3.1859 \(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx\)

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

[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

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Rubi [A]  time = 0.0146236, 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} \[ \frac{27}{32} (1-2 x)^{15/2}-\frac{4671}{416} (1-2 x)^{13/2}+\frac{10773}{176} (1-2 x)^{11/2}-\frac{8281}{48} (1-2 x)^{9/2}+\frac{8183}{32} (1-2 x)^{7/2}-\frac{26411}{160} (1-2 x)^{5/2} \]

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*}

Mathematica [A]  time = 0.016786, size = 38, normalized size = 0.48 \[ -\frac{(1-2 x)^{5/2} \left (57915 x^5+240570 x^4+424440 x^3+410320 x^2+230000 x+66592\right )}{2145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{57915\,{x}^{5}+240570\,{x}^{4}+424440\,{x}^{3}+410320\,{x}^{2}+230000\,x+66592}{2145} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0608, size = 74, normalized size = 0.94 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.3038, size = 162, normalized size = 2.05 \begin{align*} -\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{align*}

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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Sympy [A]  time = 12.7484, size = 70, normalized size = 0.89 \begin{align*} \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{align*}

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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Giac [A]  time = 1.98158, size = 131, normalized size = 1.66 \begin{align*} -\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{align*}

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)