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]

-456533/128*(1-2*x)^(7/2)+3278737/384*(1-2*x)^(9/2)-1179381/128*(1-2*x)^(11/2)+9504551/1664*(1-2*x)^(13/2)-139
2467/640*(1-2*x)^(15/2)+1101465/2176*(1-2*x)^(17/2)-161325/2432*(1-2*x)^(19/2)+3375/896*(1-2*x)^(21/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 105, 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} \[ \frac {3375}{896} (1-2 x)^{21/2}-\frac {161325 (1-2 x)^{19/2}}{2432}+\frac {1101465 (1-2 x)^{17/2}}{2176}-\frac {1392467}{640} (1-2 x)^{15/2}+\frac {9504551 (1-2 x)^{13/2}}{1664}-\frac {1179381}{128} (1-2 x)^{11/2}+\frac {3278737}{384} (1-2 x)^{9/2}-\frac {456533}{128} (1-2 x)^{7/2} \]

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 = 0.03, 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]

-1/440895*((1 - 2*x)^(7/2)*(115708576 + 619493392*x + 1740153744*x^2 + 3089723448*x^3 + 3583371246*x^4 + 26423
19225*x^5 + 1127763000*x^6 + 212574375*x^7))

________________________________________________________________________________________

fricas [A]  time = 1.04, size = 59, normalized size = 0.56 \[ \frac {1}{440895} \, {\left (1700595000 \, x^{10} + 6471211500 \, x^{9} + 8880844050 \, x^{8} + 3513142893 \, x^{7} - 3556515018 \, x^{6} - 4297543173 \, x^{5} - 970928350 \, x^{4} + 842946920 \, x^{3} + 588303696 \, x^{2} + 74758064 \, x - 115708576\right )} \sqrt {-2 \, x + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.86, size = 129, normalized size = 1.23 \[ \frac {3375}{896} \, {\left (2 \, x - 1\right )}^{10} \sqrt {-2 \, x + 1} + \frac {161325}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {1101465}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {1392467}{640} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {9504551}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1179381}{128} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3278737}{384} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {456533}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 45, normalized size = 0.43 \[ -\frac {\left (212574375 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{440895} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(3*x+2)^4*(5*x+3)^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)*(-2*x+1)^(7/2)

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 73, normalized size = 0.70 \[ \frac {3375}{896} \, {\left (-2 \, x + 1\right )}^{\frac {21}{2}} - \frac {161325}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {1101465}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {1392467}{640} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {9504551}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1179381}{128} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3278737}{384} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {456533}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 73, normalized size = 0.70 \[ \frac {3278737\,{\left (1-2\,x\right )}^{9/2}}{384}-\frac {456533\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {1179381\,{\left (1-2\,x\right )}^{11/2}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {1392467\,{\left (1-2\,x\right )}^{15/2}}{640}+\frac {1101465\,{\left (1-2\,x\right )}^{17/2}}{2176}-\frac {161325\,{\left (1-2\,x\right )}^{19/2}}{2432}+\frac {3375\,{\left (1-2\,x\right )}^{21/2}}{896} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)*(3*x + 2)^4*(5*x + 3)^3,x)

[Out]

(3278737*(1 - 2*x)^(9/2))/384 - (456533*(1 - 2*x)^(7/2))/128 - (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

________________________________________________________________________________________

sympy [A]  time = 30.13, size = 94, normalized size = 0.90 \[ \frac {3375 \left (1 - 2 x\right )^{\frac {21}{2}}}{896} - \frac {161325 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} + \frac {1101465 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {1392467 \left (1 - 2 x\right )^{\frac {15}{2}}}{640} + \frac {9504551 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {1179381 \left (1 - 2 x\right )^{\frac {11}{2}}}{128} + \frac {3278737 \left (1 - 2 x\right )^{\frac {9}{2}}}{384} - \frac {456533 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} \]

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

________________________________________________________________________________________