3.2136 \(\int \frac{(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{405}{224} (1-2 x)^{7/2}-\frac{4671}{160} (1-2 x)^{5/2}+\frac{3591}{16} (1-2 x)^{3/2}-\frac{24843}{16} \sqrt{1-2 x}-\frac{57281}{32 \sqrt{1-2 x}}+\frac{26411}{96 (1-2 x)^{3/2}} \]

[Out]

26411/(96*(1 - 2*x)^(3/2)) - 57281/(32*Sqrt[1 - 2*x]) - (24843*Sqrt[1 - 2*x])/16 + (3591*(1 - 2*x)^(3/2))/16 -
(4671*(1 - 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

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Rubi [A]  time = 0.0144368, 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} \[ \frac{405}{224} (1-2 x)^{7/2}-\frac{4671}{160} (1-2 x)^{5/2}+\frac{3591}{16} (1-2 x)^{3/2}-\frac{24843}{16} \sqrt{1-2 x}-\frac{57281}{32 \sqrt{1-2 x}}+\frac{26411}{96 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

26411/(96*(1 - 2*x)^(3/2)) - 57281/(32*Sqrt[1 - 2*x]) - (24843*Sqrt[1 - 2*x])/16 + (3591*(1 - 2*x)^(3/2))/16 -
(4671*(1 - 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

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 \frac{(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{26411}{32 (1-2 x)^{5/2}}-\frac{57281}{32 (1-2 x)^{3/2}}+\frac{24843}{16 \sqrt{1-2 x}}-\frac{10773}{16} \sqrt{1-2 x}+\frac{4671}{32} (1-2 x)^{3/2}-\frac{405}{32} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac{26411}{96 (1-2 x)^{3/2}}-\frac{57281}{32 \sqrt{1-2 x}}-\frac{24843}{16} \sqrt{1-2 x}+\frac{3591}{16} (1-2 x)^{3/2}-\frac{4671}{160} (1-2 x)^{5/2}+\frac{405}{224} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.017297, size = 38, normalized size = 0.48 \[ -\frac{6075 x^5+33858 x^4+105624 x^3+435312 x^2-909264 x+301408}{105 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(301408 - 909264*x + 435312*x^2 + 105624*x^3 + 33858*x^4 + 6075*x^5)/(105*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{6075\,{x}^{5}+33858\,{x}^{4}+105624\,{x}^{3}+435312\,{x}^{2}-909264\,x+301408}{105} \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)^(5/2),x)

[Out]

-1/105*(6075*x^5+33858*x^4+105624*x^3+435312*x^2-909264*x+301408)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.51855, size = 69, normalized size = 0.87 \begin{align*} \frac{405}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{4671}{160} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{3591}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{24843}{16} \, \sqrt{-2 \, x + 1} + \frac{343 \,{\left (501 \, x - 212\right )}}{48 \,{\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)^(5/2),x, algorithm="maxima")

[Out]

405/224*(-2*x + 1)^(7/2) - 4671/160*(-2*x + 1)^(5/2) + 3591/16*(-2*x + 1)^(3/2) - 24843/16*sqrt(-2*x + 1) + 34
3/48*(501*x - 212)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.5124, size = 149, normalized size = 1.89 \begin{align*} -\frac{{\left (6075 \, x^{5} + 33858 \, x^{4} + 105624 \, x^{3} + 435312 \, x^{2} - 909264 \, x + 301408\right )} \sqrt{-2 \, x + 1}}{105 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(6075*x^5 + 33858*x^4 + 105624*x^3 + 435312*x^2 - 909264*x + 301408)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 19.6204, size = 70, normalized size = 0.89 \begin{align*} \frac{405 \left (1 - 2 x\right )^{\frac{7}{2}}}{224} - \frac{4671 \left (1 - 2 x\right )^{\frac{5}{2}}}{160} + \frac{3591 \left (1 - 2 x\right )^{\frac{3}{2}}}{16} - \frac{24843 \sqrt{1 - 2 x}}{16} - \frac{57281}{32 \sqrt{1 - 2 x}} + \frac{26411}{96 \left (1 - 2 x\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)**(5/2),x)

[Out]

405*(1 - 2*x)**(7/2)/224 - 4671*(1 - 2*x)**(5/2)/160 + 3591*(1 - 2*x)**(3/2)/16 - 24843*sqrt(1 - 2*x)/16 - 572
81/(32*sqrt(1 - 2*x)) + 26411/(96*(1 - 2*x)**(3/2))

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Giac [A]  time = 1.52454, size = 97, normalized size = 1.23 \begin{align*} -\frac{405}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{4671}{160} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{3591}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{24843}{16} \, \sqrt{-2 \, x + 1} - \frac{343 \,{\left (501 \, x - 212\right )}}{48 \,{\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

-405/224*(2*x - 1)^3*sqrt(-2*x + 1) - 4671/160*(2*x - 1)^2*sqrt(-2*x + 1) + 3591/16*(-2*x + 1)^(3/2) - 24843/1
6*sqrt(-2*x + 1) - 343/48*(501*x - 212)/((2*x - 1)*sqrt(-2*x + 1))