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]

-3195731/640*(1-2*x)^(5/2)+1405173/128*(1-2*x)^(7/2)-4324397/384*(1-2*x)^(9/2)+9504551/1408*(1-2*x)^(11/2)-417
7401/1664*(1-2*x)^(13/2)+73431/128*(1-2*x)^(15/2)-161325/2176*(1-2*x)^(17/2)+10125/2432*(1-2*x)^(19/2)

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

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*}

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Mathematica [A]  time = 0.03, 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]

-1/692835*((1 - 2*x)^(5/2)*(369438704 + 1547888800*x + 3771434840*x^2 + 6142984080*x^3 + 6744559140*x^4 + 4795
033815*x^5 + 1995171750*x^6 + 369208125*x^7))

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fricas [A]  time = 0.58, size = 54, normalized size = 0.51 \[ -\frac {1}{692835} \, {\left (1476832500 \, x^{9} + 6503854500 \, x^{8} + 11568656385 \, x^{7} + 9793273050 \, x^{6} + 2388733575 \, x^{5} - 2741637820 \, x^{4} - 2751200080 \, x^{3} - 942365544 \, x^{2} + 70133984 \, x + 369438704\right )} \sqrt {-2 \, x + 1} \]

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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giac [A]  time = 0.98, size = 129, normalized size = 1.23 \[ -\frac {10125}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} - \frac {161325}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {73431}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4177401}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {9504551}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {4324397}{384} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1405173}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3195731}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]

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)

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maple [A]  time = 0.01, size = 45, normalized size = 0.43 \[ -\frac {\left (369208125 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{692835} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.61, size = 73, normalized size = 0.70 \[ \frac {10125}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} - \frac {161325}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {73431}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4177401}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {9504551}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {4324397}{384} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1405173}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3195731}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]

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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mupad [B]  time = 0.03, size = 73, normalized size = 0.70 \[ \frac {1405173\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {3195731\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {4324397\,{\left (1-2\,x\right )}^{9/2}}{384}+\frac {9504551\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {4177401\,{\left (1-2\,x\right )}^{13/2}}{1664}+\frac {73431\,{\left (1-2\,x\right )}^{15/2}}{128}-\frac {161325\,{\left (1-2\,x\right )}^{17/2}}{2176}+\frac {10125\,{\left (1-2\,x\right )}^{19/2}}{2432} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1405173*(1 - 2*x)^(7/2))/128 - (3195731*(1 - 2*x)^(5/2))/640 - (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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sympy [A]  time = 25.62, size = 94, normalized size = 0.90 \[ \frac {10125 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} - \frac {161325 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} + \frac {73431 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} - \frac {4177401 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} + \frac {9504551 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {4324397 \left (1 - 2 x\right )^{\frac {9}{2}}}{384} + \frac {1405173 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} - \frac {3195731 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} \]

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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