3.1882 \(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx\)

Optimal. Leaf size=105 \[ \frac{10125 (1-2 x)^{19/2}}{2432}-\frac{161325 (1-2 x)^{17/2}}{2176}+\frac{73431}{128} (1-2 x)^{15/2}-\frac{4177401 (1-2 x)^{13/2}}{1664}+\frac{9504551 (1-2 x)^{11/2}}{1408}-\frac{4324397}{384} (1-2 x)^{9/2}+\frac{1405173}{128} (1-2 x)^{7/2}-\frac{3195731}{640} (1-2 x)^{5/2} \]

[Out]

(-3195731*(1 - 2*x)^(5/2))/640 + (1405173*(1 - 2*x)^(7/2))/128 - (4324397*(1 - 2*x)^(9/2))/384 + (9504551*(1 -
2*x)^(11/2))/1408 - (4177401*(1 - 2*x)^(13/2))/1664 + (73431*(1 - 2*x)^(15/2))/128 - (161325*(1 - 2*x)^(17/2)
)/2176 + (10125*(1 - 2*x)^(19/2))/2432

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Rubi [A]  time = 0.019335, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ \frac{10125 (1-2 x)^{19/2}}{2432}-\frac{161325 (1-2 x)^{17/2}}{2176}+\frac{73431}{128} (1-2 x)^{15/2}-\frac{4177401 (1-2 x)^{13/2}}{1664}+\frac{9504551 (1-2 x)^{11/2}}{1408}-\frac{4324397}{384} (1-2 x)^{9/2}+\frac{1405173}{128} (1-2 x)^{7/2}-\frac{3195731}{640} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

(-3195731*(1 - 2*x)^(5/2))/640 + (1405173*(1 - 2*x)^(7/2))/128 - (4324397*(1 - 2*x)^(9/2))/384 + (9504551*(1 -
2*x)^(11/2))/1408 - (4177401*(1 - 2*x)^(13/2))/1664 + (73431*(1 - 2*x)^(15/2))/128 - (161325*(1 - 2*x)^(17/2)
)/2176 + (10125*(1 - 2*x)^(19/2))/2432

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx &=\int \left (\frac{3195731}{128} (1-2 x)^{3/2}-\frac{9836211}{128} (1-2 x)^{5/2}+\frac{12973191}{128} (1-2 x)^{7/2}-\frac{9504551}{128} (1-2 x)^{9/2}+\frac{4177401}{128} (1-2 x)^{11/2}-\frac{1101465}{128} (1-2 x)^{13/2}+\frac{161325}{128} (1-2 x)^{15/2}-\frac{10125}{128} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac{3195731}{640} (1-2 x)^{5/2}+\frac{1405173}{128} (1-2 x)^{7/2}-\frac{4324397}{384} (1-2 x)^{9/2}+\frac{9504551 (1-2 x)^{11/2}}{1408}-\frac{4177401 (1-2 x)^{13/2}}{1664}+\frac{73431}{128} (1-2 x)^{15/2}-\frac{161325 (1-2 x)^{17/2}}{2176}+\frac{10125 (1-2 x)^{19/2}}{2432}\\ \end{align*}

Mathematica [A]  time = 0.0229892, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{5/2} \left (369208125 x^7+1995171750 x^6+4795033815 x^5+6744559140 x^4+6142984080 x^3+3771434840 x^2+1547888800 x+369438704\right )}{692835} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

-((1 - 2*x)^(5/2)*(369438704 + 1547888800*x + 3771434840*x^2 + 6142984080*x^3 + 6744559140*x^4 + 4795033815*x^
5 + 1995171750*x^6 + 369208125*x^7))/692835

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Maple [A]  time = 0.002, size = 45, normalized size = 0.4 \begin{align*} -{\frac{369208125\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+369438704}{692835} \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)^4*(3+5*x)^3,x)

[Out]

-1/692835*(369208125*x^7+1995171750*x^6+4795033815*x^5+6744559140*x^4+6142984080*x^3+3771434840*x^2+1547888800
*x+369438704)*(1-2*x)^(5/2)

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Maxima [A]  time = 2.22928, size = 99, normalized size = 0.94 \begin{align*} \frac{10125}{2432} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} - \frac{161325}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} + \frac{73431}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{4177401}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{9504551}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{4324397}{384} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1405173}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{3195731}{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)^4*(3+5*x)^3,x, algorithm="maxima")

[Out]

10125/2432*(-2*x + 1)^(19/2) - 161325/2176*(-2*x + 1)^(17/2) + 73431/128*(-2*x + 1)^(15/2) - 4177401/1664*(-2*
x + 1)^(13/2) + 9504551/1408*(-2*x + 1)^(11/2) - 4324397/384*(-2*x + 1)^(9/2) + 1405173/128*(-2*x + 1)^(7/2) -
3195731/640*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.36473, size = 252, normalized size = 2.4 \begin{align*} -\frac{1}{692835} \,{\left (1476832500 \, x^{9} + 6503854500 \, x^{8} + 11568656385 \, x^{7} + 9793273050 \, x^{6} + 2388733575 \, x^{5} - 2741637820 \, x^{4} - 2751200080 \, x^{3} - 942365544 \, x^{2} + 70133984 \, x + 369438704\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)^4*(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/692835*(1476832500*x^9 + 6503854500*x^8 + 11568656385*x^7 + 9793273050*x^6 + 2388733575*x^5 - 2741637820*x^
4 - 2751200080*x^3 - 942365544*x^2 + 70133984*x + 369438704)*sqrt(-2*x + 1)

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Sympy [A]  time = 19.1283, size = 94, normalized size = 0.9 \begin{align*} \frac{10125 \left (1 - 2 x\right )^{\frac{19}{2}}}{2432} - \frac{161325 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} + \frac{73431 \left (1 - 2 x\right )^{\frac{15}{2}}}{128} - \frac{4177401 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} + \frac{9504551 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{4324397 \left (1 - 2 x\right )^{\frac{9}{2}}}{384} + \frac{1405173 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} - \frac{3195731 \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)**4*(3+5*x)**3,x)

[Out]

10125*(1 - 2*x)**(19/2)/2432 - 161325*(1 - 2*x)**(17/2)/2176 + 73431*(1 - 2*x)**(15/2)/128 - 4177401*(1 - 2*x)
**(13/2)/1664 + 9504551*(1 - 2*x)**(11/2)/1408 - 4324397*(1 - 2*x)**(9/2)/384 + 1405173*(1 - 2*x)**(7/2)/128 -
3195731*(1 - 2*x)**(5/2)/640

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Giac [A]  time = 1.5453, size = 174, normalized size = 1.66 \begin{align*} -\frac{10125}{2432} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} - \frac{161325}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} - \frac{73431}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{4177401}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{9504551}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{4324397}{384} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1405173}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{3195731}{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)^4*(3+5*x)^3,x, algorithm="giac")

[Out]

-10125/2432*(2*x - 1)^9*sqrt(-2*x + 1) - 161325/2176*(2*x - 1)^8*sqrt(-2*x + 1) - 73431/128*(2*x - 1)^7*sqrt(-
2*x + 1) - 4177401/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 9504551/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 4324397/384*(2*
x - 1)^4*sqrt(-2*x + 1) - 1405173/128*(2*x - 1)^3*sqrt(-2*x + 1) - 3195731/640*(2*x - 1)^2*sqrt(-2*x + 1)