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*(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

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Rubi [A]  time = 0.0151272, antiderivative size = 79, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0161117, 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 - 2*x)^(3/2)*(163888 + 388704*x + 577080*x^2 + 540000*x^3 + 288360*x^4 + 66825*x^5))/2145

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Maple [A]  time = 0.001, size = 35, normalized size = 0.4 \begin{align*} -{\frac{66825\,{x}^{5}+288360\,{x}^{4}+540000\,{x}^{3}+577080\,{x}^{2}+388704\,x+163888}{2145} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04309, size = 74, normalized size = 0.94 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.36822, size = 144, normalized size = 1.82 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 2.13258, size = 70, normalized size = 0.89 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 2.53176, size = 122, normalized size = 1.54 \begin{align*} \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}} \end{align*}

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)