3.1956 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3 \, dx\)

Optimal. Leaf size=92 \[ -\frac{3375 (1-2 x)^{19/2}}{1216}+\frac{675}{16} (1-2 x)^{17/2}-\frac{17337}{64} (1-2 x)^{15/2}+\frac{98209}{104} (1-2 x)^{13/2}-\frac{121359}{64} (1-2 x)^{11/2}+\frac{100793}{48} (1-2 x)^{9/2}-\frac{65219}{64} (1-2 x)^{7/2} \]

[Out]

(-65219*(1 - 2*x)^(7/2))/64 + (100793*(1 - 2*x)^(9/2))/48 - (121359*(1 - 2*x)^(11/2))/64 + (98209*(1 - 2*x)^(1
3/2))/104 - (17337*(1 - 2*x)^(15/2))/64 + (675*(1 - 2*x)^(17/2))/16 - (3375*(1 - 2*x)^(19/2))/1216

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Rubi [A]  time = 0.016738, antiderivative size = 92, 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} \[ -\frac{3375 (1-2 x)^{19/2}}{1216}+\frac{675}{16} (1-2 x)^{17/2}-\frac{17337}{64} (1-2 x)^{15/2}+\frac{98209}{104} (1-2 x)^{13/2}-\frac{121359}{64} (1-2 x)^{11/2}+\frac{100793}{48} (1-2 x)^{9/2}-\frac{65219}{64} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*(1 - 2*x)^(7/2))/64 + (100793*(1 - 2*x)^(9/2))/48 - (121359*(1 - 2*x)^(11/2))/64 + (98209*(1 - 2*x)^(1
3/2))/104 - (17337*(1 - 2*x)^(15/2))/64 + (675*(1 - 2*x)^(17/2))/16 - (3375*(1 - 2*x)^(19/2))/1216

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)^3 (3+5 x)^3 \, dx &=\int \left (\frac{456533}{64} (1-2 x)^{5/2}-\frac{302379}{16} (1-2 x)^{7/2}+\frac{1334949}{64} (1-2 x)^{9/2}-\frac{98209}{8} (1-2 x)^{11/2}+\frac{260055}{64} (1-2 x)^{13/2}-\frac{11475}{16} (1-2 x)^{15/2}+\frac{3375}{64} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac{65219}{64} (1-2 x)^{7/2}+\frac{100793}{48} (1-2 x)^{9/2}-\frac{121359}{64} (1-2 x)^{11/2}+\frac{98209}{104} (1-2 x)^{13/2}-\frac{17337}{64} (1-2 x)^{15/2}+\frac{675}{16} (1-2 x)^{17/2}-\frac{3375 (1-2 x)^{19/2}}{1216}\\ \end{align*}

Mathematica [A]  time = 0.021146, size = 43, normalized size = 0.47 \[ -\frac{1}{741} (1-2 x)^{7/2} \left (131625 x^6+605475 x^5+1204398 x^4+1346367 x^3+914049 x^2+372070 x+76018\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(76018 + 372070*x + 914049*x^2 + 1346367*x^3 + 1204398*x^4 + 605475*x^5 + 131625*x^6))/741

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Maple [A]  time = 0.005, size = 40, normalized size = 0.4 \begin{align*} -{\frac{131625\,{x}^{6}+605475\,{x}^{5}+1204398\,{x}^{4}+1346367\,{x}^{3}+914049\,{x}^{2}+372070\,x+76018}{741} \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)^3*(3+5*x)^3,x)

[Out]

-1/741*(131625*x^6+605475*x^5+1204398*x^4+1346367*x^3+914049*x^2+372070*x+76018)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.57969, size = 86, normalized size = 0.93 \begin{align*} -\frac{3375}{1216} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} + \frac{675}{16} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{17337}{64} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{98209}{104} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{121359}{64} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{100793}{48} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{65219}{64} \,{\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)^3*(3+5*x)^3,x, algorithm="maxima")

[Out]

-3375/1216*(-2*x + 1)^(19/2) + 675/16*(-2*x + 1)^(17/2) - 17337/64*(-2*x + 1)^(15/2) + 98209/104*(-2*x + 1)^(1
3/2) - 121359/64*(-2*x + 1)^(11/2) + 100793/48*(-2*x + 1)^(9/2) - 65219/64*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.34323, size = 201, normalized size = 2.18 \begin{align*} \frac{1}{741} \,{\left (1053000 \, x^{9} + 3264300 \, x^{8} + 3159234 \, x^{7} - 180615 \, x^{6} - 2223099 \, x^{5} - 1118224 \, x^{4} + 281231 \, x^{3} + 406155 \, x^{2} + 84038 \, x - 76018\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)^3*(3+5*x)^3,x, algorithm="fricas")

[Out]

1/741*(1053000*x^9 + 3264300*x^8 + 3159234*x^7 - 180615*x^6 - 2223099*x^5 - 1118224*x^4 + 281231*x^3 + 406155*
x^2 + 84038*x - 76018)*sqrt(-2*x + 1)

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Sympy [A]  time = 19.9555, size = 82, normalized size = 0.89 \begin{align*} - \frac{3375 \left (1 - 2 x\right )^{\frac{19}{2}}}{1216} + \frac{675 \left (1 - 2 x\right )^{\frac{17}{2}}}{16} - \frac{17337 \left (1 - 2 x\right )^{\frac{15}{2}}}{64} + \frac{98209 \left (1 - 2 x\right )^{\frac{13}{2}}}{104} - \frac{121359 \left (1 - 2 x\right )^{\frac{11}{2}}}{64} + \frac{100793 \left (1 - 2 x\right )^{\frac{9}{2}}}{48} - \frac{65219 \left (1 - 2 x\right )^{\frac{7}{2}}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(19/2)/1216 + 675*(1 - 2*x)**(17/2)/16 - 17337*(1 - 2*x)**(15/2)/64 + 98209*(1 - 2*x)**(13/2)
/104 - 121359*(1 - 2*x)**(11/2)/64 + 100793*(1 - 2*x)**(9/2)/48 - 65219*(1 - 2*x)**(7/2)/64

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Giac [A]  time = 1.86351, size = 153, normalized size = 1.66 \begin{align*} \frac{3375}{1216} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} + \frac{675}{16} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{17337}{64} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{98209}{104} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{121359}{64} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{100793}{48} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{65219}{64} \,{\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)^3*(3+5*x)^3,x, algorithm="giac")

[Out]

3375/1216*(2*x - 1)^9*sqrt(-2*x + 1) + 675/16*(2*x - 1)^8*sqrt(-2*x + 1) + 17337/64*(2*x - 1)^7*sqrt(-2*x + 1)
+ 98209/104*(2*x - 1)^6*sqrt(-2*x + 1) + 121359/64*(2*x - 1)^5*sqrt(-2*x + 1) + 100793/48*(2*x - 1)^4*sqrt(-2
*x + 1) + 65219/64*(2*x - 1)^3*sqrt(-2*x + 1)