3.1955 \(\int (1-2 x)^{5/2} (2+3 x)^4 (3+5 x)^3 \, dx\)

Optimal. Leaf size=105 \[ \frac{3375}{896} (1-2 x)^{21/2}-\frac{161325 (1-2 x)^{19/2}}{2432}+\frac{1101465 (1-2 x)^{17/2}}{2176}-\frac{1392467}{640} (1-2 x)^{15/2}+\frac{9504551 (1-2 x)^{13/2}}{1664}-\frac{1179381}{128} (1-2 x)^{11/2}+\frac{3278737}{384} (1-2 x)^{9/2}-\frac{456533}{128} (1-2 x)^{7/2} \]

[Out]

(-456533*(1 - 2*x)^(7/2))/128 + (3278737*(1 - 2*x)^(9/2))/384 - (1179381*(1 - 2*x)^(11/2))/128 + (9504551*(1 -
2*x)^(13/2))/1664 - (1392467*(1 - 2*x)^(15/2))/640 + (1101465*(1 - 2*x)^(17/2))/2176 - (161325*(1 - 2*x)^(19/
2))/2432 + (3375*(1 - 2*x)^(21/2))/896

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Rubi [A]  time = 0.0191203, 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} \[ \frac{3375}{896} (1-2 x)^{21/2}-\frac{161325 (1-2 x)^{19/2}}{2432}+\frac{1101465 (1-2 x)^{17/2}}{2176}-\frac{1392467}{640} (1-2 x)^{15/2}+\frac{9504551 (1-2 x)^{13/2}}{1664}-\frac{1179381}{128} (1-2 x)^{11/2}+\frac{3278737}{384} (1-2 x)^{9/2}-\frac{456533}{128} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(7/2))/128 + (3278737*(1 - 2*x)^(9/2))/384 - (1179381*(1 - 2*x)^(11/2))/128 + (9504551*(1 -
2*x)^(13/2))/1664 - (1392467*(1 - 2*x)^(15/2))/640 + (1101465*(1 - 2*x)^(17/2))/2176 - (161325*(1 - 2*x)^(19/
2))/2432 + (3375*(1 - 2*x)^(21/2))/896

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)^{5/2} (2+3 x)^4 (3+5 x)^3 \, dx &=\int \left (\frac{3195731}{128} (1-2 x)^{5/2}-\frac{9836211}{128} (1-2 x)^{7/2}+\frac{12973191}{128} (1-2 x)^{9/2}-\frac{9504551}{128} (1-2 x)^{11/2}+\frac{4177401}{128} (1-2 x)^{13/2}-\frac{1101465}{128} (1-2 x)^{15/2}+\frac{161325}{128} (1-2 x)^{17/2}-\frac{10125}{128} (1-2 x)^{19/2}\right ) \, dx\\ &=-\frac{456533}{128} (1-2 x)^{7/2}+\frac{3278737}{384} (1-2 x)^{9/2}-\frac{1179381}{128} (1-2 x)^{11/2}+\frac{9504551 (1-2 x)^{13/2}}{1664}-\frac{1392467}{640} (1-2 x)^{15/2}+\frac{1101465 (1-2 x)^{17/2}}{2176}-\frac{161325 (1-2 x)^{19/2}}{2432}+\frac{3375}{896} (1-2 x)^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0229051, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{7/2} \left (212574375 x^7+1127763000 x^6+2642319225 x^5+3583371246 x^4+3089723448 x^3+1740153744 x^2+619493392 x+115708576\right )}{440895} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(115708576 + 619493392*x + 1740153744*x^2 + 3089723448*x^3 + 3583371246*x^4 + 2642319225*x^5
+ 1127763000*x^6 + 212574375*x^7))/440895

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{212574375\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+115708576}{440895} \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)^4*(3+5*x)^3,x)

[Out]

