3.18.20 \(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx\)

Optimal. Leaf size=79 \[ \frac {27}{32} (1-2 x)^{15/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {26411}{160} (1-2 x)^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 79, 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 {27}{32} (1-2 x)^{15/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {26411}{160} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-26411*(1 - 2*x)^(5/2))/160 + (8183*(1 - 2*x)^(7/2))/32 - (8281*(1 - 2*x)^(9/2))/48 + (10773*(1 - 2*x)^(11/2)
)/176 - (4671*(1 - 2*x)^(13/2))/416 + (27*(1 - 2*x)^(15/2))/32

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)^4 (3+5 x) \, dx &=\int \left (\frac {26411}{32} (1-2 x)^{3/2}-\frac {57281}{32} (1-2 x)^{5/2}+\frac {24843}{16} (1-2 x)^{7/2}-\frac {10773}{16} (1-2 x)^{9/2}+\frac {4671}{32} (1-2 x)^{11/2}-\frac {405}{32} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (57915 x^5+240570 x^4+424440 x^3+410320 x^2+230000 x+66592\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

-1/2145*((1 - 2*x)^(5/2)*(66592 + 230000*x + 410320*x^2 + 424440*x^3 + 240570*x^4 + 57915*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {57915 (1-2 x)^{15/2}-770715 (1-2 x)^{13/2}+4201470 (1-2 x)^{11/2}-11841830 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-11330319 (1-2 x)^{5/2}}{68640} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-11330319*(1 - 2*x)^(5/2) + 17552535*(1 - 2*x)^(7/2) - 11841830*(1 - 2*x)^(9/2) + 4201470*(1 - 2*x)^(11/2) -
770715*(1 - 2*x)^(13/2) + 57915*(1 - 2*x)^(15/2))/68640

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fricas [A]  time = 0.99, size = 44, normalized size = 0.56 \begin {gather*} -\frac {1}{2145} \, {\left (231660 \, x^{7} + 730620 \, x^{6} + 793395 \, x^{5} + 184090 \, x^{4} - 296840 \, x^{3} - 243312 \, x^{2} - 36368 \, x + 66592\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)^4*(3+5*x),x, algorithm="fricas")

[Out]

-1/2145*(231660*x^7 + 730620*x^6 + 793395*x^5 + 184090*x^4 - 296840*x^3 - 243312*x^2 - 36368*x + 66592)*sqrt(-
2*x + 1)

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giac [A]  time = 1.55, size = 97, normalized size = 1.23 \begin {gather*} -\frac {27}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4671}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {10773}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {8281}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8183}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26411}{160} \, {\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)^4*(3+5*x),x, algorithm="giac")

[Out]

-27/32*(2*x - 1)^7*sqrt(-2*x + 1) - 4671/416*(2*x - 1)^6*sqrt(-2*x + 1) - 10773/176*(2*x - 1)^5*sqrt(-2*x + 1)
- 8281/48*(2*x - 1)^4*sqrt(-2*x + 1) - 8183/32*(2*x - 1)^3*sqrt(-2*x + 1) - 26411/160*(2*x - 1)^2*sqrt(-2*x +
1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (57915 x^{5}+240570 x^{4}+424440 x^{3}+410320 x^{2}+230000 x +66592\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{2145} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2145*(57915*x^5+240570*x^4+424440*x^3+410320*x^2+230000*x+66592)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.66, size = 55, normalized size = 0.70 \begin {gather*} \frac {27}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4671}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {10773}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {8281}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8183}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26411}{160} \, {\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)^4*(3+5*x),x, algorithm="maxima")

[Out]

27/32*(-2*x + 1)^(15/2) - 4671/416*(-2*x + 1)^(13/2) + 10773/176*(-2*x + 1)^(11/2) - 8281/48*(-2*x + 1)^(9/2)
+ 8183/32*(-2*x + 1)^(7/2) - 26411/160*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {8183\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {26411\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {8281\,{\left (1-2\,x\right )}^{9/2}}{48}+\frac {10773\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {4671\,{\left (1-2\,x\right )}^{13/2}}{416}+\frac {27\,{\left (1-2\,x\right )}^{15/2}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8183*(1 - 2*x)^(7/2))/32 - (26411*(1 - 2*x)^(5/2))/160 - (8281*(1 - 2*x)^(9/2))/48 + (10773*(1 - 2*x)^(11/2))
/176 - (4671*(1 - 2*x)^(13/2))/416 + (27*(1 - 2*x)^(15/2))/32

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sympy [A]  time = 17.22, size = 70, normalized size = 0.89 \begin {gather*} \frac {27 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} + \frac {10773 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {8281 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {8183 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {26411 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*(1 - 2*x)**(15/2)/32 - 4671*(1 - 2*x)**(13/2)/416 + 10773*(1 - 2*x)**(11/2)/176 - 8281*(1 - 2*x)**(9/2)/48
+ 8183*(1 - 2*x)**(7/2)/32 - 26411*(1 - 2*x)**(5/2)/160

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