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

Optimal. Leaf size=66 \[ -\frac {375}{208} (1-2 x)^{13/2}+\frac {1675}{88} (1-2 x)^{11/2}-\frac {935}{12} (1-2 x)^{9/2}+\frac {8349}{56} (1-2 x)^{7/2}-\frac {9317}{80} (1-2 x)^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {375}{208} (1-2 x)^{13/2}+\frac {1675}{88} (1-2 x)^{11/2}-\frac {935}{12} (1-2 x)^{9/2}+\frac {8349}{56} (1-2 x)^{7/2}-\frac {9317}{80} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*(1 - 2*x)^(5/2))/80 + (8349*(1 - 2*x)^(7/2))/56 - (935*(1 - 2*x)^(9/2))/12 + (1675*(1 - 2*x)^(11/2))/88
- (375*(1 - 2*x)^(13/2))/208

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx &=\int \left (\frac {9317}{16} (1-2 x)^{3/2}-\frac {8349}{8} (1-2 x)^{5/2}+\frac {2805}{4} (1-2 x)^{7/2}-\frac {1675}{8} (1-2 x)^{9/2}+\frac {375}{16} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {9317}{80} (1-2 x)^{5/2}+\frac {8349}{56} (1-2 x)^{7/2}-\frac {935}{12} (1-2 x)^{9/2}+\frac {1675}{88} (1-2 x)^{11/2}-\frac {375}{208} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (433125 x^4+1420125 x^3+1899800 x^2+1295695 x+421301\right )}{15015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15015*((1 - 2*x)^(5/2)*(421301 + 1295695*x + 1899800*x^2 + 1420125*x^3 + 433125*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {-433125 (1-2 x)^{13/2}+4572750 (1-2 x)^{11/2}-18718700 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-27978951 (1-2 x)^{5/2}}{240240} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-27978951*(1 - 2*x)^(5/2) + 35817210*(1 - 2*x)^(7/2) - 18718700*(1 - 2*x)^(9/2) + 4572750*(1 - 2*x)^(11/2) -
433125*(1 - 2*x)^(13/2))/240240

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fricas [A]  time = 1.40, size = 39, normalized size = 0.59 \begin {gather*} -\frac {1}{15015} \, {\left (1732500 \, x^{6} + 3948000 \, x^{5} + 2351825 \, x^{4} - 996295 \, x^{3} - 1597776 \, x^{2} - 389509 \, x + 421301\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/15015*(1732500*x^6 + 3948000*x^5 + 2351825*x^4 - 996295*x^3 - 1597776*x^2 - 389509*x + 421301)*sqrt(-2*x +
1)

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giac [A]  time = 1.09, size = 81, normalized size = 1.23 \begin {gather*} -\frac {375}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {1675}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {935}{12} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8349}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {9317}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="giac")

[Out]

-375/208*(2*x - 1)^6*sqrt(-2*x + 1) - 1675/88*(2*x - 1)^5*sqrt(-2*x + 1) - 935/12*(2*x - 1)^4*sqrt(-2*x + 1) -
8349/56*(2*x - 1)^3*sqrt(-2*x + 1) - 9317/80*(2*x - 1)^2*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (433125 x^{4}+1420125 x^{3}+1899800 x^{2}+1295695 x +421301\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{15015} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15015*(433125*x^4+1420125*x^3+1899800*x^2+1295695*x+421301)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.53, size = 46, normalized size = 0.70 \begin {gather*} -\frac {375}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {1675}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {935}{12} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8349}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {9317}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="maxima")

[Out]

-375/208*(-2*x + 1)^(13/2) + 1675/88*(-2*x + 1)^(11/2) - 935/12*(-2*x + 1)^(9/2) + 8349/56*(-2*x + 1)^(7/2) -
9317/80*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {8349\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {9317\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {935\,{\left (1-2\,x\right )}^{9/2}}{12}+\frac {1675\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {375\,{\left (1-2\,x\right )}^{13/2}}{208} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8349*(1 - 2*x)^(7/2))/56 - (9317*(1 - 2*x)^(5/2))/80 - (935*(1 - 2*x)^(9/2))/12 + (1675*(1 - 2*x)^(11/2))/88
- (375*(1 - 2*x)^(13/2))/208

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sympy [A]  time = 13.31, size = 58, normalized size = 0.88 \begin {gather*} - \frac {375 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} + \frac {1675 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {935 \left (1 - 2 x\right )^{\frac {9}{2}}}{12} + \frac {8349 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {9317 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375*(1 - 2*x)**(13/2)/208 + 1675*(1 - 2*x)**(11/2)/88 - 935*(1 - 2*x)**(9/2)/12 + 8349*(1 - 2*x)**(7/2)/56 -
9317*(1 - 2*x)**(5/2)/80

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