-1/440895*(212574375*x^7+1127763000*x^6+2642319225*x^5+3583371246*x^4+3089723448*x^3+1740153744*x^2+619493392*
x+115708576)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.52998, size = 99, normalized size = 0.94 \begin{align*} \frac{3375}{896} \,{\left (-2 \, x + 1\right )}^{\frac{21}{2}} - \frac{161325}{2432} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} + \frac{1101465}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{1392467}{640} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{9504551}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{1179381}{128} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{3278737}{384} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{456533}{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)^4*(3+5*x)^3,x, algorithm="maxima")

[Out]

3375/896*(-2*x + 1)^(21/2) - 161325/2432*(-2*x + 1)^(19/2) + 1101465/2176*(-2*x + 1)^(17/2) - 1392467/640*(-2*
x + 1)^(15/2) + 9504551/1664*(-2*x + 1)^(13/2) - 1179381/128*(-2*x + 1)^(11/2) + 3278737/384*(-2*x + 1)^(9/2)
- 456533/128*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.2912, size = 271, normalized size = 2.58 \begin{align*} \frac{1}{440895} \,{\left (1700595000 \, x^{10} + 6471211500 \, x^{9} + 8880844050 \, x^{8} + 3513142893 \, x^{7} - 3556515018 \, x^{6} - 4297543173 \, x^{5} - 970928350 \, x^{4} + 842946920 \, x^{3} + 588303696 \, x^{2} + 74758064 \, x - 115708576\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)^4*(3+5*x)^3,x, algorithm="fricas")

[Out]

1/440895*(1700595000*x^10 + 6471211500*x^9 + 8880844050*x^8 + 3513142893*x^7 - 3556515018*x^6 - 4297543173*x^5
- 970928350*x^4 + 842946920*x^3 + 588303696*x^2 + 74758064*x - 115708576)*sqrt(-2*x + 1)

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Sympy [A]  time = 23.6528, size = 94, normalized size = 0.9 \begin{align*} \frac{3375 \left (1 - 2 x\right )^{\frac{21}{2}}}{896} - \frac{161325 \left (1 - 2 x\right )^{\frac{19}{2}}}{2432} + \frac{1101465 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} - \frac{1392467 \left (1 - 2 x\right )^{\frac{15}{2}}}{640} + \frac{9504551 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{1179381 \left (1 - 2 x\right )^{\frac{11}{2}}}{128} + \frac{3278737 \left (1 - 2 x\right )^{\frac{9}{2}}}{384} - \frac{456533 \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)**4*(3+5*x)**3,x)

[Out]

3375*(1 - 2*x)**(21/2)/896 - 161325*(1 - 2*x)**(19/2)/2432 + 1101465*(1 - 2*x)**(17/2)/2176 - 1392467*(1 - 2*x
)**(15/2)/640 + 9504551*(1 - 2*x)**(13/2)/1664 - 1179381*(1 - 2*x)**(11/2)/128 + 3278737*(1 - 2*x)**(9/2)/384
- 456533*(1 - 2*x)**(7/2)/128

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Giac [A]  time = 2.34785, size = 174, normalized size = 1.66 \begin{align*} \frac{3375}{896} \,{\left (2 \, x - 1\right )}^{10} \sqrt{-2 \, x + 1} + \frac{161325}{2432} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} + \frac{1101465}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{1392467}{640} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{9504551}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{1179381}{128} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{3278737}{384} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{456533}{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)^4*(3+5*x)^3,x, algorithm="giac")

[Out]

3375/896*(2*x - 1)^10*sqrt(-2*x + 1) + 161325/2432*(2*x - 1)^9*sqrt(-2*x + 1) + 1101465/2176*(2*x - 1)^8*sqrt(
-2*x + 1) + 1392467/640*(2*x - 1)^7*sqrt(-2*x + 1) + 9504551/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 1179381/128*(2*
x - 1)^5*sqrt(-2*x + 1) + 3278737/384*(2*x - 1)^4*sqrt(-2*x + 1) + 456533/128*(2*x - 1)^3*sqrt(-2*x + 1)