3.1792 \(\int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx\)

Optimal. Leaf size=79 \[ \frac {405}{416} (1-2 x)^{13/2}-\frac {4671}{352} (1-2 x)^{11/2}+\frac {1197}{16} (1-2 x)^{9/2}-\frac {3549}{16} (1-2 x)^{7/2}+\frac {57281}{160} (1-2 x)^{5/2}-\frac {26411}{96} (1-2 x)^{3/2} \]

[Out]

-26411/96*(1-2*x)^(3/2)+57281/160*(1-2*x)^(5/2)-3549/16*(1-2*x)^(7/2)+1197/16*(1-2*x)^(9/2)-4671/352*(1-2*x)^(
11/2)+405/416*(1-2*x)^(13/2)

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Rubi [A]  time = 0.02, 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} \[ \frac {405}{416} (1-2 x)^{13/2}-\frac {4671}{352} (1-2 x)^{11/2}+\frac {1197}{16} (1-2 x)^{9/2}-\frac {3549}{16} (1-2 x)^{7/2}+\frac {57281}{160} (1-2 x)^{5/2}-\frac {26411}{96} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26411*(1 - 2*x)^(3/2))/96 + (57281*(1 - 2*x)^(5/2))/160 - (3549*(1 - 2*x)^(7/2))/16 + (1197*(1 - 2*x)^(9/2))
/16 - (4671*(1 - 2*x)^(11/2))/352 + (405*(1 - 2*x)^(13/2))/416

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 \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx &=\int \left (\frac {26411}{32} \sqrt {1-2 x}-\frac {57281}{32} (1-2 x)^{3/2}+\frac {24843}{16} (1-2 x)^{5/2}-\frac {10773}{16} (1-2 x)^{7/2}+\frac {4671}{32} (1-2 x)^{9/2}-\frac {405}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {26411}{96} (1-2 x)^{3/2}+\frac {57281}{160} (1-2 x)^{5/2}-\frac {3549}{16} (1-2 x)^{7/2}+\frac {1197}{16} (1-2 x)^{9/2}-\frac {4671}{352} (1-2 x)^{11/2}+\frac {405}{416} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \[ -\frac {(1-2 x)^{3/2} \left (66825 x^5+288360 x^4+540000 x^3+577080 x^2+388704 x+163888\right )}{2145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2145*((1 - 2*x)^(3/2)*(163888 + 388704*x + 577080*x^2 + 540000*x^3 + 288360*x^4 + 66825*x^5))

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fricas [A]  time = 0.58, size = 39, normalized size = 0.49 \[ \frac {1}{2145} \, {\left (133650 \, x^{6} + 509895 \, x^{5} + 791640 \, x^{4} + 614160 \, x^{3} + 200328 \, x^{2} - 60928 \, x - 163888\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/2145*(133650*x^6 + 509895*x^5 + 791640*x^4 + 614160*x^3 + 200328*x^2 - 60928*x - 163888)*sqrt(-2*x + 1)

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giac [A]  time = 1.20, size = 90, normalized size = 1.14 \[ \frac {405}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {4671}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1197}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3549}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {57281}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {26411}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

405/416*(2*x - 1)^6*sqrt(-2*x + 1) + 4671/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1197/16*(2*x - 1)^4*sqrt(-2*x + 1)
+ 3549/16*(2*x - 1)^3*sqrt(-2*x + 1) + 57281/160*(2*x - 1)^2*sqrt(-2*x + 1) - 26411/96*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \[ -\frac {\left (66825 x^{5}+288360 x^{4}+540000 x^{3}+577080 x^{2}+388704 x +163888\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{2145} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2145*(66825*x^5+288360*x^4+540000*x^3+577080*x^2+388704*x+163888)*(-2*x+1)^(3/2)

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maxima [A]  time = 0.53, size = 55, normalized size = 0.70 \[ \frac {405}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {4671}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1197}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3549}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {57281}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {26411}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

405/416*(-2*x + 1)^(13/2) - 4671/352*(-2*x + 1)^(11/2) + 1197/16*(-2*x + 1)^(9/2) - 3549/16*(-2*x + 1)^(7/2) +
57281/160*(-2*x + 1)^(5/2) - 26411/96*(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \[ \frac {57281\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {26411\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {3549\,{\left (1-2\,x\right )}^{7/2}}{16}+\frac {1197\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {4671\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {405\,{\left (1-2\,x\right )}^{13/2}}{416} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(57281*(1 - 2*x)^(5/2))/160 - (26411*(1 - 2*x)^(3/2))/96 - (3549*(1 - 2*x)^(7/2))/16 + (1197*(1 - 2*x)^(9/2))/
16 - (4671*(1 - 2*x)^(11/2))/352 + (405*(1 - 2*x)^(13/2))/416

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sympy [A]  time = 2.95, size = 70, normalized size = 0.89 \[ \frac {405 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {4671 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {1197 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {3549 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} + \frac {57281 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {26411 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405*(1 - 2*x)**(13/2)/416 - 4671*(1 - 2*x)**(11/2)/352 + 1197*(1 - 2*x)**(9/2)/16 - 3549*(1 - 2*x)**(7/2)/16 +
57281*(1 - 2*x)**(5/2)/160 - 26411*(1 - 2*x)**(3/2)/96

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