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

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

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Rubi [A]  time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.47 \begin {gather*} -\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 67, normalized size = 0.73 \begin {gather*} -\frac {\left (131625 (1-2 x)^6-2000700 (1-2 x)^5+12846717 (1-2 x)^4-44783304 (1-2 x)^3+89927019 (1-2 x)^2-99583484 (1-2 x)[phone]\right ) (1-2 x)^{7/2}}{47424} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/47424*((48327279 - 99583484*(1 - 2*x) + 89927019*(1 - 2*x)^2 - 44783304*(1 - 2*x)^3 + 12846717*(1 - 2*x)^4
- 2000700*(1 - 2*x)^5 + 131625*(1 - 2*x)^6)*(1 - 2*x)^(7/2))

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fricas [A]  time = 1.29, size = 54, normalized size = 0.59 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.04, size = 113, normalized size = 1.23 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {\left (131625 x^{6}+605475 x^{5}+1204398 x^{4}+1346367 x^{3}+914049 x^{2}+372070 x +76018\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{741} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 64, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {100793\,{\left (1-2\,x\right )}^{9/2}}{48}-\frac {65219\,{\left (1-2\,x\right )}^{7/2}}{64}-\frac {121359\,{\left (1-2\,x\right )}^{11/2}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {17337\,{\left (1-2\,x\right )}^{15/2}}{64}+\frac {675\,{\left (1-2\,x\right )}^{17/2}}{16}-\frac {3375\,{\left (1-2\,x\right )}^{19/2}}{1216} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(100793*(1 - 2*x)^(9/2))/48 - (65219*(1 - 2*x)^(7/2))/64 - (121359*(1 - 2*x)^(11/2))/64 + (98209*(1 - 2*x)^(13
/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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sympy [A]  time = 25.29, size = 82, normalized size = 0.89 \begin {gather*} - \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 {gather*}

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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