 9228315096*x^4 + 6628858236*x^
5 + 2753997246*x^6 + 505076715*x^7))/2909907

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{505076715\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+323646080}{2909907} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2909907*(505076715*x^7+2753997246*x^6+6628858236*x^5+9228315096*x^4+8157896208*x^3+4700947104*x^2+170682041
6*x+323646080)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.03174, size = 99, normalized size = 0.94 \begin{align*} \frac{1215}{896} \,{\left (-2 \, x + 1\right )}^{\frac{21}{2}} - \frac{59049}{2432} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} + \frac{409941}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{105399}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{3658095}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{5078115}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{3916031}{1152} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{184877}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1215/896*(-2*x + 1)^(21/2) - 59049/2432*(-2*x + 1)^(19/2) + 409941/2176*(-2*x + 1)^(17/2) - 105399/128*(-2*x +
1)^(15/2) + 3658095/1664*(-2*x + 1)^(13/2) - 5078115/1408*(-2*x + 1)^(11/2) + 3916031/1152*(-2*x + 1)^(9/2) -
184877/128*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.57497, size = 284, normalized size = 2.7 \begin{align*} \frac{1}{2909907} \,{\left (4040613720 \, x^{10} + 15971057388 \, x^{9} + 23013359226 \, x^{8} + 10299128697 \, x^{7} - 8457459318 \, x^{6} - 11546145324 \, x^{5} - 3037739768 \, x^{4} + 2155110064 \, x^{3} + 1656222432 \, x^{2} + 235056064 \, x - 323646080\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2909907*(4040613720*x^10 + 15971057388*x^9 + 23013359226*x^8 + 10299128697*x^7 - 8457459318*x^6 - 1154614532
4*x^5 - 3037739768*x^4 + 2155110064*x^3 + 1656222432*x^2 + 235056064*x - 323646080)*sqrt(-2*x + 1)

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Sympy [A]  time = 24.8812, size = 94, normalized size = 0.9 \begin{align*} \frac{1215 \left (1 - 2 x\right )^{\frac{21}{2}}}{896} - \frac{59049 \left (1 - 2 x\right )^{\frac{19}{2}}}{2432} + \frac{409941 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} - \frac{105399 \left (1 - 2 x\right )^{\frac{15}{2}}}{128} + \frac{3658095 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{5078115 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{3916031 \left (1 - 2 x\right )^{\frac{9}{2}}}{1152} - \frac{184877 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1215*(1 - 2*x)**(21/2)/896 - 59049*(1 - 2*x)**(19/2)/2432 + 409941*(1 - 2*x)**(17/2)/2176 - 105399*(1 - 2*x)**
(15/2)/128 + 3658095*(1 - 2*x)**(13/2)/1664 - 5078115*(1 - 2*x)**(11/2)/1408 + 3916031*(1 - 2*x)**(9/2)/1152 -
184877*(1 - 2*x)**(7/2)/128

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Giac [A]  time = 2.35057, size = 174, normalized size = 1.66 \begin{align*} \frac{1215}{896} \,{\left (2 \, x - 1\right )}^{10} \sqrt{-2 \, x + 1} + \frac{59049}{2432} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} + \frac{409941}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{105399}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{3658095}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{5078115}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{3916031}{1152} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{184877}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1215/896*(2*x - 1)^10*sqrt(-2*x + 1) + 59049/2432*(2*x - 1)^9*sqrt(-2*x + 1) + 409941/2176*(2*x - 1)^8*sqrt(-2
*x + 1) + 105399/128*(2*x - 1)^7*sqrt(-2*x + 1) + 3658095/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 5078115/1408*(2*x
- 1)^5*sqrt(-2*x + 1) + 3916031/1152*(2*x - 1)^4*sqrt(-2*x + 1) + 184877/128*(2*x - 1)^3*sqrt(-2*x + 1)