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

Optimal. Leaf size=66 \[ -\frac {9}{16} (1-2 x)^{15/2}+\frac {621}{104} (1-2 x)^{13/2}-\frac {1071}{44} (1-2 x)^{11/2}+\frac {3283}{72} (1-2 x)^{9/2}-\frac {539}{16} (1-2 x)^{7/2} \]

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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 {9}{16} (1-2 x)^{15/2}+\frac {621}{104} (1-2 x)^{13/2}-\frac {1071}{44} (1-2 x)^{11/2}+\frac {3283}{72} (1-2 x)^{9/2}-\frac {539}{16} (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),x]

[Out]

(-539*(1 - 2*x)^(7/2))/16 + (3283*(1 - 2*x)^(9/2))/72 - (1071*(1 - 2*x)^(11/2))/44 + (621*(1 - 2*x)^(13/2))/10
4 - (9*(1 - 2*x)^(15/2))/16

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)^{5/2} (2+3 x)^3 (3+5 x) \, dx &=\int \left (\frac {3773}{16} (1-2 x)^{5/2}-\frac {3283}{8} (1-2 x)^{7/2}+\frac {1071}{4} (1-2 x)^{9/2}-\frac {621}{8} (1-2 x)^{11/2}+\frac {135}{16} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {539}{16} (1-2 x)^{7/2}+\frac {3283}{72} (1-2 x)^{9/2}-\frac {1071}{44} (1-2 x)^{11/2}+\frac {621}{104} (1-2 x)^{13/2}-\frac {9}{16} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {(1-2 x)^{7/2} \left (11583 x^4+38313 x^3+50463 x^2+32378 x+9038\right )}{1287} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1287*((1 - 2*x)^(7/2)*(9038 + 32378*x + 50463*x^2 + 38313*x^3 + 11583*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {-11583 (1-2 x)^{15/2}+122958 (1-2 x)^{13/2}-501228 (1-2 x)^{11/2}+938938 (1-2 x)^{9/2}-693693 (1-2 x)^{7/2}}{20592} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-693693*(1 - 2*x)^(7/2) + 938938*(1 - 2*x)^(9/2) - 501228*(1 - 2*x)^(11/2) + 122958*(1 - 2*x)^(13/2) - 11583*
(1 - 2*x)^(15/2))/20592

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fricas [A]  time = 1.44, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{1287} \, {\left (92664 \, x^{7} + 167508 \, x^{6} + 13446 \, x^{5} - 128237 \, x^{4} - 51767 \, x^{3} + 35349 \, x^{2} + 21850 \, x - 9038\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),x, algorithm="fricas")

[Out]

1/1287*(92664*x^7 + 167508*x^6 + 13446*x^5 - 128237*x^4 - 51767*x^3 + 35349*x^2 + 21850*x - 9038)*sqrt(-2*x +
1)

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giac [A]  time = 1.04, size = 81, normalized size = 1.23 \begin {gather*} \frac {9}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {621}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1071}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3283}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {539}{16} \, {\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),x, algorithm="giac")

[Out]

9/16*(2*x - 1)^7*sqrt(-2*x + 1) + 621/104*(2*x - 1)^6*sqrt(-2*x + 1) + 1071/44*(2*x - 1)^5*sqrt(-2*x + 1) + 32
83/72*(2*x - 1)^4*sqrt(-2*x + 1) + 539/16*(2*x - 1)^3*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (11583 x^{4}+38313 x^{3}+50463 x^{2}+32378 x +9038\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{1287} \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),x)

[Out]

-1/1287*(11583*x^4+38313*x^3+50463*x^2+32378*x+9038)*(-2*x+1)^(7/2)

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maxima [A]  time = 0.49, size = 46, normalized size = 0.70 \begin {gather*} -\frac {9}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {621}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1071}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3283}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {539}{16} \, {\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),x, algorithm="maxima")

[Out]

-9/16*(-2*x + 1)^(15/2) + 621/104*(-2*x + 1)^(13/2) - 1071/44*(-2*x + 1)^(11/2) + 3283/72*(-2*x + 1)^(9/2) - 5
39/16*(-2*x + 1)^(7/2)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {3283\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {539\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {1071\,{\left (1-2\,x\right )}^{11/2}}{44}+\frac {621\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {9\,{\left (1-2\,x\right )}^{15/2}}{16} \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),x)

[Out]

(3283*(1 - 2*x)^(9/2))/72 - (539*(1 - 2*x)^(7/2))/16 - (1071*(1 - 2*x)^(11/2))/44 + (621*(1 - 2*x)^(13/2))/104
- (9*(1 - 2*x)^(15/2))/16

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sympy [A]  time = 17.26, size = 58, normalized size = 0.88 \begin {gather*} - \frac {9 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} + \frac {621 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {1071 \left (1 - 2 x\right )^{\frac {11}{2}}}{44} + \frac {3283 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {539 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} \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),x)

[Out]

-9*(1 - 2*x)**(15/2)/16 + 621*(1 - 2*x)**(13/2)/104 - 1071*(1 - 2*x)**(11/2)/44 + 3283*(1 - 2*x)**(9/2)/72 - 5
39*(1 - 2*x)**(7/2)/16

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